Riemannian curvature measures

Abstract

A famous theorem of Weyl states that if M is a compact submanifold of euclidean space, then the volumes of small tubes about M are given by a polynomial in the radius r, with coefficients that are expressible as integrals of certain scalar invariants of the curvature tensor of M with respect to the induced metric. It is natural to interpret this phenomenon in terms of curvature measures and smooth valuations, in the sense of Alesker, canonically associated to the Riemannian structure of M . This perspective yields a fundamental new structure in Riemannian geometry, in the form of a certain abstract module over the polynomial algebra R[t] that reflects the behavior of Alesker multiplication. This module encodes a key piece of the array of kinematic formulas of any Riemannian manifold on which a group of isometries acts transitively on the sphere bundle. We illustrate this principle in precise terms in the case where M is a complex space form.

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

“…The case of boundary and corners seems to be known for a long time, but I do not have a classical reference. But this is a special case of Theorem 3.11 in a recent paper [15]. This definition is obviously consistent with the definition for convex sets from the previous paragraph when convex sets have smooth boundary.…”

“…5 Intrinsic volumes on convex sets and on manifolds play an important role in valuations theory on convex sets [5], [21] and its generalizations to manifolds [1]- [4], [15]. They play a key role in integral geometry [8], [9], [17].…”

Classification of angular curvature measures and a proof of the angularity conjecture

“…This was first proved by Alesker [7]. For a proof that does not rely on the existence of isometric embeddings see [24].…”

Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

“…More generally, we may similarly define Weyl functors between any two categories of manifolds equipped with a geometric structure, when natural restriction operations are available for both structures. Important examples of Weyl functors are the intrinsic volumes of Riemannian manifolds, taking values in smooth valuations, and the Lipschitz-Killing curvature measures [22,25]. For a different example, a family of Weyl functors on contact manifolds with values in generalized valuations was described in [21].…”

“…In this language, the intrinsic volumes are valuations which are defined on arbitrary Riemannian manifolds and which behave naturally with respect to isometric embeddings; and the Lipschitz-Killing curvature measures are curvature measures naturally associated to Riemannian manifolds. Conversely, Fu and Wannerer [25] have recently shown in the spirit of Hadwiger's characterization that the intrinsic volumes/Lipschitz-Killing curvature measures are characterized by the Weyl principle, i.e. any valuation/curvature measure on Riemannian manifolds that satisfies the Weyl principle is a linear combination of intrinsic volumes/Lipschitz-Killing measures.…”