Titlebook: An Introduction to Riemann-Finsler Geometry; D. Bao,S.-S. Chern,Z. Shen Textbook 2000 Springer Science+Business Media New York 2000 Calc.D

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Riemannian Manifolds and Two of Hopf’s TheoremsA .. on a manifold . is a family of inner products {..}.∈. such that the quantities . are smooth in local coordinates. The Finsler function .(.) of a Riemannian manifold has the characteristic structure ..

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Finsler Surfaces and a Generalized Gauss-Bonnet Theorems, we only use objects which are invariant under positive rescaling in .. Consequently, our treatment using natural coordinates on . can be regarded as occurring on the (projective) sphere bundle ., in the context of homogeneous coordinates.

Fermentation

Variations of Arc Length, Jacobi Fields, the Effect of Curvatureiant differentiation. That is explored in a series of guided exercises at the end of 5.2. (Those exercises involve the second variation as well.) A systematic self-contained account can also be found in [BC1].

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The Gauss Lemma and the Hopf-Rinow Theoremtive there is the identity; see §5.3. Thus, for . small enough, not only does exp. [S..] makes sense, it is also diffeomorphic to S... The image set . is called a . in . centered at .. We later show why it can be said to have radius equal to ..

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The Index Form and the Bonnet—Myers Theoremonstant speed geodesic . = exp.., 0 ⩽ . ⩽ . that emanates from . = σ(0) and terminates at . = σ(.). If there is no confusion, label its velocity field by . also. Let .. denote covariant differentiation along σ, with reference vector .. This concept was introduced in the Exercise portion of 5.2.

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The Chern Connections not a connection on the bundle . over .. Nevertheless, it serves Finsler geometry in a manner that parallels what the Levi-Civita (Christoffel) connection does for Riemannian geometry. This connection is on equal footing with, but is different from, those due to Cartan, Berwald, and Hashiguchi (to

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