On convergence of intrinsic volumes of Riemannian manifolds

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