On convergence of intrinsic volumes of Riemannian manifolds

Abstract

In 1939 H. Weyl has introduced the so called intrinsic volumes $V_i(M^n),i=0,\dots ,n$, (known also as Lipschitz-Killing curvatures) for any closed smooth Riemannian manifold $M^n$. Given a Riemmanian submersion of compact smooth Riemannian manifolds $M\to B$, $B$ is connected. For $\epsilon >0$ let us define a new Riemannian metric on $M$ by multiplying the original one by $\epsilon$ along the vertical directions and keeping it the same along the (orthogonal) horizontal directions. Denote the corresponding Riemannian manifold by $M_{\epsilon }$. The main result says that ${\lim }_{\epsilon \to +0}{V}_i(M_{\epsilon })=\chi (Z)V_i(B)$, where $\chi (Z)$ is the Euler characteristic of a fiber of the submersion. This result is consistent with more general open conjectures on convergence of intrinsic volumes formulated previously by the author.

arXiv comment17 pages; typos