Abstract
The recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.
Highlights
1.1 BackgroundA valuation on a finite-dimensional vector space V is a functional μ : K(V ) → A, where K(V ) denotes the set of compact convex subsets of V and A is an abelian semigroup, such that μ(K ∪ L) + μ(K ∩ L) = μ(K ) + μ(L)whenever K, L, K ∪ L ∈ K(V ), and μ(∅) = 0
If V is a Euclidean vector space of dimension n with unit ball B, and K ∈ K(V ), as observed by Steiner [36], the volume of the r -tube Kr := K + r B around K is a polynomial in r : n vol Kr = μk (K )ωn−kr n−k
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
Summary
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. It is a very natural question to look for analogous results in pseudo-Riemannian geometry In this case, the tubes are in general not compact, but one can try to associate valuations and curvature measures to pseudo-Riemannian manifolds. The most important property of these objects is the Weyl principle, which states that for every isometric immersion M N of pseudo-Riemannian manifolds, the restriction of the intrinsic volume μkN to M equals μkM and the restriction of the Lipschitz–Killing curvature measure. D], based on the results from [12]
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