2012/02/17 by Saji, Kentaro, Umehara, Masaaki, Yamada, Kotaro
#Differential Geometry (math.DG) #FOS: Mathematics #Primary 57R45 #Secondary 53A05
paper · doi:10.48550/arxiv.1202.3854
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 Mn (n≥ 2) be an oriented compact n-manifold without boundary and TMn its tangent bundle. Let \mathcal E be a vector bundle of rank n over Mn, and φ:TMn→ \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 φ under the assumption that φ 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.