Geometry of a Quadratic Differential Form

Abstract

Relativity theory calls for the study of a differential form which is nondegenerate but of signature (+ -4. +). Such a data will be called a Lorentzian structure. Although there is yet no geometrical or physical justification, it is a natural mathematical problem to study a non-degenerate quadratic differential form with arbitrary signature. Modern mathematics requires us to say at the outset in what space the quadratic differential form is defined and the study takes place. Such spaces are called differentiable manifolds, which are, roughly speaking, spaces where differentiation makes sense and where tensor fields can be defined. The prime example is the n-dimensional number space and a general differentiable manifold behaves locally like it. From a given differentiable manifold further manifolds are constructed by taking the submanifolds or the quotient manifolds. A compact manifold is one which can be covered by a finite number of coordinate patches. A manifold with a non-degenerate (resp. positive definite, resp. of signature (-+ -+)) quadratic differential form defined everywhere is called pseudo-riemannian (resp. riemannian, resp. Lorentzian). Does such a pseudo-riemannian structure exist on a differentiable manifold? If the form is positive definite, this is so, because the set of all possible positive semi-definite quadratic differential forms has the convexity property. For a non-degenerate form of signature (p, q) (p positive squares and q negative squares in the normal form) the existence of such a form on a manifold is equivalent to the existence of a continuous field of p-dimensional subspaces of the tangent spaces.' If the manifold is compact and p is odd, a necessary condition is that the Euler-Poincare characteristic of the manifold should be zero [2], [3]. A complete topological characterization of such manifolds is not known. It follows, however, from the above