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
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
.
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
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