Crofton formulas and indefinite signature

Abstract

Abstract. We study the O(p, q)-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an O(p, q)-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, notably in the (p − 1) homogeneous case of O(p, p), yielding a Crofton formula for the centro-affine surface area when p ≡ 3 mod 4. We then exploit the functorial properties of Crofton formulas to pass to the general case. We also identify the invariant formulas explicitly for all O(p, 2)-invariant valuations. The proof relies on the exact computation of some integrals of independent interest. Those are related to Selberg's integral and to the Beta function of a matrix argument, except that the positive-definite matrices are replaced with matrices of all signatures. We also analyze the distinguished invariant Crofton distribution supported on the minimal orbit, and show that, somewhat surprisingly, it sometimes defines the trivial valuation, thus producing a distribution in the kernel of the cosine transform of particularly small support. In the heart of the paper lies the description by Muro of the | det X| s family of distributions on the space of symmetric matrices, which we use to construct a family of O(p, q)-invariant Crofton distributions. We conjecture there are no others, which we then prove for O(p, 2) with p even. The functorial properties of Crofton distributions, which serve an important tool in our investigation, are studied by T. Wannerer and the author in the Appendix.

Valuation theory of indefinite orthogonal groups

“…Therefore The same is true for O(p, q) with min(p, q) = 1, see [17]. Recently, the statement was generalized to arbitrary p, q, based on the results of the present paper [32,Theorem 2].…”

“…In a follow-up paper by the second named author [32], the Crofton formulas associated with O(p, q)-invariant valuations are studied.…”

Crofton formulas in pseudo-Riemannian space forms

“…Using the Hilbert-Schmidt inner product to identify T * 0 Sym r (R) = Sym r (R), the statement of the proposition in fact holds with all conormal cones intersected with Sym + r (R), which follows from [26,Theorem 8.1.6]. In [22,Proposition 4.9], an O(p, q)-invariant distribution was constructed on Gr n+1−k (V ). Let us briefly recall the construction.…”

“…We conclude that WF(f λ ) ∩ L E,Y = ∅, and so m 0 k is well-defined. As the proof above is independent of the value of λ, invariance under O(Q) follows by analytic continuation as in the original construction in [22], and we omit the details. Definition 5.8.…”

“…The corresponding statement in other signatures is also true, but much harder to prove. The second named author proved in [22] the statement first for certain signatures by an explicit, but difficult, computation and then used the behavior of Crofton formulas under restrictions and projections to handle the general case.…”