ON SOME CONSEQUENCES OF THE GENERALIZED GAUSS-BONNET THEOREM

Abstract

In this paper two consequences of the generalized Gauss-Bonnet theorem are obtained. It is proved in theorem 1 that if \(M^{2n}\) is a compact and orientable differentiable manifold such that \(\chi(M^{2n})\ne 0\) and \(B_{2q}=0\), then there does not exist an orientable subbundle of the tangent bundle with fiber \(R^{2q}\) on \(M^{2n}\). In theorem 2, it is proved that if \(\partial M^{2n+1}\) is a compact manifold and \(\chi(\partial M^{2n+1})\ne 0\), where \(M^{2n+1}\) is an orientable differentiable manifold, then \(M^{2n+1}\) does not admit an orientable subbundle \(E\) of the tangent bundle with fiber \(R^{2n}\) such that the restriction of \(E\) on \(\partial M^{2n+1}\) is the tangent bundle of \(\partial M^{2n+1}\). Two consequences of these two theorems are obtained.