On convergence of intrinsic volumes of Riemannian manifolds

Abstract

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