I will show a local version of Gauss-Bonnet-Chern theorem on Riemannian manifolds with boundary and its generalization are obtained by construction of the fundamental solution as follows: The fundamental solution for initial-boundary problem is constructed by symbolic calculus of pseudo-differential operators, if we introduce a new weight for symbols.

Also a local version of Riemann-Roch theorem on Kaehler manifolds can be obtained by the similar methods.

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