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

TL;DR

This paper generalizes Gauss-Bonnet formulas to an index formula for bundle homomorphisms with generic singularities, with applications to hypersurface theory and intrinsic geometry of Kossowski metrics.

Contribution

It introduces an index formula for bundle homomorphisms of tangent bundles into vector bundles of the same rank, extending previous Gauss-Bonnet results to broader geometric contexts.

Findings

Derived an index formula for bundle homomorphisms with generic singularities.

Provided applications to hypersurface theory.

Characterized Kossowski metrics as induced metrics of coherent tangent bundles.

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 22-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 $\varphi:TM^n\to \mathcal E$ 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 $\varphi$ under the assumption that $\varphi$ 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.