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The integral formula of pontrjagin characteristic forms

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1255)

Abstract

Let M be an n-dimensional oriented compact and C manifold, V the m-dimensional real vector bundle on M, and Pk(V) the kth Pontrjagin characteristic form of V. In this paper, we try to calculate the integral

$$\int {_M P_k (V)\Lambda \sigma } $$

where σ be any closed (n-4k)-form of M.

Let V⊗C be the complexification of V, \(u_C = \left\{ {s_1 + \sqrt { - 1} s_1 ^\prime ,...,s_{m - 2k} + \sqrt { - 1} s'_{m - 2k} ,u + \sqrt { - 1} u'} \right\}\)a set of smooth sections of the complex vector bundle V⊗C, N the set of singular points of uC, and N=UiNi, where Ni is the connected component of N. In this paper, we give a definition of the index of the smooth section \(u_C = u + \sqrt { - 1} u'\)of the bundle V⊗C with respect to Ni, denoted by \(I_{u_C } \)(Ni), then we prove the following integral formula

$$\int {_M P_k (V)\Lambda \sigma = ( - 1)^k \sum\limits_i {I_{u_C } (N_i )\smallint _{N_i } \sigma } } $$

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Keywords

  • Vector Bundle
  • Integral Formula
  • Chern Class
  • Complex Vector
  • Smooth Section

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Reference

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© 1987 Springer-Verlag

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Mei, XM. (1987). The integral formula of pontrjagin characteristic forms. In: Gu, C., Berger, M., Bryant, R.L. (eds) Differential Geometry and Differential Equations. Lecture Notes in Mathematics, vol 1255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077682

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  • DOI: https://doi.org/10.1007/BFb0077682

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17849-1

  • Online ISBN: 978-3-540-47883-6

  • eBook Packages: Springer Book Archive