Riemannian Geometry: A Beginner's Guide

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Editorial, Sales, and Customer Service Offices Jones and Bartlett Publishers One Exeter Plaza Boston, MA 02116 Jones and Bartlett Publishers International P.O. Box 1498 London W6 7RS England

Manuscript typed by Dan Robb

Copyright

© 1993 by Frank Morgan

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any informational storage and retrieval system, without written permission from the copyright owner.

Library of Congress Cataloging-in-Publication Data

Morgan, Frank. Riemannian geometry: a beginner's guide / Frank Morgan. p. em. Includes bibliographical references and index. ISBN 0-86720-242-4 1. Geometry, Riemannian. I. Title. QA611.M674 1992 516.3'73-dc20 92-13261 CIP

Printed in the United States of America 96 95 94 93

92

10 9 8 7 6 5 4 3 2

1

Photograph courtesy of the Morgan family; taken by the author's grandfather, Dr. Charles Selemeyer.

This book is dedicated to my teachers-notably Fred Almgren, Clem Collins, Arthur Mattuck, Mabel Riker, my mom, and my dad. Here as a child I got an early geometry lesson from my dad. F.M.

Contents PREFACE

Vlll

==i . 2. 3.

INTRODUCTION

SURFACES IN R3

11

-------zJ: •

SURFACES IN Rn

25

CURVES IN

an

1 5

5.

m-DIMENSIONAL SURFACES IN Rn

6.

INTRINSIC RIEMANNIAN GEOMETRY

7. 8. 9. 3B.

GENERAL RELATIVITY

31

55

THE GAUSS-BONNET THEOREM

65

GEODESICS AND GLOBAL GEOMETRY GENERAL NORMS

BIBLIOGRAPHY

101

105

SOLUTIONS TO SELECTED EXERCISES

NAME INDEX

113

115

SUBJECT INDEX

77

89

SELECTED FORMULAS

SYMBOL INDEX

39

117

109

---Prefaee--The complicated formulations of Riemannian geometry present a daunting aspect to the student This little book focuses on the central concept-curvature It gives a naive treatment of Riemannian geometry, based on surfaces in R n rather than on abstract Riemannian manifolds. The more sophIstIcated mtnnslC formulas folIow naturalIy. Later chapters treat hyperbobc geometry, general relatIVIty, global geometry, and some current research on energy-mlmmlzmg curves and the lsopenmetnc problem. Proofs, when given at all, emphasize the main ideas and suppress the details that otherwise might overwhelm the student. This book grew out of graduate courses I taught on tensor analysis at MIT in 1977 and on differential geometry at Stanford in 1987 and Princeton in 1990, and out of my own need to understand curvature better for my work. The last chapter includes research by Williams undergraduates. I want to thank my students, notably Alice Underwood; Paul Siegel, my teaching assistant for tensor analysis; and participants in a seminar at Washington and Lee led by Tim Murdoch. Other books I have found helpful include Laugwitz's Differential and Riemannian GeOl1tetry ftl, IIicks's Notes on Differential Geome----rtlrt'yt-[Hi] (unfortunately out of print), Spivak's Comprehensive Intloduction to Differential Geometly fSj, and Stoker's Diffelential GeometlY f S ! j f - - - . - - - - - - - - - - - - - - - - - - - - - - - I am cunently using this book and Geometric lt1easure TheOlY. A Beginnet's Guide ~, both so happily edited by Klaus Peters and illustlated by Jim Bledt, as texts for an advanced, one-semester undelgtaduate course at Williams. Williamstown [email protected]

F.M.

The central concept of Riemannian geometry is curvature. It describes the most important geometric features of racetracks and of universes. We will begin by defining the curvature of a racetrack. Chapter 7 uses general relativity's interpretation of mass as curvature to predict the mysterious precession of Mercury's orbit. The curvature K of a racetrack is defined as the rate at which the direction vector T of motion is turning, measured in radians or degrees per meter. The curvature is big on sharp curves, zero on straightaways. See Figure 1.1. A two-dimensional surface, such as the surface of Figure 1.2, can curve different amounts in different directions, perhaps upward in some directions, downward in others, and along straight lines in between. The principal curvatures "I and "2 are the most upward (positive) and the most downward (negative), respectively. For the saddle of Figure 1.2, it appears that at the origin Kl - ~ and K2 - -1. The mean curvature H - Kl + Kz - -~ 'Ihe Gauss cumature GAt the south pole of the unit sphere of Figure 1.3, Kl - K2 - 1, H 2, and G 1. Since KI and K2 measure the amount that the surface is curving in space, they could not be measUIed by a bug confined to the surface. They are "extrinsic" properties. Gauss made the astonishing discovery, however, that the Gauss curvature G KIK2 can, in prin-

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INTRODOCTION

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Figure 1.2.

At the center of this saddle, the

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Figure 1.3. At the south pole, the curvature is + 1 in all directions.

from the outset In this text, surfaces will start out sitting in an, where we can give concrete definitions of the second fundamental tensor and the Riemannian curvature tensor Only later wi11 we prove that the Riemannian curvature tensor actually is intrinsic.

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The curvature vector K points in the direction in which T is turning, orthogonal to T. Its length, the scalar curvature K = IKI, gives the rate of turning. See Figure 2.1. For a planar curve with unit normal n

For a circle of radius a, K points toward the center, and K = 1/a. For a general curve, the best approximating, or osculating, circle has radius 1/ K, called the radius of curvature. . . e curve IS ar me enze arc en

osculating circle.

CURVES IN R"

tis Figure 2.2. An element of arc length tis pushed in the direction of K decreases by a factor of 1 - K duo

Figure 2.3. The metal strip, or strake, spirals around the smokestack on a helical path.

7

8

CHAPTER 2

Figure 2.4. The smokestack had radius a 3.75 feet. Each revolution of the strake had a height of 31 5 feet

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x = (x, y, z)

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The curvature of a surface S at a point p is measured by the curvature of its slices by planes.

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13

SURFACES IN R3

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Z

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Some famous minimal surfaces. (Frank Morgan, Geometric Measure Theory, p. 68. © 1988, Academic Press. All rights reserved. Reprinted with permission of the publisher.)

15

16

CHAPTER 3

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Dornon or..). -r or examDle me Stanuaru sDnencal , . , coorumates (l) U DfoVlue local coordinates on all of the sphere of radius a except for the poles (where the longitude (j is undefined and 'P is not differentiable). The position vector determined by these coordinates is

v-{y"

7 \ - { n ... in",{"',.......

1l n ... in", ...inll

n{"',....... ",\

In general, the position is some function of the coordinates Ui' Along a curve, these coordinates are in turn functions of a sinl!le parameter .... _1.

,'-

V11 Ul'-'

'11

A

vv 111

_1

•

1

lJal ual

_1 U~11 vau v~~

SURFACES IN R3

Figure 3.4. The newly discovered complete, embedded minimal surface of Costa, Hoffman, and Meeks (Cos), (HoM), (Ho). (Courtesy of David Hoffmann, Jim Hoffmann, and Michael Callahan.)

with respect to the

Ui:

A dot will denote differentiation with respect to t: .

dx

~

.

x = -dt = .t.. x-uI

I

(the chain rule).

17

10 .1.U

L-t1A.r 1 nK -'

The arc length of a curve in the surface with coordinates u(t) is given bv t1

t

L-

r IXI at - Jr IXIUI + xzuzi at to

to 1

r

-

J

V(Xl·Xl)U~

+ 2(Xl·X2)UlU2 + (X2·x2)u~dt

\/~ ~

,

(1)

'u

.

,

.

-

-

,: " I

OIJ~

A+

~J

~,

to

_,_ .:Iv

gij

.:Iv

== Xi ·Xj = :-.:-'

(2)

uUi UUj

In other words, L = f ds, where ~

as " - ki gij aUi aUj'

~";)

t'or example, on the sphere 01 radIUS a, L - J as, where, as It turns out, rlf: z

= n 2rlm Z -t- n Z <;:in 2 m rllP -

(n 2£;., -I- n 2 <;:in z mil) rlt Z

SO gll - a-, gzz - a- sm- 'P, and glZ - gZI - U (see bxercise j.:l). ·hp

,

a

I a. .

0

L~'H

It i<;: ~n '

,

r

thp

,<;:

f'~

Ilprt

thp

ht'('f

tnt'''"

J .L

lI.T.

in thM it , thj:)t in thp r

nr

J

/

to

~

,L

for' '-'

~

2.. gijUiUj = gllUIUI + g12UI UZ + gZIUZUI + gzzuzuz - {1HU~ ~

For m::tnv

"

j:)~n Xl -

.&.

.,

+ "')~ " ,: + {1""U~ ~

~

in R 3 it i~ 7( Y \J) Thpn ~

(1, U, Zx)

to ll~e x

v ::t~ loc::tl

/

and

Xz -

(U, 1, Zy).

The following proposition gives useful formulas for the mean curvature H and Gauss curvature G. 3.5. Proposition.

For any local coordinates UI, Uz about a point p

... n

SURFA( :ES IN R ~')

maL-

A

IUf IUl.-f;

In

.':l.

.K •

. form 11 al [} lS

A

lne

J

~~

..

tV

_./

,~

v

...... -

. 1 I n " v\

n

0

'J.J.

.1

n

.....

v 'J.":'

.....

,

0

""12' I I

""22' 11-

where

Xij

=

2 d X

and

n=

dUidUj

Xl

x

X2

IXI

x

x21

,

Consequently,

H

traceg-\D 2x)·n

=

X~Xl1

-

- 2(Xl • X2)X12 2 2 XlX2

G

(1)

+

XiX22

( Xl· X2 )2

• n,

det (g-1(D 2 x). n)

=

(Xl1 • n)(x22 • n) -

-

2 2 XlX2

(2) (X12· n)2

( Xl· X2

f

Before turning to the proof of Proposition 3.5, we note that

(A) Given Hand G, you can solve easily for the principal curvatures: K=

H+YH2 -4G 2

(B) If the surface is a graph

.

(x, y, f(x, y)), then

X=

.')

')

/1

\.L

"

(i

.£2

11

Jx

\~

.

wem::lv . SO.) IS lOcauv a Q'fann

Tn::lT

''''JlJI

U.

.""

,-

tV. V. 11 • .£' UI

rn ,-

,

~

,,')

"

1 ...... /'"

~

IT

L. T

'"

J..l

.

T T -., ;-Il;-T r2'\3/2 r2 J yJ Jx

I 1 ;- T IT

/

..."

f.:1\

' ,

.£2\2 J yJ

."1 1!'\ ~

TO Tne x

.V-...

- TlX VI wnn T .IUI - T..IUI ., " ;7 . .. . x. v.

,.

lll~ Darl1~Ular 10~al ~OOrUlllal~S

v

-

I

v

"

.fl v

'" '.!' J

'"

,,\ \

'.! / / ,

nUn 0 ' '/

.

T

~

U

::IT ...n ana

DlUI -

,...r. /.IJ

1 ne

('HAPTPR 1

proposition says mat

is similar

11

to

xx f xy ] [f fxy fyy 0'

. _. .

n,..",.. '"

_.

-I

1I..T~n.

1~ ..

l..~

.v.

.'v

.. . .....

,1.

,

"'v.,"''''~,

""1' "":':

.1

lA,,,, "''1''',4>.

~""'Y

~_ ••

1~~~1

,~"~n

-I'

vv ...... y .v..,.....

~_A 1~ ..

,

...... ~

• ..,.

v

1

.....,

+\..~

-I

~+ fl •

T

~.

ax -

- ax

-

-

aU1

1-

v

aU2

~

~

vy

vy

rill~

rill

v

I =aZ- I =0,

. aZ SmceaU1

aU2

0

0

g = ITI. .::Iv ,,\..,,~...

'r\..o... h .. +\..0 ~

.&'

'£.

~

J

•••

_ . . ...

1",,10

, C'~".

...

.::Iv

"o

n ~

~

... a2 X

~

ri-Y

--.n '\ VA-

IS

I = 1-lIII

~

ri-Y ...

-

a2 X

--·n axay

- ·2n aX

. n = (JTJ) -1 IT

,

vy

VA-

~ -1 ( D 2 x)

n-{)

() " ... ,-1

-

-.n ...2

'\.

vy

vy

-

mdeed simBar to 11. Example. We will compute the curvature of the catenoid

\Jy2

+

,)2

{'{\c::h

{\f

7

v

<.;>

I

~l;;

I he

.

~?

T"

-'

WP w111

l;;avs mat

xXl

At m{\c::t

~x,

r

COSh

y, Z) -

nl;;p

thp

~nC1

7

the

7.

~cosn

WP

•

Z COS

.

.

nol~r

cosn Z sm

tJ,

= (sinh Z cos 8, sinh Z sin 8, 1)

x? = (-cosh Z sin 8. cosh Z cos 8.0) ~

All

\

A12

\

X22

<'

l;U~ll

,(.

l;U~

U,

. _.~11111.(. ~11'

.

l;U~ll

,(.

~1l1

U, V]

n V,

~11111

.(.

",.1.

~

"'v~

n

('{\nlet nc::p

",

v, v J

= (-cosh z cos 8, -cosh z sin 8, 0)

n tJ,

Z)

y

~net H

,....

"'TTRFA('F'" IN R-'

n=

....

L.~

(COS (), sin (), sinh z) . cosh z

By Proposition 3.5,

. .,

~

l:usn Z Xll

yy

n

U T

. .,

l:usn Z X22

~

·n

u.

<;>

so me CatenOla IS maeea a mInImal surrace ana G= Hence K1 -

(1)(-1)-0 = -cosh- 4 Z. 2 2 cosh Z cosh Z - 0 ~-.

cosn

K-. -

K2·

KI -

7

3.6. Gauss's Theorema Egregium. Gauss curvature G is intrinsic. Specifically, there are local coordinates Ub U2 about any point p in a c3 surface S in R 3 such that the first fundamental form g at p is I to first order. In any such coordinate system, the Gauss curvature is ;l~o.

n

1

;l~o. ~

1

1"\

2 uUI

1"\

'.I. . .

...

~

...

UU2

UUI

~

;l~o. ~

...

~

...

2 •

uU2

Remark. To say that G = I to first order means that gn(P) = g22(P) = 1, g12(P) = 0, and each rio ..

(n\-.:..2.:L(n\-O

0 .. O'J,

...

/

u~

'-L"

vUk

T

'J

'J'

~

;co tho

~

.~~

" f .,.

.

.....

.f

~&

J

""0"

;tco

.....

plane. Orthonormal coordinates on the tangent plane make the metric g equal to I to first order. We may assume that S is tangent to the x, y-plane at p = O. In x, y coordinates, .")

1 ;- T

TyT

.f .f

1

fl'J AJ

Y

-I'

f",.

-J

",n,.oo

, .• ~ 1

Y

-l;y = detD 2f= detI! =

tho

with

",t

nn thp

'0

::Inc!

f' .. J

and one computes that at 0 2 1 2 1 2 a gI2 __ a g22 __ a gn = fxxfyy ax ay 2 ax 2 2 ay2 An"

~

vl~lc!

"

th~ s::lm~

_.......... 1+

n

r

;co

.....

T t" h,.cot c

G. ..:I

tn fir40:t

22

CHAPTER 3

:

:::~

'-:... ' .:: ..." '>':'.~ :.: :: '

::

:-~

:

~

;\/\//:<. ······/i:i.;

~ :.: ......... fil!1;·h':::,''' .: .' :.........

Figure 3.5. Rolling a piece of plane into a cylinder of radius r changes the principal curvatures Kr, K2 from 0, 0 to lIr, 0 and changes the mean curvature H - Kl + K2 from (I to 17r The Gauss curvature, however , remains G K] K2 0, as Gauss's Theorema EglegiuIIl gualantees.

Remark. A full appreciation of Gauss's Theorema Egregium requires the realization that most extrinsic quantities are not intrinSlC

In R 3 , a flat piece of plane can be rolled, or "bent," into a piece of cylinder of radius r without changing anything a bug on the surface could detect. This bending, however, does change the curvature K of an arc of latitude of the cylinder from in the plane to 1/r on the cylinder; does change the principal curvatures KI, K2 from 0, to 1/r,O; and does change the mean curvature H from 0 to 1/r. The Gauss curvature, however, remains G = KIK2 = 0, as the theorem guarantees. See Figure 3.5. No kind of curvature can be detected by a bug on a curve. But if the bug moves to a surface, it can detect Gaussian curvature.

°

°

SURFACES IN R 3

23

3.7. Gauss curvature and area. An intrinsic definition of the Gauss curvature G at a point p in a surface could be based on the formula for the area of a disc of intrinsic radius r about p: area

7Tor

~7f

"

r

lJ

.J.

;- ....

UI

I'J

",.,

r \ . ,1.

VJ. V

llH~J.

n 1 -' n ..... O. ~ 011U o.v.

ro

'"

VV HI

111

EXERCISES 3.1. What are the principal curvatures Kl, KZ, the Gauss curvature G, and the mean curvature H at each of the following? At:::l nOlnt on :::l ...

H

1

_1'

of

...

AI Ifie OrH!ln lor me 2:raOfi c. AI me OrH!ln lor me 2:raOfi

D.

..

. u.

.

f"\.l 1IIC

t:. f"\.l a 6'

I.

'"

r1..~

..

~

7 -

T IX

VI -

ax " ;-

7 -

TI X

VI -

OOX

lUI ll1C

. . .. .. v

.

...

IJUllll Ull ll1C ,1

IJVllH Vll

/ IX

Z

~. 01

Ul~

• _.

lall

Z

7A

L.'fXV ;- )'1V"

-

?

T

,(.

X T L.A-

VJ

A

DV "

"

,./

1

0

n

z

-

.JV

A-

"'ty

T

T

Z

?

,., .....

Z

..JV

3.2. For the standard coordinates Ul = 0, Uz = 'P on the sphere of radius a, compute the first fundamental form [giil. Use it to calculate the length of a circle of latitude 'P = c. 3.3. Derive formulas 3.5(3) and 3.5(4) for the curvatures of a graph from 3.5(1) and 3.5(2). 'lLi

r

nf

~

in D"' h" -.;

f'(.,.\ in thp y .,._.... l~np -,. r ,-/

thp

.,._~v;c

-0

('hpl'lr

th~t

~

1'11r"p

Y

thp

J

1Ii:

hv

•

"

R ,.

X

7:::11i:

" ,.,

U~Z) l,;US U, J~Z) SIll U,

Z).

Use Proposition 3.5 to show that the inward mean curvature is given by

H=K+

.

.

1

(*)

tVl + 1'2'

wnere K IS me Inwaro curvamre or me oriQinal curve I K(T/(1;-T r I. LnecK rormUla (* I ror a sonere cemerea aI ."

-

.

ll1C

..

"1/"

-----/-24

CHAPTER 3

3.5. Do the first-year calculus exercise in spherical coordinates of ____c_o_m_puting the area of a polar cap of intrinsic radius r on a ---~sphere of radius a to ob~taHinn----------------

----T+-hen verify equation 3.7(1) for

th-P.oisi-tc~al-S1se~.~--------

This chapter shows how the theory of curvature at a point p in a 2dimensional surface S extends from R3 to Rn • As before, choose orthonormal coordinates on Rn with the origin at p and S tangent to the Xl, xrplane at p. Locally S is the graph of a function

(We will soon have so many tangent vectors v around that we are now abandoning boldface notation v for them.) The bilinear form (D 2 f)p on TpS with values in TpS.L is called the second fundamental tensor II of S at p, given in coordinates as

.

... .

26

CHAPTER 4

Again, this formula is good only at the point where the surface is tangent to the xbxrplane. If n = 3, this second fundamental tensor ~~ • ,~ .... L~ £.,

J 0.&., ..

..:I

~

.-

"' ...." -~~

_n

n

• £.£.

....

~u_

_n

~£.

~£.

..U U " ' . . . . .U ' "

L.~

Tk"" fo..... ,,"" ",4= ..,& TT

............ '"

..: _~~ .. L~ ••. : ..

_

£'U'££u

u ....'" •

~_&

n

~~

.1

11 .... L~~

...... 1 1"-&

.

<-

"U'"

..&_~_

n

&~,

_~~

••

~p""

II

IT4=

~

'1

II

\.L..

,.

....,

......

~'" " .... 11""..:1 foko

..:I .. ~

.1.'

U_ \ .L.L • • •

I

"'~

U&""

....

_.~

C

£

.11..:1

,&U'

~~~1 .... _

.L .... ' "

"'''''..........

'"

,--

fok"" ......... n U&~

'T'k~

I':!.n. , ....

...

..:I •

"'.....'U' .1 n c::. T c.L ~'" _£.£. _

.......

&U'

.1

o.&U£"

I':!.

1'I..T.

~.

&

',1.

_&U&_&

-~

..

.-

II ... "'.. I':!. ......

u..,&

~

depends on the choice of orthonormal coordinates. Let G; denote the Gauss curvature of the projection S; of S into n

n

""1

"":l

v

ff\l

n

'V

LV)

""1

"

ff\l _

'03

LV)

""1 •

Then the Gauss curvature G of S at p is n

V -

~ ~

.

~

v;

.~

simply because the dot product of two vectors is just the sum of the products of the components.

~

4.1. Theorem. Let,) De a C sUrface m K'. 1 ne Jl~~t varzatlOn OJ me area OJ ..) Wlm respect to a Compactly supporcea c vecto1Jleta v on ,,) is given oy integrating v agamst me mean curvature vector: )\~({'\

,.

J

-

r

I V.U

S

:Smce the tormula IS lmear m V, we may conSIder vanatIons m me Xl, X2, ••• Olrect!ons separately. .tor me Xl, X2 ouect!ons, ... nselT () .., I alonp ro rne \I. as tne ... <1n thp Y ....... s~vs T.pt V hp ~ sm~l1 ~nn .area dx. dx.. . at n consider an infi . we mav assume that the x." comoonent of II is diaQonal: f'rooJ.

&

...

.1

.1

.1.

&

&

KI

u

l'\

.v

n.2

To first order it is disolaced to an infinitesimal area

U Thp

<-

+ •

Iv IKI) aXI U <-

I V I K2) aX2 - U - V . H)aXI aX2'

..

~~ l~n~l

'T'k"" .L ....' "

+

'C

'U'....'U''''' .....

-

..

,~ .. ~~ .... _..:1 .. L~ ~_~ .. <0.&.............. ' " ...... . , ..

•

.1

~~

__

..'U'........

SURFACES IN R"

27

extend without change from R 2 to R n • Likewise Proposition 3.5 generalizes as follows.

4.2. Proposition. For any local coordinates Ul, U2 about a point p in a c 2 surface S in R n , the second fundamental tensor II at p is similar to 1 [P( xu) _ _ _ _ _~g-~I__#JiP(f-I-D~2XiM-)~ g-----"'-1

P(X12nt--, P(X22)J

------------------JLp(X12)

_

where P denotes projection onto TpS I and

iix Consequently, (1)

(2) 2 2

Remark.

(

)2

TpSJ... and hence P change from point to point. If

If Xb X2 give an orthogonal basis for TpS, then (3)

If Xb

X2

are not orthogonal, compute P by replacing

Example.

Consider the surface

X2

by

2R

CHAPTER 4

we

use x -

will

ana v - 1m 7 as coorainares. lnen

ffe 7

x-

-.

P

Y.

I I

x..,

-IV. 1

'eo SIn V. e"

cos V

Ill. 1I. e-

Y.-. -

Ill. 1I.

"P"

Y-.-. -

II.

"po

-.

I

-. -r e"

S1

P

~()S

Y •• -

x..,

l'llOre rnar x.

sin v I

I X. V it COS V P"

V P"

~s

x•. x..,

cos VI

sin

e"

sin

VI

VI

cos vI Sl

"P

VI

V

III 1I. 1I. 1I1 H

II

P .L.X ,L

/1

\.L

'T't ~n

~

',n

.

.1

... a ............ y<.u~... ~]

.........

........

I A \,

.J:.

~

n

~J: "'

oJ: .

.1

<.uU.u]U'"

Xu ' Xl

Xu -

~

~,

•

~~.,

,/

To compute G, first compute P(Xij)' Since Xl, P(xu) =

.........

J:~~+

\

u

~'"

/

~

Xl -

Xl 'Xl

X2

are orthogonal,

Xu "X2 X2 X2' X2

_ (-e 2x , 0, eX cos y, eX sin y) 1 + e2x Similarly, P(

) _ (0, -e2x , -~ siny, eX cosy) Xl2 1 2x '

+e

P( X22

) _ (e 2x , 0, - ~ cos y, - eX sin y) -

2x'

1+e

Hence, G-

'2 ..... ..,. AI

,-,-

r".

'"

r. .LJO~

, ~

, .:I v~u"'~

L U]

",L

..

,L ~u",

,L

un"

r

.. ~

.

1

"a~u",

..

.~1

,,2 .r-t

J:

-

a"

(J~VVL

~'V ... u~

2e2x - (1 + e2x )3' 0]

. .

1r.l

~

...a

(e 4X + e2x )

_

(1 + e2x f[(1 + e2x )2

~.

Y"~~U

~~

(_e 4X _ e2x )

1:': ...

.11.

r""'.~

,

~aua"

co

,n

. 1. • . • .

'1 .L

nIL nU-.

L

\J

,,2 u

K22

~

..

..

,,2 KU

'1 .L U

'1

n1J'.

'"

" L ..I.v, "a]"

u~

""I, ""2

u K12

'n

'1

L

nU"-.

~

.......

r' 'oJ

'T't

,L

"..,.

~ua~

~

T .. ...

~'V

~"

a .... ~ . . "'~

SURFACES IN Rn

29

EXERCISES

4.1. Compute the mean curvature vector and the Gauss curvature at each of the following: a. At the origin for the graph (z, w) - I(x, y) -

(x2

+ 2y 2, 66x 2

-

24xy

+ 59.i).

[Then compare with Exercise 3.1(b, c).] b. At a general point of {(z, w) E e 2 : w = Z2}. 4.2. Show that for the graph of a complex analytic function f, {w = f(z)} C

e2 ,

H= 0, and

In particular, the graph of a complex analytic function is a minimal surface. (Compare with the example after Proposition 4.2.)

4.3. Minimal surface equation. Show that the graph of a function f: R2 ~ Rn - 2 is a minimal surface if and only if (1 + Ify i2 )fxx - 2(fx . fy)fxy + (1 + Ifx 12 )fyy

[Compare with formula 3.5(3) for the special case n

= o. =

3.]

7

at p and S tangent to the xbx2;:.. ,xm-plane at p. Locally S is the graph of a function :T

S~

T S.1-.

v, is just the second derivative K

= (D 2f)p(v, v). .1-'

(Pf 2 m

ax~

The trace of II is called the mean curvature vector H. [Some treatments define H as (trace II)/n.] HVDersurfaces. For hvnersurfaces (n = m + 1) II is iust the unit normal n times a scalar matrix, called the second fundamental IOrrn ana alSO aenOlea oy 11. n nn, wnere n IS me ~scata'J mean . . .. . . curvature. 11 WC;; l:UUrUlIH:llC;;:S LU rIlClKC;; LIlC;; ~~

I"

~

••

IUllU

•

,

~

TT...

TT

V

~

•

n ~nb ... ,nn) 1:S C;;XonnlO.xn v, WUUC;; lUI 1
n

•

Kl

••

-r- .•• -r- Km •

Km

11 LIlC;;

unn ~

••

IUl:ClUY

Cl:S

Cl

••

UIUL

1-1- -

.•

,

~.

;:)>>./;:)v.

-

I

I

r>"

I"

LUC;;II

=

-~1" n

i=1 T l'

1

U. UJ.v

n

=

uy c

I"

I;:'

~IVC;U

(1;:'

a

1

r

1

IC;VC;I ;:'C;L

V!/IV!I, where V!= (anl/aXb ... H= -c1iv

r/

lJ \Ab

"1

• . • ,An)

,1.

t-f, LUvU

,ann/axn), and

viIV!I

(1)

5.1. Theorem. Let S be a C 2 m-dimensional surface in Rn • The first variation of the area of S with respect to a compactly supported C 2 vectorfield V on S is f(iven by intef(ratinf( V af(ainst the mean curvature vector: I

If't~)

I v· H.

- J

S

duectiOns, WhICh correspond to slidmg the surface along Itself, ('j·t~) - U, as the tormula says. Let V be a small vanatIon m the Xj ouection tm <J < ny, anO consiOer an mtImtesimal area aXl ... aXm at p, where we may assume that the Xj component ot 11 IS dIagonal:

m-DIMENSIONAL SURFACES IN Rn

- - [ :

33

:Jf-o-----.-

To fil st or del, it is displaced to an infinitesimal al ea

The formula follows.

5.2. Sectional and Riemannian curvature. The sectional curvature K of S at p assIgns to every 2-plane P C TpS the Gauss curvature of the 2-dlmenslonal surface

If v, w give an orthonormal basis for P, then by its definition the sectional CUI vatUl e is

K(P)

=

II(u, u) . II(w, w) - II(u, w) . II(u, w).

For example, if II - fam and P - el /\ ez is the sectional curvature is

XbXZ

(1)

plane, then the

Remark. For hypersurfaces (n - m + 1), for any 2 plane PkPij ei A ej, if we choose coordinates to make the second fundamental form diagonal,

n- [ : then tSl<Jsm

Thus any sectional curvature K(P) is a weighted average of the sectional curvatures Ki Kj of the axis 2-planes ei A ej' For 2

< m < n,

Rn

::::

TpS

Jet Si denote the with sectional curvature K i • Then, by X

R1

X ..• X

Rn

m,

projection of S into TpS X R i , (1), the sectional curvature K of S satisfies K = ~ K i •

-"' '1.

ru A D'T't=O thp

~-~~

II"

1T1!:11
'"

..

hp

nf hu

-I:

"I:

'"

thp

'"

~

tor

M

I

-I:

I~

ovpr

~nrt

•

-

ot

0

0

p~('h

.~

»'J

~

~vpr~ (JP

C I-

~

II

I"

\' 1n

»'J_

thp

npnt..: nt II ot

~

!:I n

~II

~

components. If m = n - 1, then II is a symmetric bilinear form called the second fundamental form. Its eigenvalues Kl, . . . , K m are called the principal curvatures. Since (D 2f)p is symmetric, in some orthogonal coordinates it is diagonal and f takes the form v",v7 - •.

.fJ

..... v=:

""

nil

.J.

r>{'V~\

.J.

~

""

/.

\'

k

k

la··1

v"

'-'"-

• • • .J.

""

k

In

...

=-.... ~..

J.

~

I~

rnen

-

I II v

~

:IU:-'

I

\

K(P) = \ ~ aikViVk)

• \

~ aj1wjwt)

-

\

~ ajkVkWj) • \ ~ ai/Viwt)

= ~ RijklViWjVkWI,

(2)

where ...,. UiL· • Ui!

I\;iJr/

::lTP. thp "} x "} k

I

.

.... OT

anC1

III II

ot II

1:-' ,;

n,.."

f( , .... ..,

n ....

= au . an

•

to '-'

TOW~ 1

1 ::lnC1 J

. n .... ..,

n

-

al2 . al2

IS

a13

a14

a23

a24

from

TOW~

I

L

I 1~

Vlil

... ...

the sectional curvature of the

thp

f( 1~

m::ln

.

4 ot

au al2 II= al2 a22

R l212 nl::lnp

/",

\..J J

Uik • U,'!

11I~t

'hll~

ot tnp.

tnP. "} x "}

J

thp.

Xl,XT

-

-

-

IV

R jikl

=

R ijlk

= -

R ijkl

(4)

(interchanging two rows or columns changes the sign of the minor), and R klij

=

R ijkl

(5)

... _nTl\

TT 1~

.. ..

Onf' r.~n

I I Il J

1"

A T I;:T TOt;' A rt;'1;: TN V"

cT

.•.

.•.

-no'

"~ fir~t

' •

V J

J

on nermutation of the last three indices:

+ R ik1j + R iljk = O. .. .

(6)

R ijkl

,

.

.

ootam a aenmuon

.

10

.'

\;UUI

:~

IIi'!

.

1 nJ.

UIl

.

maepenaem

OJ t( ..., "

IlUlt::

~

fi

lIli:1l

OJ

. 1:S

. . ..

me cnOlce or or,

I'

lIlt::

IUllU

II /\ II on 1\ 2ToS. Indeed, if {eJ gives a basis for T oS, so that < I} gives a basis for 1\ 2Tp S, then

{ek /\ el: k

TT

.u.

TTl' 1\

TTl'

\

.L.L\~k 1\

\

TTl'

.L.L\ ~kJ

~IJ

(~

\

.L.L\ ~IJ

(~

\

\.6.JUrk~r)

\

1\

\.6.JUsl~s),

uil

H.ijkl·

and \

I' \~i

A~

~

TT

.LJ.

1\ ~jJ

•. ".

TTl

"

Uik

1\ .L.l\~k 1\ ~IJ

R i~

/\ 2T. S

form on

..

l . R( [) for unit 2-v

/\

Ujk

ujl .

..

n

..J

hv the

v;-i"

:~

J

Actuallv R is determined bv the sectional curvatures p. R(P) for 2-nlanes (unit simvle 2-vectors). The Ricci curvature Ric is a bilinear form on T nS. defined as a kind of trace of the Riemannian curvature. Just as the trace of a IlIUUIA

D ~

r l' L\..ijJ Ii:>

l E

~ U

i:>Ulll ...

\..ii

~S: .. l..~ D ' ~~'

U

... .

r

~_

.. L .Ip~,

~l1v

...

1-••

vJ

61. "'U

= ),R ...

J'

yuu

I'

~_~ ~.

R.

YI'

..

...J VVvl

'Jk 'U''' "u'" ........"''''.. "'................... '" ...... '"

11

2TS

(7) ,

'J

i I'

lllIllll.. UI

,.

~

fiijkl

,

r~

,

r~

Ll\i1k1J

:~,

Ii'!

i:1:S i:1 Illi:1l11X UI

,

r~

Ll\i1k2J

Ll\i1kmJ

, [R imk2 ]

[Rimkd

[R imkm ]

then R;1 is the corresnondim! matrix of traces so the definition of Ric as a bilinear form does not really denend on the choice of orthonormal coordinates for TvS. Its application to el yields the sum £ V.l

.1. U.lV

£ V.l

.1 VU.lV

o

• 1

R 1('

(0. \ ,

.1

~ ~l'

UA.li:> 1.

fl

Ij

'11

-

-

) . f l ..

-

'Ul

i

m

.......

= 2.. I

'L.

K(el /\ eJ.

). fl ..

-

i401

IIII

~h

..

r""[J A D'T''CD .. ~~~

-~

~

~-~,

Hence for any orthonormal basis

VI, ..• , V rn

for TpS,

rn

VI .

Ric (VI)

=~

(8)

K(VI /\ Vi)'

i=2

and for any unit

V

E TpS,

v· KIC lUI

unl

r I I I

~

",

.1

-"

~'"

9.

~

1\.1U /\ WI.

I

~

I

W.LU W<:::~pJ

'T"1.

.1.

.Luu..,

..u""

n'

.L.... ~""""~

""u~

~~

1.

ua",

... ~_

au

a",

.J:

au

'0'

v~

sectional curvatures. The scalar curvature R is defined as the trace of the Ricci curvature: (10) R = ~Rii' i

Hence for any orthonormal basis v, . .... Vn> for TnS. ~

J<

L

L ~.

. r

mlm .,,,1

-J

I'" I I

11 (j])

~~

KIl"l

...

t17> • • 1. .J: .11 " • .., ~'" ..u"" "''''' .. v~ au 6._ ..1. ~ ..v .. U"" a -0 r

...

C' lJu,

'T' ~u

.J: VL

.L

'T"1.

.L

au

_

..1.

UU..,

..U""

.1

.11

I I I 1

J

rE'&'

,u""~",,

:n • ..,

KIll· 1\ IJ:I

,,",UL

.

""u~

'a .. u~"",,,, a .. a

"a .. u~""

yvu....

Remark. Historically Ric used to have the opposite sign. Some texts give the Riemannian curvature tensor R ijkl the opposite sign. Let S be a C 2 m-dimensional surface in .K • 1I T is a oirreremiaDJe runcrion on.'\ men me oerivalive v U . . . IS a veCIOrnelO. nUl II I IS a veCIOrnelO lor a nelO or malflces . . . or .val.lVe 1 LJUU .Wl:SC In 1~") lnen (ne \. .v WIll nave . lO .:>. Int: InlO 1 ~,.) IS :~ II~(J lnt: ~VVUH.:)t:t: .l unt u<: J.l emu J . .t.. lInt: Hom '" \,;Clli:1111 1l1\,;C u U~;) 111 i:1 IIIUIC O~U~1i:11 -0' scc r r Chapter 6.) In local coordinates UI, ••• , Urn in which g = I to first order at p, the coordinates of the covariant derivative of f at p are given by the several partial derivatives. For example, the coordinates li;j of the covariant derivative of a vectorfield with coordinates Ii are given

5.3. The covariant derivative.

.

~

~

~.

~,

~.

--

-

1-

UJ ~r

I'";.;

I'" ..

°li ri,l. -J

/~.

;~l n~

.

..

_ a

,

~

--...,-.

'1"7

..

-....:..'-.

~

~

,,

~

~

""""

oJl

~

~,a

,....

.-

;>

;" U

,

'r\( , y

I

~

~

~

. ."... "" ........

~.

1.\

t! 11

~

A

\"J ...

..

~ &

• .1 I L. \ • .t. v . . . . .'" "'''''''''', \V J ...,

n

!'Inri £r\ itc;:

,

~

~

,a

~IU

.. u

~,b

.,

~L v

...

'tI"

A~~ "-"

v

'" "

U''4!,

"

{

"

,

"'-

'b

U

lJ -

C1

.4

I.a-D

U

,, '"""-

,~,

b'

,-

S2

. .. UCIlVC1LlVC,

I

a.4 l"""r

~--....

c

,

I

~

,u

,

j

UJ Il

.-

a=b=c=O Inl A

C1llU \C,

.. on thp. f'lrf'lp

. .. .... Its l:uVC1IlC1lll UCIlVC1l1Vc.

I h 1 It"-

(;

~Q ..... ....,

r""[.J A D T D n

~

--~~.

~

~~

~

.....

~~

""'1' II: 1 .........

'T'h~C'

.L AAA'"

r

..]'

"1..1

..1-.0 "AU.'

'''''",,"

_

.... ~

.......... ~..,

n" ..1-.0

.n.~~IY~'" .n.~ ..1-.0 '1 ..]' .. AA.., oJ

. . . . . . . . AA_

-0"

,,~

sional surface in R;J given by

YI = Xl + 2x I X2 + Xl + 5X3, ".

-

J:"

Q

Wh~t ic;,: TT (~t , thp

~y~

-I-

y:"

J.

-I-

" :'"

y~

-I-

')Y,.Y,

:"

"'"

"oJ

. . l\? ..., -/

b. What is the sectional curvature of the xI,xrplane? ,I? ot e Wh::tf 1~ thp. ot thp. Y.. + Y"" the plane Xl + X2 + X3 = O? Q. VIve au tne components or tne Klemanman curvature tensor Use them to the :-. to narts hand c e. compute tne KICCI ana scalar curvatures. It"

J.

A

~.~.

A

..,

..,

,.,

Lonsiaer tne vectornela on K' : f - Y I + ~x + Z}J + X 'K. a. Comnllte its derivative at a £Jeneral noint in R 3 • D. \...ompute irs covariant oerivative at ~u, U, 1) on tne unn sohere. A

...,

A

5.3. Show that for the e:raoh of a function f: R n H t". .t '

div

Vf VI + IVfl:"

rlHih. -.I

",

L.

.t 'J

(1

",

~ R.

+ IVfI2)~f - ~ fifjfij (1 + IVfl:"r /:"

rl:" H rlv. rlv. .t

l

-J

div(p,q, ...) =PI + q2 +"', and At = aIV Vt - t 11 + t 22 + ....

~H .I

( t" ,.I

J.

t"

, .I

TO

,

J./'

not really a more general setting, since J. Nash [N] has proved that every such abstract Riemannian manifold can be isometrically embedded in some R . I suppose that it is a more natural setting, ut t e ormu as get muc more comp lcated.

itions behind us. Formulas et much more com licated in intrinsic local coordinates. In particular, com lications arise because the local coordinates fail to be orthogonal, as in Figure 6.1. The ul-axis is not perpendicu-

.

. .

.

40

CHAPTER 6

Figure 6.1.

For nonorthogonal coordinates, the Ur axis is not perpendicular to the uz-axis (the level set {Ul = O}). Infinitesimally, the unit vector el = MaUl is not perpendicular to the level set {dUl = O}.

components of a vectorfield

and the components of a differential cP

= (cpt. CPz,· .. ,CPm) = ~ CPidd

behave very differently, under changes in coordinates for example. To emphasize the distinction, superscripts are used on the components of vector-like, or contravariant, tensors, and subscripts are used on the components of differential-like, or covariant, tensors. Thus a vectorfield X has components Xi. Its covariant derivative has components X~j, distinguished by the semicolon from the partial derivatives X~j = iJXi/iJUj. As our first exercise in intrinsic Riemannian geometry, we will prove that the components of the covariant derivative are given by the formula X i;j -- Xi.j

i X k + £.J ~ r jk ,

(1)

k

where

rJk are the Christoffel symbols (2)

41

INTRINSIC RIEMANNIAN GEOMETRY

defined in terms of the partial derivatives of the metric gii and its inverse l i . In particular, covariant differentiation is an intrinsic notion. In formula (1), the partial derivative gives the first, main term. There are additional terms because the basis vectors themselves are turning. In both formulas (1) and (2), note how each index i, j, or k on

--- .--

... -_.'

i-ha lafi-

..._--

..

;., i-ha .," ..... a ~

i-ha ..;,..hi-

l\.T.....i-a h .....u,

•--- --0" ,

-

\'

_~ __ ~

..u n "

---

-~-

{".,

~

orr

~c hAth ~ cn'

AnT

UTlll

~nrl ~

-r

" ..... "a..

or ~nrl

T '.:J

C"

u

__

i-ha ' R'\T

-r '

--..

.....

~

___

o~

.:J

If

_.

.......

thpcp

.I

......

r~

JT\.

fAlrlc

lc ~ lCA

rA'\I~Tl~nt ......

'" 1..

hv 01

,.

'

for thp' ......

~

""

1..

it oivp''' "omp. ": Our

11

,

~

.11

.:J

~t

......

rElI'J_

.~

, ""

'

nn Tn~nl_

" "

of of S

thp

~

f\ ..... n 1 ~ .: .-:-

from ~ R'

~

metnc (the 'LevI-Clvlta connectIOn') IS symmetnc, so the larswn IS U:

-

Ti JT\.

_r

r~

JT\.

i ,-

(~\

fl

,.

T\.J

~

The Riemannian curvature is given by the formula ~.

T" •

fljkl

T""'O

'"

•

T"h

1 jk,1 T 1 jl,k T ~ \

T"hT'"

T'"

"

\'t)

Ijkl hi T 1 jll hk)'

L..

The .l{temanman curvature IS thus mtnnslc, because the connectIOn

ri

.

JT\.

illS

NAtp th~t p~{'h

1'"

..

1111

inrlpy An thp

on thp rioht ~nc1 thM ...,

.. Thp olc1

~" hoth ~

.1.. • ,'I..

lpft

in thp "l:Imp

rlln~ ovpr thp .

,

111-'

:x h

~nc1 ~ . . . '; ')(4· -hi "till holc1 for thp rpb rprl

"' R ijkl

= 2.. gihR'/kI' h

S'Ince

R i;Irl

=~

'h

J/ihR,h;/rl R i. - _R i. J'

but in general R{kl =1= - RJkl' For example, tirst Bianchi IdentIty still holds: ni ~'l

jkl

r';i

J

ni ~ 'l

ni

klJ

Bljk

R Zkl

need not vanish, The

{\

v.

IL\

\VJ

The Ricci curvature is given by the formula ~

1<.jl -

2.J 1<. jil ,

(7)

42

CHAPTER 6

and the scalar curvature by the formula

= LgjlRjl.

R

.

~.

II::

.

~

v

\.

(8)

1<:1.1111:: WII

<:I.

L

IL

'-

..

LJ<:I.SlS U

W

IS

IV

i!lVI::J

TT/

.n.\

,

v

....

"-

~

1\ W

. k

H,';/dV

wv

If S is 2-dimensional, its Gauss curvature is G Note that if g = I to first order at p, then rjk

u",

1

W.

\::7

. J, I

;

= R/2.

= 0,

y'

y'

,J

,J'

11 ..

-

-

11:

_ T

i

4- T'i

J',,," ,

J"",

J""'

'J""'

" niiii,

n

1.\.;1

. . . I.\.

R=LRJJ' ..

and K(u

A

w)

= LRjklUiwiukWI. ..

A.1l

nOln: n In Hn m~

[,["\1

or .'\ C'.c

r

Ie

rc

leI ne nHs,ec on r Ie

H" H 11"1

~

ne

or:::l n:::ll

volume

=a

r :::InOl

01

....

rm m

-

a

.l\.

m

6(m

+ 2)

n'

~

r m + 2 + ...

,

(10)

where am is the volume of a unit ball in Rm • When m = 2, this formula reduces to 3.7(1). The analogous formula for spheres played a role in R. Schoen's solution of the Yamabe problem of finding a conrormal aerormalion or a l!iven Kiemannian melric IO one or L.I conSlanI scalar curVc:l LLJ I~(

6.1. More useful formulas. There are a few more special formulas needed sometimes. The covariant derivative of a general tensor f is given by the formula

43

INTRINSIC RIEMANNIAN GEOMETRY s + ... + L fhmk fmJz.:.i lr

= fh···!s f il}1"'!s ... lr:k l} ... lr,k

l} ...

m

s f!l"'!S-l m _.~ fki fh.:.i s . + L fimk 11 .. . Ir 1 m/2 ... lr m

(1)

.. .

m

s - ~ fki r f!l···ilr-1m . l} ...

m W,rri

ii

~.'

e,'"

,~

..• '

.~,

r-"'~'

J::~1r1

V

n~n

_~ ..

n~ ..

. . _,~

~L

~,-

. . ~.

-'1~~"

..h""

A

-~

D;~~;'n'

nl

I"

nt thp

,,~"" th ~t thp

'~

L . .' - ' ' ' . ' '

•

~.~~.~.

.1

~:.'nn

~-~.

~

••

-~

~L

~

n .. n~.. _:n~ r ~ -'J "'-'-'

e' '-'"

; ..... "" ....... " ".of .. h"" ......

Y.

_

y

.-

~

_

)

.

R ' . Y

n t<

rc

r

.~

.

I~

~

::I~

.

::IT ::I

t:::tll11re ot

IIrW

v

or

tor

llii

np. t<1P.Iliil:'\

In

&

")1

I~

'-'

111:l

lntO:::t

ot

lit

~

IIVii II.

6.2. Proofs. There are two ways to prove the intrinsic formulas of Riemannian geometry: either directly from the extrinsic definitions or more intrinsically by exploiting the invariance under changes of coordinates. As an example, we prove the formula for the covariant derivative of a vectorfield both ways and then compare the two methods. Extrinsic proof of 6.0(1).

Consider a differentiable vectorfield

The ordinary partial derivative satisfies

'-

,,-

vy

Ul~

.1. Lll~

..1

'.

..I U~llVat1V~

.1 lUl~ •

'T'.L .LV 'T'

VJllV

ro

.LnD,

,,Lll~

.1. lJl~

, ~vv

U~llvauv~,

r ~lJaJl

Vi

AI, A2,' •• ,Am,

VV~

.1 lJl~

1

44

CHAPTER 6

space or range of the matrix i

A = [ax .]. au] It is a well-known fact from linear algebra that the projection matrix P = A (AtA)-lA t = Ag-1A t. (A is generally not square, but AtA is square and invertible.) In (1), the first summation over i already lies in TpS. We consider the second summation over k, m. To get the coefficient of ()/ax n in the projection, we multiply the coefficient of alax m in (1) by the n,m-entry of P = Ag-1A t, which is

to obtain

Therefore

r jki X k , X i;j -- Xi,j + ~ ~ k

where

r jki -=

~ gilXjk' X I ~ I

~ gil (~)[(XI • Xj)k + (Xl'

Xk)j -

(Xj .

xk)d

I

=

~ ~ gil(glj,k + glk,j -

gjk,l)'

I

Notice how at the final steps we passed from extrinsic quantities to the intrinsic grs,t.

Xjk

Invariance proof of 6.0(1). In this method of proof, we first check the formula in coordinates for which g = I to first order at p and then check that the formula is invariant under changes of coordinates. If g = I to first order, 6.0(1) says that X~j = X: j , which we accept after a few moments' reflection. To check invariance, we must show that if u i , d' give coordinates at p, then

INTRINSIC RIEMANNIAN GEOMETR}'

4-""'\5--

where we henceforth agree to sum over repeated indices. (Getting such formulas right-knowing whether the au il goes on the top or the bottom-is perhaps the hardest part of linear algebra, but our index conventions make it automatic.) This verification is something of a mess, but here we go. First we note that

au]1 au

I

Hence

and similar equations hold for the definition of f J" and

gjk,/, gkl,j'

tJ

g

il

tJ

-

Combining all three with

II

g

yields . f'·']k

r

1 au au II (au au- -aut) = --2 p -q gpq - jl [, kl [grs,t - grt,s + gts,r] + n , il

au au

where

because grs is symmetric.

s

au au au

~,r

ru A. P'T'P"R

"Ion

f.

1 1

-ri I ~ ik

...

;,

dU

L,

r

...

r

where &q

a U

aU

~

~

r

-&gpqg rs, q

J!'

;,

(JU

r

(JU

~

.? ~n

(J

1 '.., "T"

.

c,.

Ul~

(J! ... ,

,

Y

5sq gpq grs

-, I J!.., • "T" J!,... J I I

n.

7

rlll P :.lJli l :.lJl'" ~

il

,

rlllP rllli I rlll" I (JU

...

\

Kts r}

Krt s

aUil aU kl aUP

~

1 iJr

UJ

r

~s .Da/ vaK \Krs t

,.

= 1 if s = q and 0 otherwise • Since . (JU (JU (JU ;,

~i

1.

v",

dU' dU 2

+

.... t

v",

.L v " ,

U

h

= gpq gqr = &r, .2 (J

-r

U

r

(JU

rlll il rlll'"

rlll'

(JU

:.lJli l :.lJl'" :.lr."

..I l11U~A

e1.

111

Multiplying both sides by indices yields

auh/au"

,

~

1

Ul~ la~~

~~1111 ~IUI11

,

~U

H.

and changing primes and

2 il il hl kl au . au au a u h - r hk ',---- rt -h au m aut m au aut au m ' 1"1

i,

(2)

ow :I

VII _ "<1 ;l

v

•

v'

..L

"<1

T'll vkl lk"<1 .L

OUJr,

=

(J.

aUll

(-(JU xm) m aU

+ rJ,,(-(JU m X

m)

aU

2 il

aU aU;I a u aut au kl . = Xnm -. - - - + Xm --+xm--r~I t . lk· m m m • aUll aU au au aull au n

By (2), x .- x .J

rill.'

rill.

----

,

.... II v...." , m

v",

,

+x

1

rill. rill. -----1 ....n :l. II v",

au

il

au

n

= ----. x~ + m ll au

au

'

v",

,. riU' rill. rill. rill. ------+1. ... ,/

v",

...

v",

m

:l v",

II

J

.... m v",

il au aut m --h - - . r7mx . ll au au

By changing dummy variables in the second term h ~m, I ~n), we obtain

(m~k,

INTRINSIC RIEMANNIAN GEOMETRY

4f-7--

as desired. Remark. Of the two ploofs, the fhst has the advantages of being shOIteI and deIiving the fonllula, wheleas the second pIUVes a given fonllula.

6.3. Geodesics. Let C be a C 2 CUI ve in a C 2 m-dimensional SUI face S in Rn , with curvature vector K at a point pEe. We define the geodesic curvature Kg as the projection of K onto the tangent space TpS. Equivalently, Kg is the covariant derivative of the unit tangent vector. While curvature K is extrinsic, geodesic curvature Kg is intrinsic. A geodesic is a curve with Kg = 0 at all points. For example, geodesics on spheres are arcs of great circles, but other circles of latitude are not geodesics. (See Figure 6.2.) Shortest paths turn out to be geodesics, but there are sometimes also other longer geodesics between pairs of points. For example, nonantipodal points on the equator are joined by a short and a long geodesic, depending on

Figure 6.2. On the sphele, gleat cilcles ale geodesics (Kg 9), but other circles of latitude are not (Kg 4= 0).

AO

LnAr 1 r.K 0

"'TU

..

-,

..

WIU....U ~a~ you gO. 1 ~e pOles are lar lUl;;l ~.

. ...

1 IlC

.

..

...

01

'lUe,

~..,u..,

~.

~

.

oy

J

.. .

' au 01 Lile same WIlY

.

.y many

..

'-'

SIlunCSL l'cUIlS IIlUSL

--. UC

6.4. Theorem. A curve is a f!eodesic if and onlv if the first variation . . oJ lIS tengm vamsnes. Proof. Let x(t) be a local parametrization by arc length. Corresponding to an infinitesimal, compactly supported change ox in x(t) is a variation in length r

r

J

J

oL = 0

I (x· X)1I2 = I ~(x· X)- 1I2 2x' ox

r

- IT' ox~

r.

r

~

..

r

I K' ox-

IT' ox-

I Kg' ox

~

~

by integration by parts and the fact that ox stays in the surface. oL •

1

.

lUI .... UI VI;; I;)

a ...

au _1

.,

1

VA

...

.

•1

CUUUe, lUI;;

1

...

1

11 i1UU UIU Y 11 Kg

no

1

" auu

,1

lUI;;

•

Remark. It follows from the theory of differential equations that m any Co<, m-dimenslOnal surface, through any pomt in any dIrectIon there IS locally a umque Co<, geodesIc. Such a geodesic provIdes the shortest path to nearby pomts. A little more argument shows that if S IS connected and compact (or connected and merely complete), between any two pomts some geodesic provIdes the shortest path (the Hopf-Rmow Theorem, see leE, ch. 3J or lHe, Theorem lUAj).

6.5. Formula for geodesics. In local coordinates u" ... , urn, consider a curve u(t) parametrized by arc length so that the unit tangent vector T = U. The derivative of any function f(u) along the curve is given by 'LfJu/ (the chain rule). The covariant derivative of any vectorfield Xl along the curve satisfies ~

~

"" A ;jU

"" . A ,jU

J

~

-r "" .

J

"i

Ll.

1 jkU' A

J'

T

~ ni

"-J.1

j,k

.; ,rk jkU Ll.

."

~1)

110 11'0

r.

rn/1,1

"'T.7

1.K I

TT

1

c1. ,

-' lJy aJ.\,., "J.Vl a \ lJ a J. i covariant derivative along the curve of the vectorfield Xi = T = must vanish: I"~~ V.V\ ~J

-'

1_

1

I·

c1.

Ul~

L

0= i/ +

u

i

(2)

rJkujuk.

j.k

.

~~

tr" -.;

-" r

7_

HIP

A," !'In

-"

~

-.; r

..

::ITe: V1Ve:n hv the: '-'

" J( v l ,.

,,\

DL.

0 '-'

'> (\l.

1

•J

'./

'./ /

-

1-l'~~~--I!'In

,"n!'l(,p 1-1 fAr 'r

.

h::l

c:.

1n

•

with metric

= Y -2 Oij;

gij

that is, 1

"lC'L. _

.v

Y

.. .DOmIWISe

.

~mce

...

LVJ.l, La.

....

2

"l"L. 2~.!

.

Y

.

.

.

IS a mUIUOie . or me sranaara melflc I {! IS conan1!leS are me same m me uDDer naUOlane as on n. {!

. .. I

1

"lv-L. ..L

an:

IV;

..~,

VI

Now we compute the Christoffel symbols and curvature. j gi

= y 2 o ij •

By formula 6.0(2), 1 2 1 1 r 12 == r 21 = Y (gI2,1

2 I

~

- - v- -

,.,....

~,

rl

~

VJ

lJ

+ gll,2 -

g12,l)

~

-

-

-

-1J

'. .v, ,.., ..., 111- 1 22-

•

.1

Y

and the rest are O. By formula 6.0(4), Ri12

= - ri1,2 + ri2,1 + L(- r~lrh2 + r~2rh1) = _

lJ- 2

-

+

0

+

h (_lJ- 2

-

+

lJ-2)

-

,

= _

lJ- 2 ./

~r\

n

CHAPTHR

J\J

~ImllanV ~

I<

-

,~

~

. ,

1(11 -

~

1<,

-v

I< ~~~ -

-

~

I( 111

+ I( 121

R = - 2,

~

Y

-

II

,

~

1(22 -

Y

,

G =-1.

Thus hyperbolic space H has constant Gaussian curvature -1 and assumes its exalted place with the plane (G = 0) and the sphere (G = 1).

.

( tll'~n
h "I('J\' -,

v T

.+

.L.I'"'~t'

1'",

-.Lv,' _ (\

'),

'-'

<>nrl

OJ

-J

..

t fl'll1
~r{'

'"

'0'

,'-I. , -.Lv'"

" '.L,;'" _ (\

J

J

J

thp pnll!=1_ -...

J

.1,/.1, • +1. '-4-f' ~ . .'"' . .

'-4-'-"

.. dp.2 + py. .. x=-y dy

i = py,

Substituting for ji from the second equation in the first yields dp .Y

av I

...

10 hv '-'

'-'

-1/ 3 \Y

.\

YI'

01VP.~

II

'-'

flY

...

J:

rll

+

uy

-.I

. v

'\

/

2 2'

~

L

Y

If c = 0, we obtain vertical lines as geodesics. Otherwise, letting c = 1/a and integrating yields

(x - b)2 + y2 = a 2. These geodesics are just semicircles centered on the x-aXIS. See hi

'-'

II

IVII '-'

:::tnv two

.

.

thp.rp.

1~

:::t

.

or ~

~

In::n .[Ill the the .n:::tth . 1:'00 nTfn nC\
1~lrp

I

0

IdVldl

~

~

../ ./ ,/

-- -

,

..........

-

./

¥~

/

...·M.a1\J1\J .a,N :iF 1M ....

~

IN

KY

........

"-

""-'\

.........

((((' \\X~\\

...

~

...

Jl

,....

Geodesics in hyperbolic space Hare semicircles centered on the x-axis and vertical lines.

Figure 6.3.

It is interesting to note that the hyperbolic distance from any point (a, b) to the x-axis, measured along a vertical geodesic, is b

J

y-l dy

=

00.

y=o

Hyperbolic space actually has no boundary, but extends infinitely far in all directions.

F;.

'7

.

I"

.

---n ... 11

~

--r

'0'

~

,

~o··O£

"'., 1n

~~~

.,'J:'... "',,""

OJ J:'

-

'x'"" ..J

+h",t ... ",r",l1Al

,,,,' ~t'~~~~~.

..._& --

..

+1-.",+ ~

r

.u~

'0'

0

"'., An +hA

,

+1-.",+ .... "' ..... 11"'1

~.~~~

.u~.

0

~'XI(H( :IS~:S

h 1

A

.

fnrlH:

tpr

I .pt ./ hp thp torll"

~

hv J

::In < (() < 'l."TT' II"p thp. ~nolp ,., II < II <" '}"TT' ot

n::ltp~

hv 1/ J

I SP.P

h4l '-'

::IY1~

4

llnlt rlrrlp

""

llnlt~

~nrl

trom if)

~"

lt~

rpn-

-

52

CHAPTER 6

The torus T

Cross-sectIOn

Top view Figure 6.4.

The torus T, with coordinates 0, 'P.

n T...... ~T~T"'T......

U'

~T.,..,~

~nTT

E

~

~T

....

UH~!'

~~

,'1

~

oJ oJ

C'l. ~'T' .. l..~ .. uu'v" u n...

A.

,,'"t /

511 ['1 ~

T

\"V;:)

,2 'fI) ,

£\

sm 'P 4 + cos 'P

...

1

v,

512

521

522

['2

'''In m( L1 -lT ,

~~

.J.,

m '\

I'A'"

r.

.

v,

tilC IC:sL dlC

u.

,...,

,

1

,1

£'

1

1

lHC

a

a

cos 'P .:lt1'

that is, a 1 = cos 'P and a':' = O. Show that the covariant derivative is given by 4 + 2 cos 'P

1 U;2

'fI

;:)111

+ cos ([)

4

2 u ;1

,

,H11

.\

/..1

'fI

~v"

'fI""T

tne rest are u, c. :Show that the length ot the spIral curve ~ U s:; t s:; L7T') IS gIven by tne mtegral

~v"

tI~t)

-

'fI),

'P~t)

- t

2'7T

rt A

,.."'<" #\2 .J. 11 1/ 2 /1#

.J.

l\

'/

-~~

~

~J

.

0

d. Of course

{'()'i;.

R 212

m

,

A

= cos 'P(4 + cos 'P), \"VJ

R 22 = 4

r,

on

cos 'P(4 + cos 'P),

R 121

'f'

~v"

R 11

= R~')') = O. Show that

R} 11

.1.

'fI

+ cos 'P

,

R 12 ,... ~ ~v"

R=

1n th;:> tArn", An

..,

'0'

'fI

+ cos 'P

4

th~t th;:>r;:> "'" nn

..:I

=R

21

= 0,

,

~v"

G= 4

'f'

+ cos 'P

n"l~n Af ~n,. -.; r 'o-r 1n th;:> nl~n;:> Ar An th;:> 1 r 'r ..:I'

6.2. The sphere, This exercise will verify that geodesics are arcs of .....0,.,+

,., .... r1 +1-.,.,+ +1-.0

e'-~~

~

r.

'..1

.

~

~

••

,., ~

l'

o

••

7

1

~~

~

••-

.1.

('

~t'&&_.-

t1 ., .... rI

~.

~

~t"'-'

~

7

•• '

",f ~

••

2

1-.,.,C'

-

r ',.,1"'"

&&~~

n

..:I.' ~~.~~

-

...-

-~.

"7~+h +ho 11<"11,.,1 '.~..

-~~~.

~~_.

.1.

-,

.1

~

a,

'T'

rh.,.",lr th.,t ., or.,..,t ",~r",l.,. ~C' o~".,.n h1. th.,. O'

W

~

£ J.VJ.

',/

O'

== cot cp =

.1 Y~J. u~aJ. 5~~a,-

1

Cl COS (J

+ C2 sin (J,

£I

v

"'1

\.-,

TT.· A .J..J.111'-. 1:1 •• ~

,1 5~~U'-

Cit .J""T

\..-Oi'\.r 1 nK 0

circle is the intersection of a sphere with a plane

z=

+ C2Y·

C1 X

b. Show that the metric is given by a" sin" cp,

=

gl1

= 0,

g12

c. Compute the Christoffel symbols r 1_ = ('ot r2 = the re~t

.

.

VLi

~~

rr

1:'00

..J~

.1

thM

'Y

,..,

"

~

..

.. vv .. 'Y 'Y

•

:'ooLi

i"

':'001 V

.,

Rm= 0 sin cp cos cp iP =

45 ,.

.

..l

.....

Ii + 2 ~ot ~ ...:I ..u n...

a".

l;I.ln (fl (,{"\l;I. £f'l

(fl

~-

=

g22

/2

lD

°

~.'

~

.1.

(\

,~

on. 'Y ""vo 'Y

v,

Y'

..J

.... "'"

n"",.""

note derivatives with respect to 8, unless 8 = c. Thus w" + w = (\

~,

"" ~~

w=

C1

cos 8 + C2 sin 8.

Therefore geodesIcs are mdeed arcs of great cIrcles. II

(

..

.

thf' J.llf'm

1<.212 -

-

-

1,

- 1<.221 -

1<.i21 -

-

sm- cp,

-1<.i12 -

tne rest are u; tne KICCI curvature

. ,.,

.

.

lIl~

".., V

.

-? U

.. .

-

•

",u

fl

.

Remark. <;>

.t{12 -

~

,,::-t ;:-tT \,;Ul V

,1.

Ulal.

sm cp,

.t{11 -

-?

v,

.t{21 -

.t{22 -

. ,.. .. ..

, anu

1;

~

1111(111y {n~

uauss

'-'UI vdlUIC

Actually a simple symmetry argument shows .1

£ a.l~

a.l'-''' V.L

e.l~al.

•

"

T

'"

~

luay

.

,1.

,1.

l.ual.

UI~

...... .... • D ••J .... n .. '1 , ...... must equal its own reflection across the equator. Hence it must be an arc of the equator. ~~

"0

.. ~ .. t..~ ~~ .. ~ .. ~_ ~ .. ~ ..v ..n""

•

.oJ

~

Toward the end of the nineteenth century, a puzzling inconsistency in Mercury's orbit was observed. Newton had brilliantly explained Kepler's elliptical planetary

degree per century: Planet

Predicted Precession (per century)

ercury Here 60' (60 minutes) equals 1 degree of are, and 60" (60 seconds) equals 1 minute of arc.

43" per century? General relativity would provide the answer.

56

CHAPTER 7

,•, .

~

II ~L ..,V' ~

...

"' ..... .....

"'

,~


....-.

I•

'.'IIi'

,

'V\

.......

-..

-- -- - -

I

,.

I f\

I

~

... I

//

,

~.~

.""--;:0.. (

:r \

,

OJ I

___

lJI"

I

I

r ~~

A. ""

r \ "l.....I.. \\ \. ,/ '\,,-

,......".

1\'

I"

-J-If'

"\

~C

r/J"-

,

~

,

A

\

I

- . 4 / '/1 "Y/A \.. ~ , r'-4J.J.Y/' '\. IJ

// /

'\.

I

I I I

\ \ \

I

Figure 7.1.

'.

"Mercury's running slow"

7.1. Gellel'all'elativity. The theOly of genetaltelativity has tlllee elements. Fitst, specialtelativity desctibes motion in nee space without gmvity. Second, the Ptinciple of Equivalence extends the theory, at least in principle, to include gravity, roughly by equating gravity with acceleration. Third, Riemannian geometry provides a mathematical framework which makes calculations possible. I first learned the derivation of Mercury's precession from Spain _ _[Sp, Chapter VIII] and Weinberg [W, Chapter 9] with the help of my friend Ira Wasserman. The short derivation given here is based on a talk by my student Phat Vu at a mathematics colloquium at Williams College, in turn based on Ieffery [Ie, Chap VII] A sim-

j

~

L-

57

GENERAL RELATIVITY corn", ~

orr

-J

~ ~

ro

.. . . .. .

.

from

th",

of 'J

~

"" in IMAI .J l'

_.

,A..

.

'

.~

&.

& .. 11

u ... "'''' ,,}-'u"'''' straight line at constant velocity, For example, x = at, y = bt, Z = ct, I

eMe

_._~.

,.~,]

}-'LU LI"'~'"

"""

~..

or

x a

y b

Z

-=-=-

• ' • 'T't.~~ ~

.. u..,

~~

.. t.~

H,

LU""

i'

&~_ ~

1

.....H

_ ..... 1..

~,..

... 1,..~ ...

}-'~.u

.. ..,

~""''V

~

U

• L>LU.....

1~_~

' _1. ..,U

1

E>.. L

~"O".

~_

c 1.~_~ . . . . ""

, _1.

L'. •••

~-0.4

,11'

..

.. t.~ L"""

y

n

h

~

1

u

uu.., .....

A

".l ~

~_~~~

'r '--'

.......,.

t

"

1

r

that is, a geodesic for the standard metric

ds 2 = dx 2 + d y 2 + dz 2 + dt 2 •

(1)

It is actually a geodesic for any metric of the form ..1~2

""..,

1 ....

'T't. ... ~

l~n.~

.."" ..u

',,"

~& 'V ..

1~

1

..1.. ..... UU"'L

',~

.. .."

.. ~

L'V

L"'"

..1_2...L

".:l""~

~

~_

~

11 •

..

'1\ \ ...

•. 1

""u.. ~"" ... ~ u ...

'V ... ..,..,..

.., .. ..,

i'

.&

.. .. L,u. ."",,"

'V ..

......

... 11 ~

,

~ ..1.2 "4 ""• •

UIl lWU

.1. . . ~ ~ ~~

,,"'V'V""

"J

~

2...L

. . It:li:1l1VllY .

.

.

........

~ ..1_2...L ~ A. "L "".r "1 """"

n.:"t. ,

'0

velocity relative to one another. (Of course, in accelerating reference frames, the laws of physics look different. Cups of lemonade in accelerating cars suddenly fall over, and tennis balls on the floors of rockets flatten like pancakes.) 2. The speed of a light beam is the same relative to any inertial frame, whether moving in the same or the opposite di{ thil1. t~l't , ..., or orr '0

".I

thp

1;;.1ll'h

~I;;.

tlmp'l;;.

.I

It

r

rlown

I;;.

h"

r

"0

~t

'-'

thp

.I

Ip~rll1.

to othpr

hH1h '-'

,\

Einstein's postulates hold for motion along geodesics in spacetime if one takes the special case of (2):

ds 2 = -dx 2

-

dy 2

-

dz 2 + c 2 dt 2 •

(3)

,..n, rHA1>TPR 7

1/'\

thp.

l'h1~ 1~

w1th

I

llnit~

tA

rr1~~P

r

r

thp.

1

.

of l10ht

WP. w111

'-'

Thp. I

of '-'

hllt IAA ~~ ....or ~lon~' ·0'

ll~

~

np.w nAt

thp.

r

'.1

.

'-'

1~

th1~

ot

1~

thp

in

-.I

ot

tAr thp nAvpl

r

t~rt

that the square of the length of curves in space-time can be positive or negative, all definitions and properties remain formally the same. In particular, positive sectional curvature means that parallel geodesics converge (the square of the distance between them decreases). (See Section 6.7.) This new distance s is often called "proper time" T, since a motionless particle (x, y, Z constant) has ds 2 = dt 2 • If we replace the symbol s by T and change to spherical coordinates, the Lorentz metric becomes dT 2

=

-dr 2

r 2 dq/ - r 2 sin 2
-

(4)

7.3. The Principle of Equivalence. Special relativity handles motion - position, velocity, acceleration - in free space. The remaining question is how to handle gravity. The Principle of Equivalence asserts that infinitesimally the physical effects of gravity are indistinguishable from those of acceleration. If you feel pressed against the tpll thp flAAr Af ~ tlnn It 1~ , '\TAll .; -.I

1~

·0

-0

• it

(

1n <> ~n1"np

0

7.4.

'';

.I.

~llC't 1"n<>lrp~ J

th<>t nf

"" in

,

thp pffpl't Af

thp

O'

-.;

O'

"11111

~~

mo~t h~~lr ~lnOlp '-'

the

~t

thp.

ot

rJ"t'- _

_

~

~ol~r

.

nOlnt m::lss th~t

WP. w111

rJr- -

~

~

+

r-'rJm-

~ln-

~

mrJl-I-\

+

p"\'J

rJf-

/

Alr\ ,. /

•

~nrl

,- J

hmp.-

Thi~

i~

....or .I.

of

th~t U!h~t i~

H

11\ /

tA hp

~rp ~nrl

IlIv of

vlr\

II '-'

torm

~

p"\'J

'0

of

.

t::l kes the ~

-r

1n

of ::I

lS the ettert on the I ~l1n

'.I •

'0'

·I·hp.

~

lilrp

J

tA

.I.

~l1rh

i~ 1ll~t

1<> fnr ,riC'''' lnnlr

npl~

Thp

.

i~

r (

..J

thp

nr

nn <>

.-

noU!

of

thp

vanishes: .

..

G'k -- g'JR jk

l'

-

-RB' 2 k -- 0 •

(2)

AT

'OPT A~TTn~,

'\0

To employ thIS assumptIon, we now compute the hmstem tensor tor the metnc U). We order the vanables r, cp, tJ, t. We compute nrst the metnc 2 2 !':in 2 {fJ (7" = _ pA = pV (7, (7..,.., = - r (7"" = - r ~

~

~

~

others vanish, 11

....

t5

-A

t5

,

-e

g

-2

22

,

-2

33

,

~1

1

1 v- A 2 V t:

ro1

144

r~':\

re •

~1 ??

1

')1\

11

ro?

,

r4

14

,

r

VV U\:;I\:; /\

. ,., re • sm
~.

,

ro"l

1 12 1 13

= cot (/).

"Y'

omers vamsn;

then the Christoffel svmbols 1

-2 ~~u

t5

1 """" -1

1 = ?v .

ro?

,

':sIll

133

others vanish.

I

,.

r

lU\:;U :'lUlU\:;

U/\f U/,

cp l,;,U::' cp,

..

Ul lU\:;

~

\,) I

......

~UlvaLULI;;

...2 H121 1 R 212

--

I

2' f\ '. 1/ 2 v \

n4

~"343

--

02 "'''4L4

~11l

2

'I-' [:;

,

-A

A

.\

....

v !:.e 2

...4 1\.141

R 3232

A

2};.,rA'e- , 1 .\

7'>1 .l\.313

1 R 414

1 -1 I /\, 2/

...3 H131

= 1 - e -A , ...2 H323

, 2

~~u

03

1 -1

"'''4.54

L

I

v

....

R~42

~1l1

2

1

/1

2 41/ ,

/1 "} \.21/ "}\.2/\

T

= ~V'( -re- A), -A,

;1

'I-' \..L

[:;

h

'f',

+ ~V,2 -

A(V"

1 " 21/

~V' A'),

v-A.

,

then some components of the Ricci curvature R 11

= rIA

J}

-

'kk

I

*

+ v' A

~ v"

-

1 ..L ! ..n-Af ~

,

~

R 33

= sin'" cp R 22 ,

J}

_

I

,,/\

I

*v'

-

n-

A

~

J

L,

,

!ov-Af,f' -l-!, /2 _ ! , / ~I -l- ') .. -1,/\

,

'"

'"

/

'"

,

R= _'Jr- 2 -l-p-A(lJ"_'Jr- 1 .\/ -!,!.\/ -l-!1./ 2 -l-'Jr- 1 ,/ -l-'Jr- 2 ). '"

'"

and hnally some components of the Einstem tensor

ni =

.

lil ..,

lii G4 .A

oijR. - ! R?i~

~

r

~

J

"

+e

.., li3 -

e

'~-r 1

1

'~-2V

= r- 2 + e- A(r- 1 )..'

~ince li4 -

v, e

v - r ".),

+ 2r _

, -1

yr

r-

1

~A

- 2r

1

~v

+ 4 V A'

1

-

4

..,

V ~),

2)

lOr some constant 'Y (just ChecK that

60

CHAPTER 7

dy/dr = 0). Consideration of a test particle with

°

velocity and large

r (see Exercise 7.1) leads to the conclusion that y = 2G M, where M

is the central mass and G is the gravitational constant. Therefore e- A = 1 - 2GMr- 1.

Since G~ = G1 = 0, A + v is constant. Since the metric should look like the Lorentz metric for r huge, we conclude that A + v = 0. Therefore e = e- A = 1 - 2GMr- 1. (4) V

°

Now G~ = G~ = automatically. We have obtained the famous Schwauschild metric dT 2 = -(1 - 2GMr- 1)-1 dr 2 - r 2(d'P 2 + sin 2 'PdfP)

+ (1 - 2GMr- 1) dt 2 •

(5)

Notice that if M = 0, the Schwarzschild metric (5) reduces to the Lorentz metric 7.2(4). Notice too that as r decreases to 1/2GA1, dT 2 blows up: shrinking the sun to a point mass has created a black hole of radius 1I2GM"!

7.5. Relativistic celestial mechanics. Now we are ready to see what differences general relativity predicts fOr Mercury's orbit. The physics is embodied in the fOur equations fOr geodesics 6.5(2) in the Schwarzschild metric 7.4(5). Four equations should let us solve for r, 'fI, fJ, and t as functions of T. Actually, instead of the first equation for geodesics involving d 2 r7dT 2 , we will use the identity dT 2 -

(1

2G Mr ') 1f'!fi)2 2

2

S

--------=-F-r---b:1si+.lnc--w-cp

r2~f-2-----------+('F'\I)~-

2

2

+ (1

2GMr

1)(;;)~-=__±1.-.- - - - -

To compute the three other geodesic equations, we proceed from 7.4(4) to compute A'

= -v' = -2GM(r 2 - 2GMr)-1,

A"

= -v" = 2GM(r 2 - 2GMr)-2(2r - 2GM),

and then, from 7.4(3), r~l = -GM(r 2 - 2GMr)-1

61

GENERAL RELATIVITY T"'1 J. 22

r~3

\J.

= (2GM 1/ 1 2 \ J.

44

T"'1 J.

,

",,)II

"")II -1,

./1

,o.,\JHJ.

r) sin 2 cp

"")lI~ -1,/"")lI~

'\""\J1Uf

r 212 -- r 313-- r- 1 n2 '+' .1 33 . ~11l

\"v~

,

-2,

'+'

r~3

= cot cp r14 = GM(r 2 -

2GMr) -1.

Hence the last three geodesIC equatIOns Lcompare to

(:).~{L)J

are

2

.'"

a cp dr 2

-l-

'Jr-1 ar acp _ drdr

c;:tn

En {'r\C;: En

(aft\ _ () \drJ

(Tn ,.

Of

.'"

aft + ?r- 1ar aft + ? ('ot m acp aft = () dr L dr dr drdr .'" LlJ1VJ. ar at a °t + L L r - 2GMr dr dr dr

(TTl)

()

(TV)

The solution of eauations I throul!h IV will I!ive Mercurv's orbit. Assuminl! that initially dCDIdr and cos CD are 0 bv (II) CD remains 7T12. Thus even relativistically the orbit remains nlanar. The other three eCll1ation~

-

~

1

~

P

LlJ1VJ.r

",I do\L

11 dr\L

~1

-~ d,..)

')

LlJ1VJ.r

-~ d,..) T ~1

r d20

Lr

d,..2 T

+ 2GM(r 2

[

dr'" T

.1..

TTT .

.., . i1Ull~

•

TT TI

i111U. V

.

1dt \L

')~ d J -

.1 dr dO d,..d,.. -

U,

1,

~1

)

~111·)

. .

.7

a

~1

_

2GMr)-1

..

ac = O. dr dr aT

(IV')

YlClU~ A~

(a constant),

r"'-=h 7

(III")

UT

At

(1 - 2GMr

1 ) -:-

UT

= f3

(a constant).

(IV")

62

CHAPTER 7

Therefore I' becomes 2

_,-2(1 - 2GM,-1) + /32h- 2 = h- 2(1- 2GMr- 1), Putting' = -1 yields _,-4(:~)

(I")

U

(dU)2 ~ 2GM( u' ~ _1_ u2 + /31 dO 2GM ror

11n. 11,

som~

u 1" 1//.( iM rJ~

01

mil":

np.

Irlu\

ann

1 n~

:'\111 ('P. Tnp. TOOTS S'

r,'\l"UC

11,

'.f\

,.

7~TM(JJ

-

' f ,.

. I

IUUI

V

1

...

r1

..L

UJ\U

r:!.ltAf •.

IU,

.... •

..L

rnp. rn1Tr

•.

....'"

.. ,1

I

. -+-l

•

\ JJ. I I

. 6.

I

\1

·1IL.

~L./J

\

U.,

V.I..., . . . . "

rJ -

..L

_~A'£\~

.1. : .. ~~ .. 1-

..1..

JJ

....L.,

UIIU

.£V".1.""

-+-

U2J

-l

~"'£\""

4:. •..

TIVI

£..\Jll'.L

'"'Ir"lt,ff •

" L~

/

\U1 V

"1"'.

1

-~...-. .1

-6./ \

\

AU ,

1m Tn I /.

I JJ.. \1 JJ -

JJ. HJJ -

\UUI

I

11,

ann

11....

2

1- I -

+ /30)

U

A(

1

....'"

+

.

p ('o~

1

.11.'

H\

" WILli

U1

.-1

t

\1Te)'U2 a=

_-1

t

\1

e),

i:1IlU 1l1t;i:11l

~ (~1 + ~J = 1 ~ e2'

For one revolution, 2'77"

~O =

I Vi I

6=0

1 1 + GMr (3 + e cos 0) r 1 elsin le(1 - cos O)i 1 e(1 + cos 0)

2'77"

=

1 1 + GMr (3 + e cos 0) dO

6=0 = 27T + 67TGM/l = 27T + 67TGM/a(1 - e2).

01 dO

n01

,--

0"-'

GENERAL RELATIVITY

The ellIpse has precessed 67TUMla(1 - e<-) radIans. The rate of preceSSIOn III terms of Mercury's perIod T IS A,.,.,.r:71.Jf /~

e

U\ .1

rn'

2,

)l.

or. back in more standard units (in which the soeed of lil:!ht c is not .1),

67TGM ~2~/1

~

~'\~

n

2 \ 'T'

~

J~

ramans.

Now ~

.1

M

I:. 1:.'7 v

1 (\-11 ..... 3/1,'" <'0,..2

~.~

~~

o~-

~

mass of sun

=

...

~ ~

.1

1. 99

=

~~ 1~",1. .. .......

<

UbU

X

'2 (\(\ v

1 (\8 ..... In,..,..

..,.~~

~"

... ~~"'I ~

r

..

~~.~

=

.... •

_~ .. ~ L . . . .""

~~ '-'L

..1. un,.".....

r

=

..

-

...

,

cr

3600".

..

.1-,..

':n

0.206,

~-J'-"

'21:.(\ 1'1 _

degree 'AU5

=

lU

5.768 x 10 m,

=

36525 days, ..,,,,,

L

..

,

QQ (\ A",,,,,

L'£~L~_~J

century

lUI'•. , . • . 1

~~~

u u '-''''''''',

e = eccentricity of Mercury's orbit £

no

10:5U kg,

a = mean distance from Mercury to sun

'T'

U~

0

. ...

, .......

..t.~ . . . 1.

u n. . . .u .....

.t.

L'-'

43.1"/century, in perfect agreement with observation.

_..

EXERCISES ,..,

•

t

1 • .1-

11

Cl MUClll •1

Ult::'

~UJJ.

it::'it::'VQll

r

.....

t::'\{'

..

lllCl~~

.

•

•

11

.Y Cll It:M Cl

1ft

" lfI u,

UICll ~

~}

,

HUlll

UH1t

tilt::'

/ At\2

+ (1 - yri){ ~)

\U'/I

(l -

,

. n

1 ~JJUW

I.J

/ A ..\2 'i) 'i{

- .

... .

t·

uu~t:

~

llUlll

-(1 - yr

I

.

W -1)

=

1

(I')

\UTI

dt = B. dT

(IV")

,.. ..

n~

rH A ........ J.l 7

t We have

not made the text s assumption that y - L
~lllci~ th::.t

,(dr\2,

11

\.1-

y'

-1\2 )

£)-2/1 f.J

\.1-

y'

-1\3 }

\atl

.

~Ince

. . ..

we are assuming tnat InitIallY anal -

.

aeauce tnat

U,

,., / A ...\

[(r 1) == ( =-) = (1 - I" I)L - (1 - y ro 1) 1(1 - I" \dtJ

If,

with ['(ro 1 ) = y(l - 1"( 1 ). Of course, classically ~m (dr/dt) 2, ,.. . r .1 ~y'Ual~

L ,.

lU~

IUllel1l

~U~I~Y,

1 Ul~

•

1 IV~~

VI

}'III

1

~U~I~Y,

:1111<1

GMm(l/r - 1/ro), so J

f(r ,.

-1

l) ,

=

I dr 2 1\ , J. I

=?ri

~4-1

./:'1\'0-1\J

'"',.. 11.,1 ~'-JJ."~

A • ...

.

1

- - -

\

\Ull

VV.lI.J..lJ

Mr.1 \'

4-1-

4-1-

..., l..l.l"-'

for large ro, conclude that 'Y = 2GM.

.

\

I

'0 1 ~

·0'

a~J

or

.

,11

LJ

,, ,,, ,,, , ,,

,,

..

I

I

1"......

,,

I

I

..

, ,,

I

I

I........ .,

• rt ... ,.....,. ............... ,...,.. ... ,i .:==:=~ .,- I-----I';-'---j~ - ...... "" .........., ,,

,

~========t,===t,===~

I"""l

I-----+-:--I,~ , Jb.e.~.~'em·· , ,,,

:

,,,

I

I

(\~~ ~~

.. 1-.~ ~~~..

1

...., ....,...,.. ...., ......,'n

1

~

J

.... tJLV

'LU'-'''

X 1

U"

'.'hl>

V

.'

'-'L ,,'-'

...:I

1

UHU

J

.& VI.

"..,1.'

.J'. ~UJ.J.U'-''-'~.

I.

.1.

LH~

• ,1 . LHU. . '!

, pt

I ' ~ •• ~~.

1/

hp

"nf"

tho

n

rlier

in

n

C:'

{T.

1 ot

tho

IL"

g

o I~

r

r I

I

I

.t<

d.t<

l7 .,.

'.r •

ThoYl

L7T.

K

I

0

ror examDIe lOr a OlSc III me Diane v T L.7T L.7T. ror me u. .rJer r a l o e unl sere L7T T II L7T. . l"'lOIice mar mis fOrmUla imnties mar IT is imrinsic as announcea . nv

l T~: I.... S

r 11'

I~

~

TnT I

.... ,v

I

I

rlH

h~<;:

thp

IK~

.11

.

~

n~

1£1

Wlln a TOT-

IT ~

~

I

T

IT

ro

~11

I UTlih

tPTrn thp

n~

IO(,~

T

OVPT iH< hv (-jTPP.n's

"

11".n

~v

rv.

'0

I

~

...

In

.,

(1 \

~<;:

rn~v

"

In

hp

x to It thp

'-

'/

~~

vv

c n A r l t K lS

.

•

, I

al I

1t -

~

~all

/

j£

/

/

I II

./

/ .." /'

I

~

R

/"

J"-.

/

11t-a~

/

/

fa

/

/

I

.-I'

.I

••

An mtenor angle a contnbutes

]figure 8.1.

a to

'TT' -

JaR Kg,

angles are treatea separatelY,

r r I G + I Kg + k( 'IT -

:R

~

,

,

ai)

= 2'IT.

(2)

.

In parucUlar, lOr a geooeslC TrIangle D,

r

I <.i +

'IT -

al + a2 + a3,

~j)

J

b. 1. a

u....

r

,1.

t't' J

..I . ' Ul"" .....

..

v

-.-

"''''lu6

•1.

.....

• • 1.'

,

J: .

..J ..... u v

,

..

..... '

...

' u LV .... ¥ v u.... ,

,..,

LU,",

"" ....1 vaLU.,", a",

G On thp nnlt

• 1.

LuaL .LV.1 a ~1.

D . " . .J.I]

"''''Ul LV

,1.

"''"' u.1a]

'"

v u u.'"'

= lim f

-.-

oIt:::.

al

al

+ a2 + a3 -

'IT

area 6

r:'

+ a2 + a3 =

thp ~rp~ A rlf 1\. 'IT

+ A,

(4)

THE GAUSS-BONNET THEOREM

\

\

• .'leUI'l;

\

"""-.. n,.. 0.......

I I

\

II \

T""'

.I·VI

() I

//

\. a

angles al + a2 + a3 'TT/2 = 'TT + 'TT12.

//

=

r

1.1

•

'-'

,UI~

;:lULU VI

UI~

'TT + A. Here 'TT/2 + 'TT/2 +

the basic formula of spherical trigonometry. For example, for the geOdeSIC tnangle 01 t'lgure ~.L WIth one vertex at the north pOle, two on the equator, and three nght angles, 7T

7T

7T

7T

-+-+-='1T+222 2 1 ~.

1'2\ ~~ 10"1'7 \'" I ...

~'-'-

•

n

• ...J

...J

8.2. The Gauss-Bonnet Theorem. The Gauss-Bonnet Theorem is a global result about a compact, 2-dimensional smooth Riemannian manifold M. It relates a geometnc quantIty, the mtegral of the Gauss curvature, to a topologIcal quantIty, the .Euler charactenstIc x. l'or ~n"

of M

ulith

V

I.

~r1~~~

l'lnrl

I-i

f~{'p<;:

v

68

CHAPTER 8 ~

,

,/

--

.-- I '-'., I I

.",

............

I

"""""'"""-

\

I I

/

,

\

. .. - ---

~

4

"",

•

I I

"

\

I I

I

.........

\

\. \. \

...

,

~•

, I

I

.. ...

~,

-".' 'r;u, '" "',J,

.

lJ'

'"'T

\ "11'"

J;'

UlIU

r:..

J

V

V

I I I

...

'T" •

..

")

o..;,Ul"'l

Tho

r--~

T>

? .......

Lt. ....

"

np

1<+ I<

eo:~Veo:

n~r

"

-.

f

.

T"' .

lIa~

1") ..J.. Q

""'''' th"t Jf r~ •J

nv

leo:

J;'..J..

I

G

(1)

= 27TX·

M

For example, consider the unit sphere, triangulated by the equator and two orthogonal great circles of longitude. (See Figure 8.3.) The Euler characteristic is

+ F = 6 - 12 + 8 = 2.

X=V - E

-

f

A_- ")_., '/\

~

( )np

of f, 1 / \ leo:

A

leo:t1r leo:

. I·

of thp.

~

th~t

~nt1

leo:

~

fOT

"

~

-

thp 1-<.111PT

II~tl0n

of

A A

.

of

{1 ~

A second remarkable consequence ot {I) IS that

~

J

.

v

,

( j IS

dent of the metric, depending only on the topology of M.

/

~

/{1 ~

Indepen-

THE GAUSS-BONNET THEOREM

On each tnangle

69

ts., the Gauss-Bonnet formula 8.1(2) becomes

J J G= -

Kg

+ Iai -

7T.

L.

Now add up the formulas for all the triangles. The first term contributes I M G. Since each edge OCCUl s twice in opposite directions, the various I ilL. Kg cancel. The angles around each vertex sum to 27T, so the angle term contributes 21Tll. The last term contributes TTF. faces, E

1

G

;F. Therefore

= 27TV - 7TF= 21T{V -

~F+ F) = 27T(V -

E + F)

= 27TX.

M

8.3. The Gauss map of a surface in R 3 • The Gauss map of a surface M in R3 is just the unit normal n: M ---+ S2. Consider such a surface as pictured in Figure 8.4, tangent to the x,y plane at the origin pI, with principal curvatures Kl along the x-axis and Kz along the y axis. For the purposes of illustration, suppose Kl < 0 and K2 > O. We want to consider the derivative Do, called the Weingarten

~~ ~1

--------I-IDft-iD

gJ~c-l-l-II.------

Tbis identity bolds in any orthonormal coordinates The Jacobian of the Gauss map equals the Gauss curvature: det Dn

= KIK2 = G.

If M is a topological sphere, n has degree 1 (covers the sphere once, algebraically), and

--------~G M

area image D

411'

21TX.

-,...

III

. I

,~

, I

..

-,~

"'

-

"' "

,• ,. I

--

I

\ .....

...

111 r

~

... ......

-.

= l

\

I

\

I

\

f--------------IJ~IJ~w~I~/' 1

/1

THE GAUSS-BONNET THEOREM Wp h~H'P J;'.......

-' thp .1

.....

""

-r

~ ... D3 , ... ],"'''' -' &&. ~-

2 ."

'-'

"/")

-,

/\

O'

;n »3

.1.

':l

-r

", ... r1

,..,

XA

I'

fnr

'1'1-

n':ll1CC_ D

... 'A'

0"

the uauSS-lSonnet Theorem tor any closed surtace m

For a hypersurface M n in

8.4. The Gauss map of a hypersurface. Dn-t-! ~-

U1~.], ~"'

-

, .}.,"" ... .],""

... ;"

r!", .."" ~"'_ .... 114 n. HA

. . •

u.~t'

..

.1

T~

ern OJ

..

•

~

K .

,,,,1

"'.

AU~'

l'

'"

........

.1'

,1'

;....

-~~.

r~·&&'

.],"" ••• -

'" .

..:I

'Q

.....

.v

U'~t'

U

Kl

= -II,

Dn= _

U

K

n_

anu me Jacooian or n is (-1<.) ••• (-1<..)

1t thp (';'!'I11":": .£.

if

(T

n is even the.J

= (- nne;

1~

1<. ' •• 1<.

::I~

of n is

-

the Euler

oPt II

A~

tor

.L

X-V-~'+F-'"

!'Inn r

G

v

= ~ areaS n 2

(1)

'

M

a generalization of Gauss-Bonnet to hvpersurfaces bv H. Hopf in 1925 rHol. (If n is odd y = 0.)

8.5. The Gauss-Bonnet-Chern Theorem. Amazingly enough, a generalization of the Gauss-Bonnet Theorem 8.4(1) holds for any evendimenSIonal smooth compact Riemannian manifold M. An extnnsiC proof was obtamed by C. H. Allendoerfer lAll j and W. Fenchel Lt'enj around lYj~, an mtnnsiC proof by ~.-~. Chern lChj m lY44. Tf N!'I..:h'..:

npn

in

T""'

.1

lQ"4

11'

~omp R n .1

Tt

,1 'Q

rNl ":!'Iu": th!'lt pupru M {'!'In hp Pnlhpn_ L'

~

''''

.'"

-A 11 whv i.. it not £'::Illpn thp n!'ll1 ...._.... ?"'n N::I~h'~ .1 W::l~ not

.

.£.

-' 1

-

until

72

CHAPTER 8

The formulation and proof require a definition of G in local coordinates. It is

R

1 G = 2nlZ 'd

n. et gii

iI i2hh

R

Rio

i3 i4l3i4'"

in-linin-linE I

0

oi

i ' °i 0

nE 1

n,

where E il "' in = ± 1, according to whether iI, ... , in is an even or odd permutation. For example, for a 2-dimensional surface tangent to the xI,xz-plane at 0 in R n , with Xl, Xz as local coordinates, det gii = 1, and 1 I"

10

L

0

0

0

\

0

LL! thp

nt thp nnhl

(T!:IIll~~

thpTp

('hprn

ll~prI

L

thp

l!:IIn

(T

!:II~

~

nf ~

HI'

tn ... 'JI 'J II

1~

-.I

A{'tll!:llllv

~

thp

I !:II

Tnnt nt thp Ulnr" nn

hhpT '.I

L

&

l f lt~

I ~pp h

!:IIlnno thp {'llrvp -0

::It ::I nOlnt on ::I {'nrvp {'::In hI'"

"hv n;.l

r

~

-r

l
,th::lt l
'\ll \I

A

L ' - /,

thp

IPI

';.I

~

::Irp, ::III "n::lr::lllpi ~

M

In !:II

'I l
It It
!:II ('llT'lp i~ !:II

It M

o

l
i f !:IInri nnhl -.I

'V l
r

•

a curve, and e IS the angle from a parallel vectorfield X to the umt tangent T, then the geodesic curvature Kg = d8/ds. If 'Y is a closed curve, the result X(1) of parallel-transporting X around the curve will be at some angle a from the starting vector X(O). (See Figure 8.5.) By the Gauss-Bonnet formula 8.1(1), 21T -

JG J =

Kg

=

J~:

=

21T - a,

so a = f G. Hence the Gaussian curvature may be interpreted as the net amount a vector turns under parallel transport around a small closed curve. More generally, in a higher-dimensional Riemannian manifold M, R iik1 may be interpreted as the amount a vector turns in the ei,erplane under parallel transport around a small closed curve in the ek,erplane.

THE GAl ISS-BONNET THEOREM

73

Figure 8.5. Geodesic curvature Kg dO/ds, '",here 0 is the angle from a parallel vectorfield X to the unit tangent T By the Gauss-Bonnet formula, the angle a from the initial X(O) to the final X(l) equals [ G. For

example, heading east along a circle of latitude in the northern unit hemisphere involves curving to the left (think of a small circle around the north pole). For latitude near the equator, this effect is small, and a parallel vectorlield ends up pomtmg slIghtly to the right, Le., at an angle a of almost 21T to the left. Sure enough, the enclosed area also is almost 21T, the area of the whole northern hemisphere.

\Ve have already seen the infinitesimal version of this interpretation of Riemannian curvature in formula 6.1 (3): ;¥"i

_

The left-hand side describes the effects on X of moving In an infinitesimal parallelogram: first in the k direction, then in the 1direction, then backwards along the path that went first in the I direction, then

74

CHAPTER 8

in the k direction. R 1kl gives the amount the j component of the original vector X contributes to the i component of the change. 8.7. A proof of Gauss-Bonnet in R3 • Ambar Sengupta has shown me a simple proof of the Gauss-Bonnet formula 8.1(1) for surfaces in R 3 • Then, of course, the Gauss-Bonnet theorem 8.2(1) follows easily as in Section 8.2. The oroof with a simole oroof of the formula for a l7eonesic rianl7le on Ie 1] n1 Sl lere l'I. 114 1

&

&

~

~

&

a

aue IO

HarnoI

I

&

..

11S[- 'V'ne

-tI I

a..,

a" - 7T -t- A

-t-

mL~.

see I LO. L

K

.

Ie

Of'

~

III

.

0

.';0 III.

&

ne

I

Tor a

I~

. ..

nv

I ~rp~

"

W

-/

-I-

I' J

J"...

v

'5

I

,-'

/

J

aR ~.

"

.L 'luau y, .1.' Uli3"- ~

1

a

au

.~-

,1.

Vll all Y

Y..... "

r

I

U

/V

'1'

r<

or<

"

\~/'

-'" ,

/V,

I

11\

-

r&~

I

-

~&

r I'

111 •

1

•

--,

L.7T.

1

/\

~

D

'J

'J'

~

""of "¥".,, ~&

_&-

&

It &

"nrf

_&&~

Qt:..

,," 'n

O'

+.,,""

0& _ _ L

,1

a

....3

111""

h" th¥".". n¥"."t

i'V.

r lUI

-

Kn

]:'",..h ..... ".¥ ""of n¥"."t r'&& ~-_&&

,1

LU~

1Y.l.

r

,~

T"" ~~

1

'0

-~Ub&

__ UL

1"n".<-

&_U_~

T

~l'

T ' ,,,'th

~

I

' &U&

angles ai' The lunes L i intersect in 6.; the lunes Li intersect in a congruent triangle 6.' on the back. The lune L i has area proportional to ai; consideration of the extreme case ai = 7T shows that area (L i ) = 2ai' Since U L i is congruent to U Li, each has area 27T. Hence

27T = area(UL i) =

L area(L i) -

2A

= 2(al + az + a3)

- 2A.

Therefore al + az + a3 = 7T + A, as desired. From piecing together geodesic triangles it follows that for any geodesic polygon on the sphere, area(R)

+ L( 7T - ai) = 27T.

To deduce (3), consider the Gauss map n: M ~ SZ. Since the Jacobian is the Gauss curvature G, n maps the region R to a region R' of area A = f R G. Let y(t) (0 < t < 1) be the curve bounding R. Let X(t) be a parallel vectorfield on y, so X(t) is a multiple of n. Along the curve

THE GAUSS-BONNET THEOREM

75

of the angles of a geodesic triangle equals 1r + A, as may be proved by viewing the triangle as the intersection of three lunes L l , each of area 2aj.

y' (t) == n° y(t) on the sphere, the unit normal is the same, so X(t), bodily moved in R 3 to the sphere, is still parallel. Let a be the angle from X(O) to X(l). Then JaR Kg - 27T and JaR' Kg - 27T both equal -a (see Figure 8.5). Therefore

JG + J R

aR

Kg -

27T = area(R') +

J

Kg -

27T = 0

aR'

by (2), proving (3). Here area(R') denotes the algebraic area of R', negative if G is negative and n reverses orientation. Similarly, JaR' Kg - 27T must be interpreted so that, for example, if G is negative, it switches sign too.

II

I

_

r----_--+-I_-----.I I

I

I I

f - - - - - - - - l - ' - - - - - - = ... _~~--__f'----------------_____j

I

I

~

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I

I

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Our streamlined approach has avoided a deep study of geodesics or even the exponential map. This chapter discusses geodesics and some theorems that draw ~lobal conclusions from local curvature hypotheses. For example, Bonnet's Theorem 9.5 obtains a bound on the diameter of M from a bound on the sectional curvature. Chee~er and Ebin provide a beautiful reference rCEl on such topics in global Riemannian geometry. Let M be a smooth Riemannian manifold. Recall that bv the theory of differential eauations. there is a uniaue geodesic through every Doint in everv direction. Assume that M is (geodesicallv) comvlete-that is l!eodesics mav be continued indefinitelv. (The geodesic may overlap itself, as the equator winds repeatedly around me spnere.) 1 nis conai1:ion means mar JV1 nas no Dounaary ana no missing poims.

9.1. The exponential map. The exponential map hXpp at a pomt p m M maps the tangent space 1 pM mto M by sendmg a vector v m 1 pM to the pomt m M a distance Iv I along the geodesIc from p m the dIrection v. (:see t'lgure Y .1. ) For example, let M be the umt CIrcle m the complex plane C,

7R

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_ .-

~ .. ~

. . . ~.--v

~ . . . .~ . .' ~~ .... .. ~ ~ ....." . " ~.;,.. .. , . n . . ,.

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,

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.;..;.----

4i~lft.C~:.j··.qZ'·':;}";1[).;1:f{f;::

L.~':?f.::;/;.~'.'

(-:-..,'.,'. ........ '. ..

..... ......

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o

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.; :'?,,::~:,:~:;:Z

..'... :. .. . .

... :.: :.~..:.'~ :.:',::/; i:~:i~~:~5~;!~\

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.~

'.

••• "

. :' :~. -', .: ~.': ;',::' ~..:':':':',::.:::::~~:~~~~;~~f~llf!,f:f

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The exponential map .bXPp maps Vt:::l .1Vl 0 me oin a lis ance v a onl! e geodesic in the direction v.

fl'lgure 9.1.

~

p

r.

,

tty)'· Im:n

1, 1 piVl

. -

Fxn, fiv) =

piy

(See FIgure Y.2.) As a second example, let M be the Lie group SO(n) of rotations of R n , represented as

'.

r

C'llot . . \ lJ 'V'

lH

'H J

A

A.

At

...J...J

T

ICL. ICLICL

A

u"'~

J. "-. .U

ICL

~ 1 ,-- ~n2 .LJ ' - . ,

The tangent space at the identity matrix I consists of all skewsymmetric matrices J •."

J,

nI -

~

A . A

-

A~

because differentIatmg the definmg relatIon AA' =1 YIelds 1A' +AI = 0, that IS, A'= -A. (See Figure 9.3.) The exponential map on T[SO(n) is given by the exponential matrix function familiar from linear algebra: ..12

A

EXPiA) = e =I+A

+ICL

+

.l! "P",r "' ... "

-'"

.1.

.. -

... ",f-AC' nf- ... -~

h", . .""

~-~

fr<.

y. \

., , ...

r

,.

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f-h""

.

.

,

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.

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, ...

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r

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GEODESICS AND GLOBAL GEOMETRY

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\

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= {iy},

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M=Sl

t'or M me cirCle, me tangent space and the exponential map is EXPI(iy)

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"U_A_

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UA~~.

tho

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,

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The Hopf-Rinow Theorem says that as long as M is connected, there is a geodesic giving the shortest path between any two points, In particular, Expp maps TpM onto M,

Q?

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Tho

£'\f ~n(YI\ Tn

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rl-

_1'- ' /

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."

1

1

t~n (1pnt~ ~ kin (1 p~{'h ~li{'p 0

0

or

,,-I

a"

.c;: UT~

£'\f

. , 1

an

nn

-'

UT~

-' th~

f"

_1 £_

h.., th~ 'J '0

UlP

rat~

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component, any curve headmg m the same dIrectIOn and any tangent

Figure 9.3.

For the special orthogonal group SO(n), the tangent space is T[SO(n) = {A: AI = -A}, and the exponential map is Exp/A) = eA.

vectorfield starting with the same tangent vector will give the same result.

J

I

J'

rI-i

, ANn GTORAT

TI-'.' II 11-'.'

I ne

nT Tne t',.

II.

TI-'.t 1M 1-'.

('\1

K Y

IS P"lVen nv

t', .

~

J

K(E i , E j ) = ~[2Ef' 2EJ - (EiEj

+ EjEi ) . (EiEj + EjEi )].

Since for matrices A . B = trace (AB l ) and (ABY = BlA l , 1

K(Ej, E j ) = :4 trace (4E i E i EjEj - EiEjEjE~ - E·EE~E~ - EE·E~E~ - EE·E~E~) , ~

-

+..",..'"

..

U~_ _

(A ,

r. ·c;c;c;

/

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T'l

•

Li

Li ~,.....,

\r1L1 '-' )

/,.....~

I

T

I.l a",~ \ '-'Llr1

"

-

·r . r. .

r·

-

-

r. . r . r .r ./ /~ ~,

\r1L1}

l.la"'l;;;

~1~I~J~J

~ ~,

/

_1

allU

r:. r: r:. r:

r: r: r: r:

'~I~I~J~J

l.la"'l;;; \Llr1},

/

...

1 c1.

\

TT

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.l.ll;;;l1"'~

1

K(E i , E j ) = :4 trace (EiEjEjEi - EiEjEiEj - EjEiEjEi 1

=:4 I,... L' .•

trace ([E i , Ej][Ei , EJll),

thp

"'.1

+ EjEiEiEj )

,... ,... ,.. .. ,... .J

r

- JJ

,..,

1

K./ t',.

t',.1 J

T

•

. III. allY

T

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•

LIt;; l!.IUUV, lUI Ul

K(v, w)

II t',. '

.. =

t'- :I r J~

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.

.111dl

V. W.

~ I[v, w]1 2 .

For example, for SO(3), take

,....

1 -"\/" a-

1 D

LJ2

0

-1

0

1 .1

1"\

1"\

li

II

II

'1"\

1"\

1

V

.L

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U

U

1

0

0

0

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0 -1

1

U

v

v

V

~

v2

1 E3 = - 0 v2 u

-J

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I

1.1

a",~

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CHAPTER 9

~ \.~

---

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p

.... ,

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p

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I J ~/

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18. J.

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VVU~l~

r,

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CJ IS Ult:: SUUUI pUIt::, UUL ~~,

•

lIIaPS Ult::

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... ~ ..- CIICIt:: 11 V1 7TJ UIllU WJ, ~"t::t:: rlgult:: ';1.'+:) r ") "), lOr me SaQOle iZ x T V tor rHmre ';1.J un me omer 1::11 Exnn is a 210bal diffeomomhism. Fxn..v F M i~ c~l1~C1 to n if Fxn.r f~il~ to Anoint n ' he a .'rr ·that i~ if the l ' at v man D Exn..v is ..1 .1' .l!lT to v M v F TM Thi~ wh~n ..1 .1 _'-1 '::1.. whpn to 7pro Mnl=M or ~

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83

GEODESICS AND GLOBAL GEOMETRY

~.~

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,,-.

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U

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r· . . .: . •. . ;...1

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. .

,. .' .. . . ,.... .... . '.

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., .

Ji'1

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=

-x 2 + y2}, Expo is a

. j.J JU03JU.

1:.

j/

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elL

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..

f;...••.. •••..•• / :i~.·\;

For the saddle {z

1.

UJ

~.:======~

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1:. J,JVJJJ

...l

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:'.

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Figure 9.5.

•

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-: .. '. '

:

L····· l,! IlII

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7

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elJ\,.o

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,

vanishing at p and q, for which the second variation of length is zero. In other words, let 'Ys(t) result from letting a finite piece of geodesic 'Yo(t) flow a distance sIJ(t) I in the direction J(t). Let L(s) = length 'Ys. Then L"(O) = O. Note. It turns out that once a geodesic passes a conjugate point, 1 H

."

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J' v ' u y.

. , ....

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1

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c," "

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Theorem. Let M be a smooth Riemannian manifold. If the sectional curvature K at every point for every section is bounded above by a constant K o, then the distance from any point to a conjugate point is at least 7T/'V'i
,

•

Ulf VI

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R4

rHAPTFR Q

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r.

.

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Figure 9.6.

Geodesics on the cylinder originating at

p stop being shortest paths at the cut point q,

point on a sphere, where infinitesimally close geodesics focus (see +h". 1'I..T",+". U&_ •

~~_

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GEODESICS AND GLOBAL GEOMETRY

/'

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Figure 9.7.

".

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points. The unit sphere with K = 1 and diameter 7T' (distance is measured on the sphere) shows that Bonnet's Theorem is sharp. We now give a proof sketch beginning with three lemmas. The first lemma relates the second variation of the length of a geodesic to sectional curvature K.

.

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ror examDle leI 'Y oe me eauawr on me unn snnere or r1l:wre 9.7. Let W be the unit uDward vectorfield. Then the circle of latitude 'Y. a distance s from 'Y has lenQth L(s) = 27T'cOS s. Hence ~

["(0)

...."'''"'

~,',~~~

"

V

1

......

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.L. 1

:1'

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= -27T = -

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sectional distance decreases). Note that by scaling, for the variational vectorfield aW of

or

au

CtiAl'TJ::.K '::I

len2:m a 1

I I -n-KII

(/1\ -

W \

I 'J \

/

J

The second lemma considers variational vectorfields of variable length. Lemma. Let y be an initial segment of the x-axis in R 2 of length L(O). Consider a smooth vertical variational vectorfield f(x)j. Then the initial second variation of length is given by L(O)

J f' (X)2 dx.

L"(O) =

(3)

0

Flowing sf(x)j produces a curve of length

Proof.

L(O)

J [1 + s2f' (X)2]

L(s) =

112

dx

0

1 10\

-

f1

.1 ~

L~

_.

.f'{ ...\ ... .1 ••• 1 A ...

C' ...

... v

J

J -- •

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UlIlerenllallon vielOs I;) J. . . 1 ne ImrQ lemma SIaIeS wnnom runner Droor me reSUlI or com- IWO lemmas I L, ;) J. oinin2: me eneCIS or lne nrSI

Ipt

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hp n hnitp niprp rlf r

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with 1Jnit (J

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r

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87

GEODESICS AND GLOBAL GEOMETRY

gonal, parallel unit vectorfield on 'Y. Take as a variational vectorfield I

\

".,.

(sin

t )W, which vanishes at the endpoints of 1'. By (4), the initial £ I , -' J

\

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of l' (extending W by parallel transport). Averaging over all such choices permits us to replace the bound on K by a bound on its average, the Ricci curvature. The theorem of Myers concludes that if Ric > (n - l)Ko, then diam M < 7T/V K o. 9.6. Constant curvature, the Sphere Theorem, and the Rauch Comparison Theorem. This section just mentions some famous results on a smooth, connected, complete Riemannian manifold M. Suppose that M is simply connected, so every loop can be shrunk to a point. Suppose the sectional curvature K is constant for all sections at all points. By scaling, we may assume K is 1, 0, or -l. If K = 1, M is the unit sphere. If K = 0, M is Euclidean space. ..

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in 1951. and generalized to ~ < K berg in 1960.

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88

CHAPTER 9

.

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.

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. Tt "!l"" , fnr PV!l

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1

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Ing: Let M 1, M 2 be complete smooth Riemannian manifolds with

.-

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..,~~

P2

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~~

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z..~11

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images in Mt, M 2 • Then length (11)

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Euclidean space, or hyperbolic space, all of which have well-known trigonometries. Thus one obtains distance estimates on the other .~

1..1

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In nature, the energy of a path or surface often depends on direction as well as length or area. The surface energy of a crystal, for example, depends radically on direction. Indeed, some directions are so much cheaper that most crystals use only a few cheap directions. (See Figure 10.1.) This chapter applies more general costs or norms to curves and presents an appropriate generalization of curvature.

convexity of is equivalent to the convexity of its unit ball {x: (x) y:

$

I}.

For any curve C, parametrized by a differentiable map [0, 1] ~ R n , define

90

n.",

AT NnRMO;;:

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/ /

,/ "//

,~

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convex, there is a linear function or I-form

10.2. Proposition.

Among all differentiable curves C from A to B, the straight line L minimizes (C) uniquely if is strictly convex.

Since the unit ball of is convex, there is a constantcoefficient differential form cp such that Proof.

cp(v)

with equality when v

<:

(v),

B - A. (See Figure 10.2.) If

=

Figure 10.1. Crystal shapes typically have finitely many flat facets corresponding to surface orientation nf InU!

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by Stokes's Theorem, so C is -minimizing. If is strictly convex, the inequality is strict unless C 1 is also a straight line from A to B, so C is uniquely minimizing.

10.3. Proposition. A nonnegative homogeneous C 2 function on R n is convex (respectively, uniformly convex) if and only if the restrictions <1>( 8) of to circles about the origin satisfy

<1>"( 8) + <1>( 8)

0

<

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10.5. The isoperimetric problem. One famous isoperimetric theorem says that among all closed curves C in R n of fixed length, the circle bounds the most area - that is, the oriented area-minimizing surface S of greatest area (see, for example, [F, 4.5.14]). In other

art:a oJ ::::: Lt.,.". ~I

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L).

Given a convex norm cP or R n, we seek a closed curve Co of fixed cost cP (Co) which bounds the most area, so any area-minimizing surface S with given boundary C satisfies

with eaualitv for C = Cn. AlmQren's methods rAIm eso. Section 91 usinQ Qeometric mea. .. . .. .. . :SUIt:

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upHrnat lSUperl ,.c;;u ..... curve t:XIM:S.

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ne

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where n is the unit normal obtained by rotating the unit tangent T 90° counterclockwise. The dual norm '1'* is defined by 'I'*(w) I ~V

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hold for optimal isoperimetric hypersurfaces in all dimensions.

10.6. Theorem. Let 'I' be a norm on R 2 . Among all curves enclosing the same area, the boundary of the unit 'I'*-ball B (Wulff shape) minimizes r.> '1'(n), R

.,

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Remark. Careful attention to the inequalities in the proof recovers the result of J. Taylor [T2] that the Wulff shape is the unique minimizer among measurable sets [BrM].

a

2

6

Fi ure 10.3.

The unit
In general dimensions, optimal isoperimetric curves are not well understood. Obvious candidates are planar Wulff shapes. The following new theorem says, however, that optimal isoperimetric curves are not generally planar.

10.7. Theorem. For some convex norms in R 3 metric curve is non lanar.

eri-

96

CHAPTER 10

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, 1

'-'

CI by two segments in the directions b o, b I , in the order that keeps the projection PC' of C' onto P out of the interior of C. (See Figure 10.5.) Then <1>(C') = <1>(C). Let S' be an area-minimizing surface bounded by C'. Then area S' > area PS' > area S. Consequently, area S' <1>(C')2

--->

area S . <1>( C)2

Remark. By approximation one obtains examples that are also smooth and elliptic. For length, optimal isoperimetric curves are circles of constant curvature. For general <1>, the generalized curvature at least satisfies an inequality.

IIiX

J-Inr n

I

-~

rlJrllP

I

I

thp v

opn_ (J

())-

m{ r

\ (1\ \~I

GENERAL NORMS

---"

(

~

.-.--

-......... ,

(:

97

- -......... --

.L

•" ,. ••

,,

,. ,. "---.. ..............

..

~

...............

---

~

.-.-- .-.--

r

Figure 19.5. Obtain C' from C by replacing Ct with segments in the directions ho, hI' Then «P(C') (q, but area 5' > area S

Remarks. For the case where is length and Co is the unit circle, (1) says IKI s; 1. The smoothness hypothesis on Co is unnecessary; still the conclusion implies that Co is Cl,l. If Co bounds a unique smooth area-minimizing surface So with n the inward normal to Co along So, D 2<1>( K) actually must be a constant multiple of n. In particular, a planar optimal isoperimetric curve has constant generalized curvature:

Proof. Let f: [0, a] ~ R n be a local arc length parameterization of Co. Consider compactly supported variations Sf. Then

o:> o( area S :> -

a( C)2)

JISfl ds + 2a<1>( Co) JD

2

<1>( K) . of ds

by 10.4. Therefore

_ _ _ _ _ _ _IDDE<2mnJ(~K01)I~<~~1~~~~(C~Q~)~============= 2a(Cu) 2 area So

l1..

norm is called crystalline if the unit -ball is polytope.

Figure 10.6. Length-minimizing networks meet in threes at equal angles of 120°.

10.9. Conjecture. If is crystalline, then an optimal isoperimetric curve is a polygon.

10.10. cI'-minimizing networks.

A network N is a finite collection

10.12. Theorem [LM, Theorem 4.4]. Let be a differentiable norm on Rn • In -minimizing networks, n + 1 segments can meet at a point, but never n + 2. It turns out that all such junctions locally can be "calibrated" and classified.

GENERAL NORMS

99

//

/~

/ [,\. / /.':".,\. / /'" ::. -::.\.

/ /': :: :::'. :\. ..... \. // ': ... \. // .: '" . /F:···.· .. : : .... : \. / /'. r /\. . . .' .:'. '::.',::' ..,,' ,\.

,,

:."

.

.

\\ 1\ \ /. \ \ /

..

"

/:\ 0L ",'~

,

, ,

'

.......

.'

~

e.... :§

1\, I, II\. ,I, '

.

\ \

'.-"

. .....i I

.:,':::' I \ E{:\;i,;,

#,;:,,,~:,.::,,,\

1E::;:,::,:;i:.::::,',

II:::.';~

:'.~:.~':":::.~:;';,.:::.~.

IV:1 1/;':.:1 11-',:-/ If)., II:;" 11.1 IU

"':'::;::::,',\':':::'\

\.',::::'::'::;;::'; V::'~:i'\::'

\::,,!:.:;,:;

\::':':::

V,,::,

II

'\ \\

Figure 10.7. Soap films meet in threes at 120 angles in an attempt to minimize area. 0

Thp npvt ~

'C

'n

-,

-0

-~

ul1th 'T

•

tn

1 (l 11

,

,...1~""

~

~

Y UfUUP, vvuuams, su 1988, [A3, A4]). Consider piecewise differentiable, uniformly convex norms

J.UIl;UIIl;UI

rTT .1"

/

\ ;:)vv

..

1-1. J.a I } . T"'

,L,.

r'"

.

.

.

~ ;,)lVl.t\.L_

rr'"

.1

.y l l v I '"-'v'"' I .'r.. Vll

\

..1'

..I'

..I

.1

, ' • 1. ~ l '

1.

..l'..l

Vl.ll.

UIU

llVl.

y.

10.14. Theorem (M. Conger, Williams '89; [Con]). Consider piecewise differentiable, uniformly convex norms

IUU

CHAPTER 10

The proof shows that six segments meeting along orthogonal

. . ~n 0 , 3 "''''0 m --",,,"0C' ... _............ --... - ....

••

,no ~

-

m ..... 'rho .........

f ........

... ...,

.

n .......... ho........ f

1

-~

0

..., ...

possible competitors requires cleverness as well as persistence in the r~~~~'

Conger conjectured that the sharp bound for the number of . in a m_ . . . . in On iC' '),., For non-uniformly convex norms, if the unit -ball is a cube ~-+~

0

In V

thp

rl

,

-0

thp

.

lC;:

~aa

-

ro_ . .

-

......

~

1

~a_

Id;nCT of thp ?rl -0

•• ,CT

....,

~

~~

~&

aaa

__

a ....

thp

-~-+~

~~

_,. r '

\J.l1th

an

pac;:l.1 'J

~

r

Thic;:

exhIbIts the upper bound (see LFLM, Intro. and 2.1J). ~vf ~ ~ It ~ "1 hv M A 1.C. (Wl111~tn~ 'Q()) ~nr1 , AL J has studIed dIrected networks ot one-way streets. ~

to

O'

/

r .L

.•. lhlu .J

rA1

L'

- - ..... - . .. -• .-.. ..... -_. -........ •

_ ...........' .- .-._-L .-

\..J~J

4

4

.-I

~

~~I.~"'''''

'-J •

• • • LA...

....:

"~_l ~

Curvature vector: 2,0(1)

= dT/ds

K

Metric of surface x(d): gii

(u 1 , u2 ,

' , ,

== Xi

give parameters on surface;

Inverse matrix

3.4(2)

' Xi

== ax/aui )

Xi

gii

Arc length of curve u(t):

f V'2

f!::li

I

ui dt

3,4(1)

J

I'

"

U

=

r

;

I

. .,

UU lULl ~

YI1J~

1-

,.,

...,.u

TT

....~

•... 1

",{}ILl' X?, Xl?

-"-" .<Wol.<Wo"

Ya\.u.l~

-l.

......

,../

? X?Xll

n.

,...

........"''''

......... ' " .....

~

W~'

/

4-_~~",

U " ' •.u

In

1J- \

,

,...

..J

\J

u~\.

TT J.J.

-r

? Xl X??

'0

'-

\.<Wol

-

,.,. )( 11

I"

.<Wo~1

n.. "'2

..,

I

X11 • 0 II X..,.., • 0 I

IX1.., •

or

(T

.2 .2 A1 A 2

/

.

\J"1

'\2 A2)

'" ""1 ,- 'j'

For the graph of a function f: R2 ~ R, H = (1

G=

+ f;)fxx - 2fxfyfxy + (1 + f;)fyy (1 + f; + f;)312

fxx!YY - f;y (1 + f; + f;f

3.5(3) 3.5(4)

102

SELECI'ED FORMULAS

2-dimensional surface x(u\ u2 ) in Rn : Mean curvature vector H

trace II

H

4.2(1)

Gauss curvature G - det II G

2 2 -X ( l ' X2 )2 XIX2

4.2(2)

where P denotes projection onto TpS.l.. For the graph of a function f: Rn - i ~ R,

div (p, q, ... ) (Exercise 5.3).

=Pi + q2 + ... , and af= div Vf= f11 + f22 + ...

For the level set {f

c} of a function f: R n

~ R,

Chnstoffel symbols: 6.0(2) Riemannian curvature tensor: 5.2(3) 6.0(4) (ajk are components of second fundamental form II in orthonormal coordinates. )

Ricci curvature: J

JI

6.0(7)

i

Scalar curvature: 6.0(8) (If S is 2-dimensional, G = RI2.)

C;:J:;T

-,

;"'.... -

&~&

.&

K(v

A W)

-,

TT A.C;:

,1.

= II(v, V)· II(w, W) = ~ gihR~1 ViWjVkW I.

wy

II(v,

5.2(1) 6.0(9)

Xi:

x~; = X~; + ~ r:kx"

.

., L <:11

U

6.0(1)

k

.

.

ii;

II

-+

'r\

7 ,.~ Jjj Jj '"

h"\l7\ - /

,

I'"

j,K

Gradient:

Vl = gijf.j Laplacian: Af ~

oij ( f .. v

,~

"J

,,,""

~&

Covariant derivative of a vectorfield

.......

... ""' ....

r~f

)

'JJ, . /

1 '\ /..J v U\o-L5

a '\

uUi

(~oijf,) V

V

~

'J

Bibliography [A]

C. Adams, D. Bergstrand, and F. Morgan. The Williams SMALL undergraduate research project. UME Trends, January, 1991.

[AI]

Manuel Alfaro. Existence of shortest directed networks in R2 • Pacific J. Math. (in press).

[A2]

Manuel Alfaro, Tessa Campbell, Josh Sher, and Andres Soto. Length-minimizing directed networks can meet in fours. SMALL Undergraduate Research Project, Williams College, 1989.

[A3]

Manuel Alfaro, Mark Conger, Kenneth Hodges, Adam Levy, Rajiv Kochar, Lisa Kuklinski, Zia Mahmood, and Karen von Haam. Segments can meet in fours in energy-minimizing networks. J. Undo Math. 22 (1990), 9-20.

[A4]

Manuel Alfaro, Mark Conger, Kenneth Hodges, Adam Levy, Rajiv Kochar, Lisa Kuklinski, Zia Mahmood, and Karen von Haam. The structure of singularities in
[All]

C. B. Allendoerfer. The Euler number of a Riemannian manifold. Amer. J. Math. 62 (1940), 243-248.

[AIm]

F. Almgren. Optimal isoperimetric inequalities. Indiana U. Math. J. 35 (1986), 451-547.

[BG]

Marcel Berger and Bernard Gostiaux. Differential Geometry: Manifolds, Curves, and Surfaces. New York: Springer-Verlag, 1988.

[Br]

John E. Brothers. The isoperimetric theorem. Research report, Australian Natl. Univ. Cent. Math. Anal., January, 1987.

[BrM]

John E. Brothers and Frank Morgan. The isoperimetric theorem for general integrands, preprint (1992).

L

•J

~-

II In

.

IChl L

tUtU Tt"\n 1) A DUV

nrnnf nf thp (';'::111<:.<:.. . lOr cioseo Klemanman mamlOlOs. Ann. aT Mam. S

-s

I::U•• r r " . •1

'"-'v'" 1

T"'

T

......

oJ.

A

(~hprn

.

r".

r.. •

11n.::"7\ A'l1 \ ...... .., I '

rr....... L

~K.,

..1r

.U'"

'U'u

.

...

n .. lJ

11\

LJUH.

~v

d.

l ... " " . ,

/4/-

IIcn

r

A

~I

J ..

I'

(1~441

4:J

·0

-J

~,

thp<:.i<:.

(

-

~ ...

.

ll).{l)

I f' UP ~

Un

~

ll).{l) ~

IlOSI

l . losta. ImerSOes mlmmaS comDietas em K oe Q:enero urn e curvatura total finita. Doctoral thesis, IMPA, Rio de Janeiro, Brazil, 1982. (See also Example of a complete minimal immersion in R3 of genus one and three embedded ends, Bull. Math. Soc. Bras. Mat. 15 (1984), 47-54.)

[CR]

R. Courant and H. Robbins. What Is Mathematics? New York: Oxford Univ. Press, 1941.

[F]

Herbert Federer. Geometric Measure Theory. New York: Springer, 1969.

[Fen]

W. Fenchel. On total curvature of Riemannian manifolds: I. J. London Math. Soc 15 (1940), 15-22.

[Fer]

P. Fermat. Oeuvres de Fermat. Paris: Gauthier-Villars, 1891.

[FLM]

Z. Fiiredi, J. C. Lagarias, and F. Morgan. Singularities of minimal surfaces and networks and related extremal problems in Minkowski space. Proc. DIMACS Series in Discrete Math. and Camp. Sci. 6 (1991), 95-106.

[GM]

Thomas Garrity and Frank Morgan. SMALL Undergraduate MathI)

.

IH::II L

M A r; rJl.

r~~

(}n IVI arn. 14

rY.J:l

1\.T . •1 • 'v'.."

l" ....

...

.

Il-lT l'

I

SIAM

..._...

~

.

T\'rr

lH.

" ' v ........... , •

,

~710.

.

r"

.......J J ...............

10'::<:: 1.... ,.+ ... +~ ..;... +\ , ...... 1"· ....1· ~~

,~-

.

•

IHOI ~.,

-,; T.

...

..

~.

., ... rI

~+~+~

..,..

A

n .. A

'J

mall'nrm

r ....

with

rH:;~~,

.•

C

·Lr:1.

~

T Y.J: .1. "' . . . . . ." .......

-..T

L:1:

C

n. uinerendal ueUmelry, Lie uruufJs anu .Jym-

~lgurUur

ffU:UtL

0

. I l'100

,

Ifit: 1

f1 , 001-,\

( -J

Npw Ynrk'

.a

Hnnlr"

In~

r In;;_ 'r

1l).{"'1

David Hottman. (he r-alded dIscoverv ot new emhedded :sUnact::s. lYlam. InleuiKencer ~ \ .l~o II, O·-":'.l. ...........

U.

n

• n,

... .....

n3

i1UU VY. ',1-

....

n.

~.

....

~

~~~.

..J

,1-

J"'\.

.

..J .

...

T

' ..... 5"........ v .." ............. " " " ......... " •

"1 110Q<::\ H\O 1"., , ... ~'I' ... ~ ......•

~ .~~

.. , r"

..... .......

,

..

BIBLIOGRAPHY

[L)

Detlef Laugwitz. Differential and Riemannian Geometry, Boston:

networks meet oni in threes. J.

Theory: A Beginner's Guide. shortest networks and

tational Geometry). ACM Press, 1991.

[Sp]

Barry Spain. Tensor Calculus. New York: Wiley, 1953.

361-365.

168

BIBLIOGRAPHY

-----lS1l

J. J. Stoker. Differential Geometry. New York: Wiley-Interscience, 1969.

[Tl]

Jean E. Taylor. Crystalline variational problems. Bull. Amer. Math. Soc. 84 (1978), 568-588. Jean E. Taylor. Unique structure of solutions to a class of nonelliptIC vanatIOnal problems. Proc. Symp. Pure Math. 27 (1975), 419427.

~

[Wu]

Steven Weinberg. Gravitation and Cosmology. New York: Wiley, 1972. G. Wulff. Zur Frage der Geschwindigneit des Wachsthums und der Auflosung der Krystullflachen. Zeitschrift fur Krystallographi und Mineralogie 34 (1901), 449-530

C'~ ,,+';~ ...... c-.

C" _ . __

"'..- ..-.

~

'l1

.'" ..-""'" .

!II

1/n

h

?n ?h 4nh ?(n -I- h\,

..

.

n~

-

+Ifl fl\ J \'

H - 15l ~ -

K

'I

,

+ ll~ -

1/n

(nr -

r IT -

. .,

1 'l,,)

.. . ~

~

..-

'-.:

.L..J .L ~ """'.... '-"'.... u

~~

1/n 2 ? / n

1/n

LV 'T: ______ _ ___

_ .-:l

~

""'" ""'" '-'"

+~

UL.lVII~

\JV

- 1/n

I

.")AI I

l-24

l1SJ'

1/n 2 -?In\, ~

")..1 2

,

1" (\(\(\

-

,

H±VH2 -4G

..,

lUU,LJU •

• 1~

.... T

U.

1 'VL\.; LUaL

~1 • 1~ .-L ,y-plaU\.; 103 UVL LU~

.1_ UI~

...

~1

....

pIau\.; aL U, 03V

H= 4v2 G = 6. Hence K=3v2. v2. e. :SwItCh vanables anCl use -1

...

-'

,

H - U,

,

'f

2

...

j.~~j,4).

2 <:~/,2 ~\2 _ -1/(1 -I- ,,2 -I- ,2\2

,

J(1 -I-

1 111 ~

)'

~2

.,....'"

.... ,

.\

1 111

...

'" J

-'

.... 2

2\ )' J'

f. Use 3.5(3 4) and implicit differentiation. H

' .j x

~

+

(Slx'"

~

10 . [ Y

+

(

jo

\ 01 ,2

1 £:. .. 2

•v .......

K

~

UZ

+ 16y'" + z"'y''''

, ~

'-.:

""'" U

:l~U,

1~,,) . 1112

.:II

~ _

..-

...v)'

'" H±YH2 -4G

.....,

\

.2/

,

...

U03~

~/ ...

1'\

.J • .J\.J,.,~).

-------tiO

SOLOTIONS TO SELECIED EXERCISES

3.2. 21T a sin c. (It is a circle of radius a sin c.) (-fx, -fy, 1)

3.3. x

(x, y, f(x, y)), D

H

3.4.

VI + f~ + f;'

(1 + f;)(1 + f;) - f;f;

G

fxxfyy - f;y

Xz

(I' cos 6,t' sin 6, 1),

D

,

1

(/sin 6,/cos 6, 0),

X6

= -(cos 6, sin 6, -/,)(1 + /'2)-112 ( [ N' COS,[ (J N' SII1" • (J

(teas

X66

(J,

[sin

6)

(J,

(inward),

( [r, SII1,[ . (J cos" (J 6) rt

XZ 6

6),

rta

f

3.5. A =

21Ta sin cp a dcp

= 21Ta2(1 -

cos:.) a

= 21T,2 - 1:-...!. ,4 + ... 2 a 12

cp=o

4.1.

(x, y, x 2 + 2l, 66x 2

8. X

(1, 0, 0, 0),

Xl Xl2

(0, 1, 0, 0), xn

X2

= (0,0,0, -24),

By 4.2, II

24xy + 59l),

X22

(0, 0, 2, 132),

= (0,0,4, 118).

(0, 0, 6, 250), G

?xij

= Xij'

15,008.

Note that answers are sums of answers to Ex. 3.1(b, c).

b.

x

= (x, y, x 2 - l, 2xy),

Xu

= (0,0,2,0),

X12

Xl

= (1,0, 2x, 2y),

= (0,0,0,2),

6

By 4.2, H

p

Since

are orthogonal,

P(Xll)

Xl> X2

something

Xll

_ (-4x, 4y, 2, 0) - 1 + 4x 2 + 4l'

X22

X2

= (0, 1, -2y, 2x),

= (0,0, -2,0).

6. FOI G, we need Pxij •

111

.:lVLU I LV..'1,

° 2)

(-4v -4x

r~X12)

1

A~2

...L

.....

4.2.

.0/1

A

V,~

~

...

(4x -4v, -2

A 2 ' r~X22)

...L

1

•.1

~

I'

..L..L..L

IV

2

~

2\-3

A

...L

V,~

= (z,f(z)), with z = U1 + iU2' x, = (l f'(7)) x., = (i if'(7)) x"

...L

Q)

A 2 •.1

AI.12\-3

0/1

T.l I

A~2 "

'1"'1 I

x

(U,!,,(z»,

Xu -

P(O)

II

l!"(Z»,

Xl2 - (U,

~~

x.,

c~_ r" • .,~ _~~rI

~.

~

..... ,

something

~

(X.,

X22 - (U,

n

~~

°

=

= ix,)

-r(z».

D.,

.~--- ~

~/r

Since x, x" soan a comolex subsoace P iust oroiects onto the orthol!onal comolex subsoace soanned bv (-777.), 1) E C2 . (-£if" f") P(xu)

'P(X22)

I J:' 12\ '

11 ,~

=(-

P(x,,)

if'f", if")

,~

+ IJT

1

.,.,1 +"12{1

r"-

-- ..,. AI

']I

~

..

'II

H

I I

.

11"1 2\-3

.J.

I ,

II

'.1.

"La>

~

II

I f

~

•• 1.J.

A "/1 \

..... '~I J ..".... ~

= Pv(l + If.1 2 + If.,I2)-l,

wh~r~

v = (1 + I/ynlxx - 2(fx' fy)fxy + (1 + Ifxl")fyy E Rn

.

~.

J

-2

.~

H

hand if H

~.l.

~.

~

u,

"

=0

..

ww

~

U

\

VI.

r.., ..... ""'a>'" V,

~41

.,.

. .. 2

(0,0)

~L,

~L,

~U,

.t:::\

,-, VI

L)

(0,2)

I"

"\

,-,-,

= (0,4) . (10,2) -

-

L)

I"

f\\

I"

f\\

,-,

~I

,-,

~I

11

d-' 1"

1'\ .I., f\\ VI "'" • II(U, W) 1

\~,

II(U,

W)

(0, - V2) . (0, - V2)

If\

n

{Xl

1 \

\V, V, .1./'

= 6. 11

One orthonormal basis for

..

".....

,

.

• \...IU lUt; UlUt;1

(10,2)

.1 1. • • v ... ~.~ .~

K = II(U, U) • II(W, W)

.

= f01.

(2,0)

(0,0) 1"1

1::-

(2,6) U)

..

~

c1UU lUt;

v E ker P n Rn -

I ~y21·v

~~,~l

-2.

UU;;U I I

..... "v .., I

8. 11

I-

C R2 X Rn

1

n\

+ X2 + X3 = o} is U = \.1., ~' v I ,

1 1 "l

..L..L,,",

J.

~VLU

J. V

J.Vl'l~

/1

1

\~,

~,

"

...

"-)

V·

"V.

n

r

W

VO

d. R 1212 = 12, R 1213 = 12, R 1223 = 0, R 1313 = 32, R 1323 = 20, R 2323 = 20. Rest by symmetries. Redoing b,

and e, we have the following:

C,

b. R 1212 = 12. r-

II. -

1'.

-II~ -

1

A

:;f(nt.,.

VI

'v2

V ~~

,'\ I,.. v",-, v3

1

-r

o.

:;f(·n.,.,. 11'\

" wI v,

~I

w2

~I

v

.... ,'\ IE -""' v v.

IE

v, w3

~D

~D

~D

--'-D

~D

o~ 'l~Lj

o~'LH~

.j~·LH.j

l~~ 'l~~l

o~ ·l.j~l

1

1

1 12

1

1 12

-

R 2112 + 6R 2113

"3 R 2313 +

-

R 2121 + "3 R2323

2 - 2 + 32/3 + 1 - 2 - 20/3 + 1 - 2 - 20/3 + 1 + 20/3

44

20

0

"fl

~"

1"

1..,

C..,

~-

...

" ~

R - .1.1

-

U

",y

U

I

U

I

~v""

()

()

I dT dT dT I II I rlY rlll rl.,. I

0,1)

,~

1"" '1""1""

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+

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+

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= !_Q:

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,. ~~a~u

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r •

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1"

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t:<' .. 1""...

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~

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11.

rk

k tim~~

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v

-

.,.

. . IIli1P,

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~ii

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a~u.l'-',

U.l,",au ,",U.l ..

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.,

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seCnOnal curvawre. "'''' ,",U.l

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I I

.,

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--

Gauss curvature, 1, 13, 26, 71, 72 bmstem tensor, )lS Riemannian metric, 18 Christoffel symbols, 40 mean curvature vector 26. 27 32

TT .l~

~.J

),l

.cxpp

lij

.• ,"" 11 R9

"1'1'

1.., ,

U.l~~

riC'-

G

,

'u.~U.l,",

, 'l1

'+~

c

oJ

scalar curvature IK I, 1;

K

'.{

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&

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unit normal, 5, 11, 32;

\,;Ul V4LUIC VC\,;LUl,

,...

.luap,

norm,

.

''{L!.

,-

0'

~

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&

Kg

q>

£(\ V7,

'+ I

'"71 f

~

ts~

,..,

,-

n

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Rjkl

Riemannian curvature tensor, 34, 41

R.

RiC'C'i

J

RiC' ~'i .:11

SO(n)

ortho~onal ~roup,

-

.. n~+

T

..... .£ ...

_&&&'

.....

,,

78, 79

11 ........ L1.. 7

C'Vlt.6nAT .., ...

"" .,.

.

lime,

c"(~pr:1

T<'

It -

p

"T"

""

nl

l.pJ

V

v /\ w

v

AA

11(111111'1

.

-c

-,..

,:)0

- --

, 11 ,...~

"11'11

-r

,

~J

velocity, 5 plane spanned by v and w, 33 . •

y 1.-

II I" I I

•

I( . .

"1

c;

I

..

. .

'\

1'\

i

1,.,

Ai

palLla.l

X

time denvatlve axlat, 1"1 components of vectorfield, 40

Xi

¥~.,/

Ll I V ,.,

.

U~llVa.L1V~

I,., I I I

-I ,I

.

.

UA/UU,

; I I V ,., I I V l:

40

J./

--Name

IfttIe~x---

Alfaro, M., 100 Allendoerfer, C. B., 71 Almgren, F., v, 93

Kepler, J., 55 Klingenberg, W., 87 Knothe, H., 94

Berger, M., 87 Bianchi, L., 35 Bonnet, P. 0., 67 Bredt, J., iii, viii Brieskorn, E , 91 Brothers, J. E., 94

Laugwitz, D., viii Levy, A., 98 Lorentz, G., 57

Cheeger, J., 77 Chem, S.-S., 71, 72 Cockayne, E. J., 100 Collins, c., v Conger, M. A., 100

Mattnck, A. , v Meusmer, J., 15 Michelson, A., 57 Morgan, B., v MOIgan, F., v Morley, E. W., 57 , Murdoch, T., viii Myers, S., 87

Ebin, D G., 77 Einstein, A , 56, 57, 58 Enneper, A., 16 Euclid, 50 Euler, L., 13, 15

Nash, J., 39, 71 Newton, I , 55

Penchel, V'I., 71

Rauch, II., 87 Riker, M., v Robb, D., iv

Gauss, C. F., 2, 21, 67 Gromov, M., 94 Harriot, T , 74 Hicks, N. J., viii Hildebrandt, S., 91 Hopf, H., 71 Jeffery, G. B., 56

Peters, K., viii Playfair, J., 50

Scherk, H , 15 Schoen, R , 42 Schwarzschild, K., 58 Selemeyer, C., v SengupLa, A., 74 Siegel, P., viii Smale, S., 91

---lt6

NAME INDEX

Spain, B., 56 Spivak, M., viii Stoker, J. J., viii

Vu, P., 56

Taylor, J. E., 94 Tromba, A., 91

Wasserman, I., 56 Weinberg, S., 56

Underwood, A., viii

... • ...... ... I nflPV ....., ~ '-'J "'" "'" '" ••• ~

.'11 nlPf'T

.

A _~ 1~~~+h

, 10

~

1 f\1

~~,

~.-

~""'~1

~_.

Bending, 22 .'."

... ".

,..,')C.... ,

Co

"

~..

~

..

'H~"

'"

,~

~ nf l'nTVP '"

47 101

,~

,

77

,

Curves, 5' see also curvature of curves, geodesics

S2A

~Ul vaLUl~

Calibration, 99 Catenoid, 15, 20 f""1...,~n

.... 1~

, 17

.~

r'

.

.•..• r

i

All

I""

,

complete, II conjugate pomts, I:SL ana seCtIOnal curvature, ~,j and shortest l!eodesics 83 Connection 41 Contravariant, 40 Convex norm, 89, 92 1 A

"

f""....... +..,'..

_.

v

, 17

~

l

Gauss curvature mean curvature, principal ~Ul ~.

,

.

~~alal

.....

~1~~1 ~Ul

i J'

co

~~

v

5~

hI

Fermat problem, 98

,

"'Ul

'aLUl~,

~Ul vaLUl~,

curvature

~

-

1 5' see also

UI Vi:l'

n

13 '. Euler's Exnonential man Exn~ 77 82 and sectional curvature 83 ' ~

Xy

.

~.

.....11IPT

Covariant, 40 Covariant derivative, 36, 42, 103 along curve, 41:S (

,_...

Ellipsoid, 23 Enneper's surface, 16 eqUivalence pnnciple, 56,

")n

'70 ~

'"

Ulameter, I I Jjonnet·s meorem, ~4 Mvers theorem 87

, •. , ..,J ,

,

,

-

K, J

"""" ""'" "",.., :7.:1, :7.:1, :71

tangent vector T, 5 Cut points, 83 Cylinder, 22, 84

1 ll,)

nf 0",,,,,1.

• RR

f"'.

V~~LVI

' ......VI II, .:1

'.T ••

r.

_v

_ _A

rl

A1

Dl ... ~\' h~l~ ~f\ __ ,

....

~4' ~~

generalized D~4>(K), 93, 97 geodesic Kg, 47 radIUs of, 5

Area, and Gauss curvature, 22

Too'

_._._

~~~

.

Pir"t

.

<'.

,..,

Too

1R

• fnnn

r

'-Jau~~'

proof in R3, 74

.

/'"

,

UJ

118

SUBJECT INDEX

Gauss-Bonnet theorem, 65, 6769,71 proof in R 3 , 74 Gauss-Bonnet-Chern theorem, 71 Gauss curvature G, 1, 13, 19, 26, 27,42, 101, 102, 103 and area, 22 constant, 50, 87 of geodesic triangle, 66 of hypersurface, 71 in local coordinates, 72 and parallel transport, 72 of projections, 26 ~~ . . n n _~ _ _ ~~~~~

u'~Y

,t;o

aa,

'71

,

~

Laplacian, 103 Length, 15, 18, 101 Lunes, 74, 75 Mean curvature H, 1, 13, 19,32, 101, 102 Mean curvature vector H, 26, 27, 32,102 Mercury, 55, 60-62 Metric ds 2 = giiduidui, 18, 57, 101 I to first order, 21, 42, 78 Lorentz, 57, 60 -"x

~

n!:ll1~~'~

-0

?

·0

fI"

71

Jacobi field, 82

IVllmma su aces ,

-

~

,

,

'

'6'" ..'-'.. r

'\1

,

11 1

.,.

'C

.. "'~.

'~'~"""

"

'J'

..

4':1

,/v

......

.....r

TO

JU

curve

:J

J

~

, ....

00

1 {\{\

O~

..

...

''-'.L'''~,

(

~~~

U

OSl

hT~t

lS4

..

~

,

"

. curve, 5

4

,

\.}~

':1':1 r.~

~r

,77

.L, JU

in'] ,

nt

Nlannanan

)l,lVU4J, I I

, .:Ji:,

11

LV

'J

--

"n

~~

~

-J

...

LJ'

U

LV 11

~ ...

no'

1.1

n

~r. ~~4~t;,

-

.. vv

",

ax

, ".,. , "II

oounaea

~~ , .:J'7,

inn

7Sl

lUL.

(

~

no

,

.. ' ' - ' U

l'Inl1

,

,

.....r

'\Sl Sl'\

IOrmUla lOr

.

l'li:tMJ

A'" ...... .c.

. .

,.

, 00, /'+

,

./

~' 01

I

lJeneral reIarivltv :>:> -o,j

,...

fill

L'+'

.~

r .;... .,nh

surface, 18, 19, 102 Gravity, 56, 58 Helicoid, 15, 23 Helix, 8 Hopf-Rinow theorem, 48, 79 Hyperbolic geometry, 49, 87 Hypersurfaces, 32, 33 Injectivity radius, 84 Intrinsic, 2, 18, 21, 22, 39-54 Isoperimetric problem, 93-98

.......

,

...........

j:1.JIVU," .:JV\n"

_~

_r.

7

Osculating circle, 5 Parallel transport, 72, 74 and second variation, 85, 86 Precession, 55 Pressure, 14 Principal curvatures, 11, 13, 19, 26,34 Principal directions, 13 Projection, 44 Proper time T, 58 Rauch comparison theorem, 88

... n.'

':Ie:

....~n..."'. "'....

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n .. ],-,J.~

__....

1/\':1

C'

"'VoJ

CI

uvap

VI

....

,1':1 ,

....

,

O.'~~."n 1 ~'"-_

T'II'

c ... ,

0,., ,

0 ' ~~."n ' • ~,.

,11

oJoJ,

._&.."

e'.

.. I.

'oJ

...~

':1,1

..... '"

,oJ

,

roo.f

C'

partials, 43 and parallel transport, 72-73 D

.•.

, A" , 1"

'"

';1.7

HI

"':1

.LV,

.... oJ,

,.,e:

e:"'7 oJ

e:':I

£"'7

,oJoJ,

0"

'oJ,

oJ,

e: e: oJoJ'

,1"'7

V

,

£0

£/\

VV,

V."

0"'7

v .... ,

V

,

V

0"'7

..

'J'

~"'7 ~

, "'1

.£.

:... DJ .u _, , 11 in Dn ,,1:\

,

,

0..1

.• fn.,..... TT

£-

1 0

~~

-oJ -' -'

, "~J'

roof C

~-

~

'

._......

1/\ · ... v

Steiner problem, 98 Stokes's theorem, 92 Strake, 6-10

, '1~ , A" , 1 fl'1

•

CI

....... ...,y..

~.

_&&-

C'

/7

"'J.'-'

...... .-:1 '

c:

nn

"'':1

,111/\" .... , ... v ....

",n.-:l

~1.

""y"_"_,

,oJ

£ v

,

J.J.. . . . ~,

';C:L

,1':1

"'.....

'

.. . .

,..

ll~

SUBJECT INDEX

, 1" ,

, ';1..11

..,,,

._, --,

J

31,79 SectIOnal curvature, 33, 42, 77,

~

, '1..1 , ..11

'oJ

- ..... Tangent vector T to curve, 5 'T

<''"''''',..,.

"'r

~

11

-p~'

.~

Torsion, 41 'n.,11<'. , ..

If I'''{

}{4 }{~ -~

0

~fl

..I'

ot

R7

nf I'11M,-=>

RLi

. .

J,l

111

.

m~n R~

of

T~

V

h. 7

L1R., O'::l. RI:\

nf I'11M,-=>

..J .£ .

11 ?f\ 1? ~nti ~("~I~T

47

J

'h

.

XX

sphere theorem, '61 weIghted average ot aXIs

wemgarten map Un, Wultt Shape, ~4

.~.~

:"I. 11 l'P

Rh.., R7

Y

•

4/

b~,

'11