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

Abstract

In this paper angular curvature measures are investigated. Our first result is a complete classification of translation-invariant angular smooth curvature measures on R n . Subsequently, we use this result to show that the class of angular curvature measures on a Riemannian manifold is preserved by both the pullback by isometric immersions and the action of the Lipschitz-Killing algebra. The latter confirms the angularity conjecture formulated by A. Bernig, J.H.G. Fu, and G. Solanes.

On the extendability by continuity of angular valuations on polytopes

“…As was already observed in [9] for k = n − 1 there is no condition on f except continuity. Theorem 1.2 directly implies the classification of angular curvature measures that has been recently obtain by the author [44].…”

“…Proof of Theorem 1. The converse follows from the fact proved in [44,Theorem 1.4] that to each restriction of a 2-homogeneous polynomial to the image of the Plücker embedding corresponds a constant coefficient curvature measure. Globalizing this curvature measure we get an element of A k .…”

“…Other important examples of such valuations that admit a continuous extension are the Hermitian intrinsic volumes arising in Hermitian Integral Geometry [14] or more generally the constant coefficient valuations (see, e.g., [14,19]). Moreover, valuations of type (3) are of interest in connection with the Angularity Conjecture [15,44]. The problem of characterizing those functions f for which (3) admits a continuous extension was explicitly stated by Fu [19,Problem 2.3.13].…”

Curvature Measures of Pseudo-Riemannian Manifolds

“…Alesker and Fu [11] (see also [8] and [32]) have introduced a product structure on the space V ∞ (M ) of smooth valuations on a manifold M , which is compatible with the filtration (3). It has led to several deep applications in the integral geometry of isotropic spaces [6,9,19,53,54,59]. The product satisfies a version of Poincaré duality, which gives rise to the notion of generalized valuations on a manifold.…”