An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications. (English) Zbl 1372.57039

Saji, Kentaro; Umehara, Masaaki; Yamada, Kotaro

J. Math. Soc. Japan 69, No. 1, 417-457 (2017).

Let \(M^n\), \(n=2m\), \(m\geq 2\), be an oriented compact \(n\)-manifold without boundary, \(TM\) its tangent bundle, \(\mathcal{E}\) a vector bundle of rank \(k\) over \(M\) and \(\varphi : TM\rightarrow \mathcal{E}\) an oriented vector bundle homomorphism. The authors obtain an interesting index formula for the bundle homomorphism \(\varphi\) (Formula (1.4)), under the hypothesis that \(\varphi\) has only certain kinds of generic singularities. In the last part of the paper, several applications of this index formula to hypersurface theory are presented. Other papers by the authors directly connected to this topic are [the authors, Kyushu J. Math. 62, No. 1, 259–280 (2008; Zbl 1155.57027); Ann. Math. (2) 169, No. 2, 491–529 (2009; Zbl 1177.53014); Math. Proc. Camb. Philos. Soc. 146, No. 3, 731–746 (2009; Zbl 1173.53039); Osaka J. Math. 47, No. 2, 591–607 (2010; Zbl 1209.57020); C. R., Math., Acad. Sci. Paris 348, No. 11–12, 665–668 (2010; Zbl 1235.57018); J. Geom. Anal. 22, No. 2, 383–409 (2012; Zbl 1276.57032)].

Reviewer: Dorin Andrica (Riyadh)

MSC:

57R45	Singularities of differentiable mappings in differential topology
53A05	Surfaces in Euclidean and related spaces

Keywords:

wave front; coherent tangent bundle; Morin singularity; Gauss-Bonnet type formulas; index formula

Citations:

Zbl 1155.57027; Zbl 1177.53014; Zbl 1173.53039; Zbl 1209.57020; Zbl 1235.57018; Zbl 1276.57032