Integral geometry of complex space forms

Andreas Bernig, Joseph H.G. Fu, Gil Solanes

Producció científica: Contribució a una revistaArticleRecercaAvaluat per experts

54 Cites (Scopus)

Resum

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. © 2014 Springer Basel.
Idioma originalEnglish
Pàgines (de-a)403-492
RevistaGeometric and Functional Analysis
Volum24
DOIs
Estat de la publicacióPublicada - 1 de gen. 2014

Fingerprint

Navegar pels temes de recerca de 'Integral geometry of complex space forms'. Junts formen un fingerprint únic.

Com citar-ho