Titlebook: Comparison Finsler Geometry; Shin-ichi Ohta Book 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer

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Efficient Sparse Representation and ModelingIn this chapter, we consider the natural energy functional (for functions) and the corresponding Sobolev spaces. Then we introduce the . in a way that its associated harmonic functions are minimizers of the energy functional. We also show the . as the first analytic comparison theorem.

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https://doi.org/10.1007/978-3-031-01519-9This chapter is devoted to a geometric application of the improved Bochner inequality, the . (also called the .). This is one of the most important geometric applications of the Γ-calculus. The asymptotic behavior of (nonlinear or linearized) heat semigroups for large time will play an essential role. A related analysis also shows the ..

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Finsler ManifoldsIn this chapter, we begin with Minkowski normed spaces which appear as tangent spaces of Finsler manifolds, and recall Euler’s homogeneous function theorem as an important calculus tool throughout the book. Then we give the definition of a Finsler manifold, followed by some examples and a naturally induced (asymmetric) distance structure.

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Covariant DerivativesIn this chapter, we revisit the geodesic equation and give an appropriate definition of . of vector fields (associated with the Chern connection). Our argument will be heuristic and is motivated by a Riemannian characterization.

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The Nonlinear LaplacianIn this chapter, we consider the natural energy functional (for functions) and the corresponding Sobolev spaces. Then we introduce the . in a way that its associated harmonic functions are minimizers of the energy functional. We also show the . as the first analytic comparison theorem.

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Bakry–Ledoux Isoperimetric InequalityThis chapter is devoted to a geometric application of the improved Bochner inequality, the . (also called the .). This is one of the most important geometric applications of the Γ-calculus. The asymptotic behavior of (nonlinear or linearized) heat semigroups for large time will play an essential role. A related analysis also shows the ..

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Comparison Finsler Geometry978-3-030-80650-7Series ISSN 1439-7382 Series E-ISSN 2196-9922

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Sparse Matrices and their Applicationstion for the energy functional. To this end, some important quantities such as the fundamental and Cartan tensors are introduced. We will see that the metric definition of geodesics coincides with the variational definition as solutions to the geodesic equation. We also prove the Finsler analogue of the Hopf–Rinow theorem.