An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications
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- Saji Kentaro
- Department of Mathematics, Faculty of Science, Kobe University
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- Umehara Masaaki
- Department of Mathematical and Computing Science, Tokyo Institute of Technology
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- Yamada Kotaro
- Department of Mathematics, Tokyo Institute of Technology
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Abstract
<p>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.</p><p>Let Mn (n ≥ 2) be an oriented compact n-manifold without boundary and TMn its tangent bundle. Let ℰ be a vector bundle of rank n over Mn, and φ: TMn → 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.</p><p>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.</p>
Journal
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- Journal of the Mathematical Society of Japan
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Journal of the Mathematical Society of Japan 69 (1), 417-457, 2017
The Mathematical Society of Japan
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Details 詳細情報について
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- CRID
- 1390001205115581568
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- NII Article ID
- 130005310476
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- NII Book ID
- AA0070177X
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- ISSN
- 18811167
- 18812333
- 00255645
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- NDL BIB ID
- 027859115
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- Text Lang
- en
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- Data Source
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- JaLC
- NDL
- Crossref
- CiNii Articles
- KAKEN
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- Abstract License Flag
- Disallowed