On the Curvatura Integra in a Riemannian Manifold

Abstract

In a previous paper [1] we have given an intrinsic proof of the formula of Allendoerfer-Weil which generalizes to Riemannian manifolds of n dimensions the classical formula of Gauss-Bonnet for n = 2. The main idea of the proof is to draw into consideration the manifold of unit tangent vectors which is intrinsically associated to the Riemannian manifold. Denoting by R' the Riemannian manifold of dimension n and by M2' the manifold of dimension 2n 1 of its unit tangent vectors, our proof has led, in the case that n is even, to an intrinsic differential form of degree n 1 (which we denoted by II) in M2'. We shall introduce in this paper a differential form of the same nature for both even and odd dimensional Riemannian manifolds. We find that this differential form bears a close relation to the Curvatura Integra of a submanifold in a Riemannian manifold, because it will be proved that its integral over a closed submanifold of R' is equal to the Euler-Poincare characteristic of the submanif old. The method can be carried over to deduce relations between relative topological invariants of a submanifold of the manifold and differential invariants derived from the imbedding, and some remarks are to be added to this effect.