Some conjectures on intrinsic volumes of Riemannian manifolds and Alexandrov spaces

TL;DR

This paper proposes conjectures on bounds and extensions of intrinsic volumes for Riemannian manifolds and Alexandrov spaces, exploring their properties, limits, and potential compactifications.

Contribution

It introduces new conjectures on intrinsic volume bounds and their extension to Alexandrov spaces, and suggests a framework for their continuity and limits.

Findings

Conjecture that intrinsic volumes are bounded by constants depending on dimension, diameter, and curvature.

Proposes that intrinsic volumes can be extended to certain Alexandrov spaces with curvature bounds.

Discusses known cases and implications for the space of Riemannian manifolds.

Abstract

For any closed smooth Riemannian manifold H. Weyl has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of the manifold, upper bound on its diameter, and lower bound on the sectional curvature. Furthermore we conjecture that intrinsic volumes can be defined for some (so called smoothable) Alexandrov spaces with curvature bounded below and state few of the expected properties of them, particularly the behavior under the Gromov-Hausdorff limits. We suggest conjectural compactifications of the space of smooth closed connected Riemannian manifolds with given upper bounds on dimension and diameter and a lower bound on sectional curvature to which the intrinsic volumes extend by continuity. We discuss also known cases of some of these conjectures.