On convergence of intrinsic volumes of Riemannian manifolds

TL;DR

This paper proves that the intrinsic volumes of a family of scaled Riemannian manifolds converge to a product involving the Euler characteristic and the intrinsic volumes of the base manifold, confirming conjectures about their behavior.

Contribution

It establishes a convergence result for intrinsic volumes under a specific metric scaling in Riemannian submersions, supporting broader conjectures in the field.

Findings

Intrinsic volumes of scaled manifolds converge to a product involving Euler characteristic.

The convergence result aligns with and supports existing conjectures.

Provides a new perspective on the behavior of geometric invariants under metric deformations.

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 $\varepsilon >0$ let us define a new Riemannian metric on $M$ by multiplying the original one by $\varepsilon$ along the vertical directions and keeping it the same along the (orthogonal) horizontal directions. Denote the corresponding Riemannian manifold by $M_\varepsilon$. The main result says that $\lim_{\varepsilon\to +0} V_i(M_\varepsilon)=\chi(Z) V_i(B)$, 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.