A sampler of Riemann-Finsler geometry by David Bao, Robert L. Bryant, Shiing-Shen Chern, Zhongmin

A sampler of Riemann-Finsler geometry by David Bao, Robert L. Bryant, Shiing-Shen Chern, Zhongmin

By David Bao, Robert L. Bryant, Shiing-Shen Chern, Zhongmin Shen

Finsler geometry generalizes Riemannian geometry in precisely a similar manner that Banach areas generalize Hilbert areas. This publication provides expository bills of six vital themes in Finsler geometry at a degree appropriate for a different themes graduate direction in differential geometry. The participants think about matters relating to quantity, geodesics, curvature and mathematical biology, and comprise quite a few instructive examples.

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Injectivity and range of the area definition Let us start the study of the injectivity and range of the Busemann definition of area by describing the unit ball of the (n−1)-volume density in terms of a well-known construction in convex geometry. Busemann area and intersection bodies. Consider R n with its Euclidean structure and its unit sphere S n−1 . If K ⊂ R n is a star-shaped body containing the origin, the intersection body of K, IK, is defined by the following simple construction: if x ∈ R n is a unit vector, let A(K ∩ x⊥ ) denote the area of the intersection of K with the hyperplane perpendicular to x, and let IK be the star-shaped body enclosed by the surface {x/A(K ∩ x⊥ ) ∈ R n : x ∈ S n−1 }.

The representation of length as a mixed volume gives an easy proof of the following monotonicity property of length in two-dimensional normed spaces. 6. If K1 ⊂ K2 are nested convex bodies in a two-dimensional normed space, then (∂K1 ) ≤ (∂K2 ). The proof is left as a simple exercise to the reader. The following related exercise is, perhaps, somewhat harder. 7. Show that a Finsler metric on the plane satisfies the monotonicity property in the previous proposition if and only if its geodesics are straight lines.

The following result, stated in [Busemann 1961] gives an important characterization of totally convex k-densities in terms of what Gromov [1983] calls the compressing property. 27. A k-density φ on an n-dimensional vector space X is totally convex if and only if for every k-dimensional linear subspace there exists a φdecreasing linear projection onto that subspace. 13. 28. The mass∗ k-volume densities of an n-dimensional normed space X, 1 ≤ k ≤ n − 1, are totally convex . Proof. 27, it is enough to show that given any k-dimensional subspace W , there exists a linear projection P : X → W that is mass∗-decreasing.

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