An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications

Abstract

In a previous work, the authors introduced the notion of ‘coherent tangent bundle’, which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss–Bonnet formulas on coherent tangent bundles on $2$-dimensional manifolds were proven, and several applications to surface theory were given.

Let $M^n$ ($n\ge 2$) be an oriented compact $n$-manifold without boundary and $TM^n$ its tangent bundle. Let $\mathcal{E}$ be a vector bundle of rank $n$ over $M^n$, and $\phi:TM^n\to \mathcal{E}$ an oriented vector bundle homomorphism. In this paper, we show that one of these two Gauss–Bonnet formulas can be generalized to an index formula for the bundle homomorphism $\phi$ under the assumption that $\phi$ admits only certain kinds of generic singularities.

We shall give several applications to hypersurface theory. Moreover, as an application for intrinsic geometry, we also give a characterization of the class of positive semi-definite metrics (called Kossowski metrics) which can be realized as the induced metrics of the coherent tangent bundles.