Separable Quadratic Differential Forms and Einstein Solutions

VOL. 11, 1925

MA THEMA TICS: E. KASNER

95

some sort. In particular, equations of the type (3), such as the equation of demand (7), may in part be replaced by equations of the type (4). 1 Irving Fisher, The Purchasing Power of Money, New York (1922). 'The corresponding problem for more than one compartment, in the case of competition, is discussed by C. F. Roos, in an article not yet published. Mr. Roos incidentally treats a hysteresis problem in which the functional character of (3') is preserved. ' Amer. Math. Monthly, 31 (1924) pp. 77-83. Compare the fundamental hypothesis with the statistical study of Irving Fisher, The business cycle largely a dance of the dooUar, Quar. Pub. Amer. Statis. Assoc., December (1923). 4 Cournot, Rcherches sur les principles mathitnatiques de la thAorie des richesses, Paris, 1838. See also G. C. Evans, A Simple Theory of Competition, Amer. Math. Monthly 29, (1922), pp. 371-380.

SEPARABLE QUADRATIC DIFFERENTIAL FORMS AND EINSTEIN SOLUTIONS By EDWARD KASNER DEPARTMENT OF MATHIMATIcS, CoLuBLu UNIvERsITY CommUniCated November 11, 1924

A quadratic differential form in n variables we shall call separable or summable if it can be reduced by any transformation of the variables to the sum of two forms, one involving only h of the variables, the other involving the remaining k variables, where n = h + k. Such a form we shall call separable of type (h, k). We may of course have separable forms of type (hi, h2,. . . h7) where n = hl + h2 +... + hr. This means that the form can be reduced to the sum of r forms involving completely independent variables. The various types for a given n are not necessarily mutually exclusive. There are possibilities of certain special forms belonging to more than one type. If the form is euclidean it is of type (1, 1, . . , 1) and vice versa this type is obviously euclidean. This is the extreme case of separability, the n-ary form being then transformable into the sum of n independent unary forms. If n = 4, the possible types are

(a) (b) (c)

(2, 2) (3, 1)

(2, 1, 1) (1,1,1, 1) (d) We have examined the question: When can an Einstein manifold be one

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MA THEMA TICS: E. KASNER

PRoc. N. A. S.

of these separable types? For (a) we have shown' that there is one and only one solution, namely:

ds2=-

dX2 +dX2 dx23 +dX2 +4 1

x2

(1)

x2

For (b) we find that no solution exists, the proof being quite easy. For (c) no solutions exist except of course those which are of the trivial euclidean type (d). Hence we have the theorem: If an Einstein manifold is to be separable it must be either cuclidean or else equivalent to (1). This latter form defines an algebraic2manifold, namely a quartic spread offour dimensions imbedded in a flat space of six dimensions; the finite equations in-stx cartesian co6rdinates being (2) X1 + X2+ X -= 1, X + X2 + X 1. This is the intersection of two spherical hypercylinders. The separability so far defined has been complete. There is another more general and more difficult theory of incomplete or dependent separability For example a form in four variables xl, X2, X3, x4, may in certain cases even though not separable in the first sense, be reducible to the sum of three binary forms

-Q'+ Q"+Q"'

(3)

where Q' involves only xi, x2, Q" only xi, x3, Q"' only xi, X4. A special important case of (3) is ds2 = a(x)dx2 + j#(xl)dx2 + _y (xl)dx2 + 5(xl)dx2 (4) Here Q' - 1adxl + (3dx2, Q" = s adxI+' + ddx, Q"' = j adx2 + adX4. Einstein solutions of the above type actually exist and have all been determined elsewhere.3 They may be represented as curved spaces of four dimensions immersed in flat space of 7 dimensions. The finite equations represent three surfaces of rotation with a common axis. Every form in n variables is of course separable if we allow a sufficient number of forms in a smaller number of variables, provided the variables in the various forms added are not required to be independent. For example we may decompose the given form into the sum of not more than jn(n + 1) unary forms. We shall examine elsewhere the maximum number of binary forms that are requisite, and other generalizations. 1 Science, 54, 1921 (304-305); Geometrical theorems on Einstein's cosmological equations. Amer. J. Math., 43, 1921 (217-221); Math. Ann., 85, 1922 (227-236); also an article in press "An algebraic solution of the Einstein equations," Trans. Amer. Math. Soc., 27 (1925). 2 This means algebraic in finite form, not merely that the potentials gik are algebraic functions. ' Same references; also Bull. Amer. Math. Soc., 27, 1920 (62); and a detailed discussion in a forthcoming paper "Solutions of the Einstein equations involving only one variable," Trans. Amer. Math. Soc., 27 (1925).