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

Finsler geometry generalizes Riemannian geometry in precisely a similar method that Banach areas generalize Hilbert areas. This ebook provides expository money owed of six vital themes in Finsler geometry at a degree appropriate for a different themes graduate path in differential geometry. The participants examine matters concerning quantity, geodesics, curvature and mathematical biology, and comprise various instructive examples.

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Extra info for A sampler of Riemann-Finsler geometry

Example text

Notice that for any basis w 1 , . . , w k of W , we have that µm∗ (w 1 ∧ w 2 ∧ · · · ∧ w k ) = |ξ 1 ∧ ξ 2 ∧ · · · ∧ ξ k (w 1 ∧ w 2 ∧ · · · ∧ w k )|, and that if w 1 , . . , w k is dual to ξ 1 , . . , ξ k , then µm∗ (w 1 ∧ w 2 ∧ · · · ∧ w k ) = 1. By the Hahn–Banach theorem, there exist covectors ξˆ1 , . . , ξˆk ∈ X ∗ such that (1) |ξˆi (x)| ≤ 1 for all x ∈ B and for all i, 1 ≤ i ≤ k; (2) the restriction of ξˆi to W equals ξ i for all i, 1 ≤ i ≤ k. We may now define the projection P : X → W by the formula k ξˆi (x)w i , P (x) := i=1 and show that it is µm∗ -decreasing.

3. Using the Hahn–Banach theorem and the notation above, show that µm∗ (a) is the supremum of the numbers |ξ 1 ∧ ξ 2 ∧ · · · ∧ ξ k (a)|, where ∗ ξ 1 , . . , ξ k ∈ BX . In the study of volumes and areas on Finsler manifolds, we shall also need to work with k-densities and smooth k-densities on manifolds. For this purpose we introduce the bundle of simple tangent k-vectors on a manifold M , Λ ks T M . This is a subbundle of algebraic cones of the vector bundle Λ k T M , and if we omit the zero section it is a smooth manifold.

In this case, B ∗ is the octahedron with vertices (±1, 0, 0), (0, ±1, 0), (0, 0, ±1) and WB ∗ = B ∗ . VOLUMES ON NORMED AND FINSLER SPACES 39 Let K be the cuboctahedron with vertices (±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1). The dual ball K ∗ is the rhombic dodecahedron with vertices ±(1, 0, 0), ±(0, 1, 0), ±(0, 0, 1) and (± 12 , ± 12 , ± 12 ). A simple calculation shows that WB ∗ = W K ∗ . In fact, if L is any centered convex body that lies between the cube and the cube-octahedron then WL∗ = WB ∗ .

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