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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.Comment: 68 pages; minor change
Integral geometry for the 1-norm
Classical integral geometry takes place in Euclidean space, but one can
attempt to imitate it in any other metric space. In particular, one can attempt
this in R^n equipped with the metric derived from the p-norm. This has, in
effect, been investigated intensively for 1<p<\infty, but not for p=1. We show
that integral geometry for the 1-norm bears a striking resemblance to integral
geometry for the 2-norm, but is radically different from that for all other
values of p. We prove a Hadwiger-type theorem for R^n with the 1-norm, and
analogues of the classical formulas of Steiner, Crofton and Kubota. We also
prove principal and higher kinematic formulas. Each of these results is closely
analogous to its Euclidean counterpart, yet the proofs are quite different.Comment: 17 pages. Version 3: minor clarifications. This version will appear
in Advances in Applied Mathematic
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