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

ISBN-10: 0521831814

ISBN-13: 9780521831819

Finsler geometry generalizes Riemannian geometry in precisely a similar method that Banach areas generalize Hilbert areas. This booklet provides expository money owed of six very important issues in Finsler geometry at a degree appropriate for a different issues graduate direction in differential geometry. The individuals reflect on concerns on the topic of quantity, geodesics, curvature and mathematical biology, and comprise numerous instructive examples.

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**Example text**

While it seems unlikely that the wedge body of the unit ball in a Minkowski space is the ball of a Minkowski space, it is not hard to show that the wedge body of a polytope is a polytope. 11 [Thompson 1999]. The mass∗ area definition takes crystalline norms to crystalline norms. The question of determining the range for the mass∗ area definition is completely open. Is it possible that any centered convex body is a wedge body? 2. Area of the unit sphere. In this section we give the higher-dimensional analogues (as far as we know them) of the theorems of Sch¨ affer and Gol¸ab discussed in Section 5.

16 (Minkowski). The Holmes–Thompson (n−1)-volume density of an n-dimensional normed space X is itself a norm in Λn−1 X. VOLUMES ON NORMED AND FINSLER SPACES 25 In order to prove the convexity of the Holmes–Thompson (n−1)-volume density, we shall first give an integral representation for it. This representation depends, in turn, on two classical constructions: the Gauss map and the surface-area measure. Our approach follows [Fernandes 2002]. Let X be an n-dimensional vector space and let φ be an (n−1)-volume density on X.

26 ´ J. C. ALVAREZ PAIVA AND A. C. THOMPSON Since the surface-area measure dS∂B ∗ is nonnegative, µht (a + b) = ε−1 n−1 |a(ξ) + b(ξ)| dS∂B ∗ ξ∈Σ |a(ξ)| dS∂B ∗ + ε−1 n−1 ≤ ε−1 n−1 ξ∈Σ |b(ξ)| dS∂B ∗ ξ∈Σ = µht (a) + µht (b). 18. Let X be an n-dimensional vector space and let φ be an (n−1)volume density on X. Show that if φ is a norm, then compact hypersurfaces cannot by minimal. 3. Convexity properties of k-volume densities. We now pass to the more delicate subject that Busemann, Ewald, and Shephard studied extensively under the heading of convexity on Grassmannians.

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

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