Integral geometry of complex space forms

By A. Bernig, Joseph H. G. Fu, G. Solanes. 2012

93 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b3ebdd84f959eefa10968fb93e58d8ab25dd4403

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Difference bodies in complex vector spaces (2012, background): “ Then, we always get a ﬁnite dimensional real vector space (see [6]). Its dimension, a basis and the arising integral geometry have been studied intensively. For some references on this direction see [4, 10, 12, 13, 14]. Some other important particular cases of valuations are given, for instance, wh”
Kinematic formulas for tensor valuations (2014, background): “onstants dΦ ij depend on Φ and the chosen bases. Hence, kG s 1,s 2 (Φ)(K,L) = X i,j dΦ ij Φi(K)⊗Ψj(L) (7) for K,L∈ K(V). 3.3. Relation between algebraic structures and kinematic formulas. We refer to [15, 16, 17, 18, 20] for the relation between algebraic structures on scalar valued valuations and i”
A homogeneous decomposition theorem for valuations on convex functions (2019, background): “Here strong structure and classification theorems have been established over the last seventy years (see [1, 2, 6, 7, 19–21, 28] for some recent results and [22, 23, 36] for information on the classical theory).”