Gil Solanes

Integral geometry of complex space forms

We show how Alesker’s theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex Euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.

Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces

We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.

The Gauss-Bonnet theorem and Crofton-type formulas in complex space forms

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in any complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives the Gauss curvature integral in terms of the Euler characteristic, and some hermitian intrinsic volumes. The other one, which is shorter, involves the measure of complex hyperplanes meeting the domain. As a tool, we obtain variation formulas in integral geometry of complex space forms.

Classification of invariant valuations on the quaternionic plane

Abstract We describe the orbit space of the action of the group Sp ( 2 ) Sp ( 1 ) on the real Grassmann manifolds Gr k ( H 2 ) in terms of certain quaternionic matrices of Moore rank not larger than 2. We then give a complete classification of valuations on the quaternionic plane H 2 which are invariant under the action of the group Sp ( 2 ) Sp ( 1 ) .

Kinematic formulas on the quaternionic plane

We introduce different bases for the vector space of Sp(2)Sp(1)-invariant, translation invariant continuous valuations on the quaternionic plane and determine a complete set of kinematic formulas.

Integral geometry and valuations

Part I: New Structures on Valuations and Applications.- Translation invariant valuations on convex sets.- Valuations on manifolds.- Part II: Algebraic Integral Geometry.- Classical integral geometry.- Curvature measures and the normal cycle.- Integral geometry of euclidean spaces via Alesker theory.- Valuations and integral geometry on isotropic manifolds.- Hermitian integral geometry.

Total Curvature of Complete Surfaces in Hyperbolic Space

We prove a Gauss-Bonnet formula for the extrinsic curvature of complete surfaces in hyperbolic space under some assumptions on the asymptotic behaviour.

Integral geometry of equidistants in hyperbolic space

We generalize the classical formulas of integral geometry, by getting integral geometric formulas for the intersection of a fixed compact hypersurface of hyperbolic space and a moving totally umbilical hypersurface. In particular we compute the mean value of the volume, the total mean curvatures and the Euler characteristic of these intersections when the totally umbilical hypersurface moves over all the intersecting positions. Analogous formulas are given for totally umbilical hypersurfaces contained in totally geodesic planes of ℍn.

Regularized Riesz energies of submanifolds

Given a closed submanifold, or a compact regular domain, in euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamards finite part and analytic continuation), and show that they give essentially the same result. We prove that some of these energies are invariant under Moebius transformations, thus giving a generalization to higher dimensions of the Moebius energy of knots.

Dual Curvature Measures in Hermitian Integral Geometry

The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space CurvU(n)∗ of dual unitarily invariant curvature measures. Building on the recent results from integral geometry in complex space forms, we describe this algebra structure explicitly as a polynomial algebra. This is a short way to encode all local kinematic formulas. We then characterize the invariant valuations on complex space forms leaving the space of invariant angular curvature measures fixed.