A Geometrical Determination of the Canonical Quadric of Wilczynski

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A GEOMETRICAL DETERMINATION OF THE CANONICAL QUADRIC OF WILCZYNSKI By E. B. STOUFFER DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KANSAS

Communicated February 3, 1932

In the study of the projective differential properties of curved surfaces a canonical development for the equation of the surface and a geometrical determination of the associated tetrahedron of reference are of fundamental importance. Wilczynski' was the first to solve this problem. The vertices of his canonical tetrahedron can be easily located as soon as a certain quadric surface Q upon which they lie is determined. Green2 and others3 have obtained similar canonical developments which take their simplest forms when the vertices of the associated tetrahedrons lie upon Q. The quadric Q is commonly called the canonical quadric of Wilczynski. It was located by Wilczynski by means of a unodal cubic surface osculating the curved surface. The introduction of this canonical cubic surface complicates the situation considerably, especially since its determination before Q is known is a rather involved process. Recently Bompiani4 obtained Q-apparently without recognizing it-by a geometrical process somewhat simpler than that of Wilczynski but little related to the problem of determining a canonical tetrahedron. In the present paper the canonical quadric of Wilczynski is defined geometrically in an exceedingly simple manner by means of the axis of Cech, a line which is covariantly related to the curved surface and which is easily located geometrically. Since most of the preliminary facts used in this paper are found in the memoir by Green,2 we shall use his notation in order to facilitate reference. Accordingly, we refer our curved surface S to its asymptotic net and take the associated system of differential equations in the form Yuu + 2byv + fy = 0, Yvv + 2a'Yu + gy = 0.

Let the two asymptotic curves passing through a point y of the surface be denoted by Cu and Cv. The tangents at points of C, to the curves v = const. generate a non-developable ruled surface R(u) and the tangents at points of Cu to the curves u = const. generate a similar surface R(v). The points

P Yu= -Y, -y where

a

= yv- ay,

(2)

and ,B are functions of u and v, lie on the tangents to C. and Cv,

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respectively, and the line 1 determined by them lies in the plane tangent to S at y. The tangent planes to R(u) at p and to R(v) at a intersect in a line 1' which passes through y and the point Yuv - aYu - fly.. Green calls either of the lines 1, 1' the reciprocal of the other. If ,B = au/2a', a = b./2b, the lines I and 1' are the directrices of Wilczynski of the first and second kind, respectively, and if = -b/4b, bu a = - a'/4a', they are the canonical edges of Green of the first and second kind, respectively. These two pairs of lines have been located geometrically in several different ways. For example, in the former case the point p is the harmonic conjugate5 of y with respect to the points where the flecnode curves of R(u) intersect yp, and in the latter case the point p is the pole of the tangent at y to C, with respect to the osculating conic6 of Cu at y. The corresponding points a may be located in a similar manner. The directrix of the first kind of Wilczynski and the canonical edge of the first kind of Green intersect in a point called the canonical point and their reciprocals lie in a plane called the canonical plane. The intersection of the canonical plane and the tangent plane to S at y is called the canonical line. The harmonic conjugate of the directrix of the second kind of Wilczynski with respect to the canonical line and the canonical edge of the second kind of Green is the projective normal of Fubini, the pseudo-normal of Green. For it fi = - 1/2(b./b + a'/a'), a = - '/2(a'/a' + bv/b). Again, the harmonic conjugate of the projective normal with respect to the directrix of Wilczynski and the canonical edge of Green, both of the second kind, is the axis of Cech. It is given by ,B = - '/6(bu/b -a'/a'), a = - '/6(a'/a' - bv/b). Both the projective normal and the axis of Cech have' other geometrical definitions. Green obtained a general expression representing a group of canonical developments, including that obtained by Wilczynski. by choosing as the vertices of the canonical tetrahedron the four points

(3) -ay, r= Yu- ayu -yD +acy, where a and f, are functions of u and v which need only be properly assigned in order to obtain any one of the several developments. Since any point X in space is defined by an expression of the form xly + X2p + X30 + X4T, the coordinates of X may be taken to be (x1, X2, X3, X4). In this new coordinate system the general expression for the several canonical developY, p= Yu

-

fy

= Yv

ments has the form

t7n + 2/3bt3 + 2/3a'n3 + 1/6(4bg + bu)t4 + 2/3(bv - 2ba)t3 ( + 1/6(4a'aI + a) + + 2/3(a'- 2a'3)'Z3 where t = x2/xl, v = x3/xl,b = x4/xl. Cutting off the development after v

the first term we have

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(5) O, which is the equation of the canonical quadric Q of Wilczynski. Equation (5) is independent of a and ,3 and the quadric is the same for all developments (4). The lines yp, yoa, pT, CT lie on the quadric. For given functions a and ,B the points p and a generate two surfaces Sp and S. The tangent at p to the curve u = const. on S, and the tangent at a to the curve v = const. on So intersect the corresponding line 1' in the points Pv - ap = T- vy and au - Oa = r - auy, respectively. The harmonic conjugate of these two points with respect to y is the point r -1/2(au + iv)y. This point is r and lies on Q if a and a3 are chosen to satisfy the condition XIX4

- X2X3 =

au +

v

(6)

=0.

But this condition is satisfied if 1' is the axis of Cech. THEOREM. Let 1' denote the axis of Cech associated with a point y of a curved surface and let p and a denote the points where its reciprocal intersects the tangents at y to the asymptotic curves Cu and Cv, respectively. The tangent at p to the curve generated by p for u = const. and the tangent at a to the curve generated by a for v = const. intersect 1' in two points such that the harmonic conjugate of y uith respect to them is a point r on the canonical quadric of Wilczynski. The lines pr and fIT thus geometrically characterized lie on the

quadric. Equation (4) shows that the equation of any non-singular quadric surface which has contact of the second order with S at y is of the form

x1x4 - x2x3 + Ax2x4 + Bx3x4 + The lines xi = x2 = Oand xl = C =

=

Cx4

= 0.

X3 = O lie on this quadric only if A = B

0.

THEOREM. The canonical quadric of Wilczynski at a point y of a curved surface S is the unique quadric which has second order contact with S at y and which contains the lines pT and ar associated with the axis of Cech. With Q thus located, any geometrically characterized pair of reciprocal lines determines by their intersections with Q a tetrahedron and a corresponding canonical development. For example, the directrices of Wilczynski give the development used by Wilczynski, the canonical edges of Green give that used by Green, and the projective normal and its reciprocal, or the axis of Cech and its reciprocal, give similar developments. The osculating cubic surface which Wilczynski found it necessary to introduce in order to locate Q is easily determined with Q already known. It is, in fact, the cubic surface for which (0, 0, 0, 1) is a unode and xi = 0 the uniplane and which has third order contact with S at y. This cubic surface, which is of course different for each canonical development,

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gives an easy method of characterizing the unit point of the coordinate system. If we impose the conditions, for example, that the coefficients of the first, second and third degree terms in (4) shall all be unity, then the unit point lies on Q and its projection from (0, 0, 0, 1) upon X4 = 0 lies on a tangent of Segre. This point, the two points of intersection of the tangent with the above cubic surface, and the point of intersection of the tangent and the line xi = X4= 0 have a cross ratio equal to -2. The three choices of the tangent of Segre correspond to the cube roots of unity which appear when the above conditions on the coefficients are imposed. 1

Wilczynski, Trans. Amer. Math. Soc., 9, 79-120 (1908). Green, Ibid., 20, 79-153 (1919). 3 Stouffer and Lane, Bull. Amer. Math. Soc., 34, 460 (1928). 4 Bompiani, Rend. Acc. Lincei, [6] 6, 187-190 (1927), and Math. Zeitschrift, 29, 678-683 (1929). 'Fubini e Cech, Geometria Proiettiva Differenziale, 1, 148 (1926). 6 Stouffer, Bull. Amer. Math. Soc., 34, 301 (1928). 2

DYNAMICAL SYSTEMS OF CONTINUOUS SPECTRA By B. 0. KOOPMAN AND J. v. NEUMANN DEPARTMENTS OF MATHEMATICS, COLUMBIA UNIVERSITY AND PRINCETON UNIVERSITY

Communicated January 21, 1932

1. In a recent paper by B. 0. Koopman,I classical Hamiltonian mechanics is considered in connection with certain self-adjoint and unitary operators in Hilbert space t (= 22). The corresponding canonical resolution of the identity E (X), or "spectrum of the dynamical system," is introduced, together with the conception of the spectrum revealing in its structure the mechanical properties of the system.2 In general, E(X) will consist of a discontinuous part (the "point spectrum") and of a continuous part. The case of a pure point spectrum, and the other extreme, that in which the inner product (E(X)f, g) is, for every f and g in ID, the Lebesgue integral of one of its derivatives, may readily be treated by known analytical tools.3 The present paper is devoted to the case where E(X) is continuous (X # 0), but without (E(X)f, g) being necessarily equal to the integral of its derivative. It will further be assumed that the system is nonintegrable in the sense that any f in t such that U1f = f almost everywhere on Q must be almost everywhere constant. In other words, we are assuming the following hypothesis: C. E(X + 0) - E(X -0) = 0, for X # 0: If Eo = E(+ 0) -E(- 0), then Eof = almost everywhere constant.