The integral formula of pontrjagin characteristic forms

Abstract

Let M be an n-dimensional oriented compact and C^∞ manifold, V the m-dimensional real vector bundle on M, and P_k(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 u_C, and N=U_iN_i, where N_i 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 N_i, denoted by $I_{u_C }$(N_i), then we prove the following integral formula

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

.