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

Finsler geometry generalizes Riemannian geometry in precisely an analogous approach that Banach areas generalize Hilbert areas. This ebook offers expository bills of six vital themes in Finsler geometry at a degree appropriate for a unique issues graduate path in differential geometry. The members ponder concerns regarding quantity, geodesics, curvature and mathematical biology, and comprise quite a few instructive examples.

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Choose a basis of W , w 1 , . . , w n−1 , such that φ(w 1 ∧ w 2 ∧ · · · ∧ w n−1 ) = 1 and consider the support hyperplane of the unit sphere of (Λn−1 X, φ) at the point w 1 ∧ w 2 ∧ · · · ∧ w n−1 . This hyperplane is 24 ´ J. C. ALVAREZ PAIVA AND A. C. THOMPSON given by an equation of the form ω = 1, where ω is an (n−1)-form with constant coefficients. In other words, ω ∈ Λn−1 X ∗ . We claim that the linear projection n−1 (−1)i ω(x ∧ w 1 ∧ · · · ∧ wˆi ∧ · · · ∧ w n−1 )w i P x := i=1 is φ-decreasing.

A definition of area on normed spaces assigns to every ndimensional, n ≥ 2, normed space X a normed space (Λ n−1 X, σX ) in such a way that properties (1)–(4) above are satisfied. For simplicity, we shall speak of the Busemann, Holmes–Thompson, and mass∗ definitions of area to refer to the definitions of area induced, respectively, by the Busemann, Holmes–Thompson, and mass∗ volume definitions. Definitions of area in normed spaces are related to important constructions in convex geometry such as intersection bodies, projection bodies, and Wulff shapes.

ALVAREZ PAIVA AND A. C. THOMPSON 28 The form φˆ dξ1 ∧ · · · ∧ dξn does not depend on the choice of basis in X. Up to a constant factor, we define the form φˇ as the contraction of this n-form with the Euler vector field, XE (ξ) = ξ, in X ∗ : φˇ := −1 φˆ dξ1 ∧ · · · ∧ dξn XE . 4(2π)n−1 It is known (see [H¨ ormander 1983, pages 167–168]) that φˆ is smooth on X ∗ \ 0 and homogeneous of degree −n − 1; therefore φˇ is a smooth differential form on X ∗ \ 0 that is homogeneous of degree −1. 25. Let (X, φ) be an n-dimensional Minkowski space.