On convergence of intrinsic volumes of Riemannian manifolds

Let $$\pi :M\rightarrow B$$ be a Riemannian submersion of two compact smooth manifolds, B is connected. $$M(\varepsilon )$$ denote the manifold M equipped with new metric which obtained from original one by multiplying $$\varepsilon $$ along vertical subspaces (i.e. fibers) and keeping unchanged (orthogonal to them) horizontal subspaces. $$V_i(M(\varepsilon ))$$ ith intrinsic volume. The main result this note says that $$\lim _{\varepsilon \rightarrow +0}V_i(M(\varepsilon ))=\chi (Z) V_i(B)$$ where $$\chi (Z)$$ denotes Euler characteristic fiber .