Multivectorial Curvature

VOL. 17, 1931

MA THEMA TICS: M. S. KNEBELMAN

43

correspond to their inverses. Hence such a group admits k + 1 (k + 1)/2k automorphisms, where k is an arbitrary natural number. These automorphisms are distinct whenever k > 1. For k = 2, 3 and 4 there result groups which admit three-fourths, two-thirds and five-eighths automorphisms respectively. The group of inner isomorphisms of all of these groups is the dihedral group of order 2k. When k = 2 or 3 it has been proved that all the groups which admit (k + 1)/2k automorphisms have such a group of inner isomorphisms but this is not necessarily true when k = 4, as was proved above. In fact, in this case the group of inner isomorphisms may also be abelian and of order 8 and of type (1, 1, 1) as well as of order 16 and of type (1, 1, 1, 1). Hence the groups which admit five-eighths automorphisms are composed of various categories of elementary groups which have this useful common property. 1 These PROCEEDINGS, 15 (1929), 369; 16 (1930), 86.

M ULTIVECTORIAL CUR VA TURE By M. S. KNEBELMAN DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY

Communicated December 12, 1930

Throughout what follows I consider an n-dimensional space V. with a Riemannian metric which I assume to be positive-definite-a restriction which is imposed for simplicity's sake and which may be removed by a slight change in the definition of length. In a previous paper' I defined an r-cell determined by r( . n) linearly independent vectors x,, (a = 1, ..., r) indicating the vector and (i = 1, n) indicating the component. It is the object of this note to define the curvature of an r-cell or of an r-plane and to obtain some of its properties. Let the space V, be referred to an arbitrary coordinate system x and let y be a Riemannian coordinate system with origin at the point xo. Then the equations of any geodesic through xo are yi = x1(xo) t, x' determining the direction of the curve. Consider the set of all geodesics through xo whose directions are linearly dependent on the r independent directions x1(xo). Any r independent geodesics in this set will have the equations yi

( 1) C! xita= (XO) vp the determinant C! being different from zero. The above equations define a subspace Vr of V, whose induced metric,2 bao is given by (2) b.,o gij x. =

=

xjo

MA THEMA TICS: M. S. KNEBELMA N

44

PROC. N. A. S.

gij being the components of the fundamental tensor of V. and the summation convention being used for all repeated indices as well as for subscripts of indices. If we denote the components of the Riemann-Christoffel tensor of V, by Rijkl and those for the subspace V, by Ba,B,#a, then it can be shown easily that Bap-

(3)

Rijkl xit4X.

=

The space V,, being non-singular at xo and the vectors being independent, b = bp > 0; in fact, b is the square of the content' of the r-cell. If b"I denotes the normalized cofactor of bap in b, then baP. b . Ba7pj is the scalar curvature of Vr. We now define the r-vectorial curvature of the space V, for the orientation x, a = 1, .. ., r(l < r s n) to be one-half the scalar curvature of the r-cell determined by X. The factor one-half in the above definition is not essential and is used merely to make this definition agree with that of Riemannian curvature (r = 2). We shall denote this curvature by K(t) or simply by K if there is no chance for confusion. From (3) it follows that K is a scalar invariant and it can be shown to remain unchanged if instead of the orientation X.4 we use a linearly dependent one. Hence if we define an r-plane to be the totality of r-cells whose orientations are dependent on x$, we may properly speak of the curvature of an r-plane; in this r-plane we may choose the set X.a to be mutually orthogonal unit vectors; that is, gijj4,= 5aap With this choice we have

I

r

K(,)

=

1/2 eg3P sa,3

BaaP1j%

E

= al