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" Introduction to modern Finsler geometry / "
Yi-Bing Shen, Zhejiang University, China, Zhongmin Shen, Indiana University-Purdue University, Indianapolis, USA.
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BL
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| Record Number
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890735
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| Main Entry
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Shen, Yibing.
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| Title & Author
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Introduction to modern Finsler geometry /\ Yi-Bing Shen, Zhejiang University, China, Zhongmin Shen, Indiana University-Purdue University, Indianapolis, USA.
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| Publication Statement
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Beijing :: Higher Education Press ;Singapore :: World Scientific,, [2016]
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1 online resource
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| ISBN
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9789814704915
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: 9814704911
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9789814704908
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9789814713160
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9814704903
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9814713163
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| Contents
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Intro; Contents; Preface; Foundations; 1. Differentiable Manifolds; 1.1 Differentiable manifolds; 1.1.1 Differentiable manifolds; 1.1.2 Examples of differentiable manifolds; 1.2 Vector fields and tensor fields; 1.2.1 Vector bundles; 1.2.2 Tensor fields; 1.3 Exterior forms and exterior differentials; 1.3.1 Exterior differential operators; 1.3.2 de Rham theorem; 1.4 Vector bundles and connections; 1.4.1 Connection of the vector bundle; 1.4.2 Curvature of a connection; Exercises; 2. Finsler Metrics; 2.1 Finsler metrics; 2.1.1 Finsler metrics; 2.1.2 Examples of Finsler metrics; 2.2 Cartan torsion.
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2.2.1 Cartan torsion2.2.2 Deicke theorem; 2.3 Hilbert form and sprays; 2.3.1 Hilbert form; 2.3.2 Sprays; 2.4 Geodesics; 2.4.1 Geodesics; 2.4.2 Geodesic coefficients; 2.4.3 Geodesic completeness; Exercises; 3. Connections and Curvatures; 3.1 Connections; 3.1.1 Chern connection; 3.1.2 Berwald metrics and Landsberg metrics; 3.2 Curvatures; 3.2.1 Curvature form of the Chern connection; 3.2.2 Flag curvature and Ricci curvature; 3.3 Bianchi identities; 3.3.1 Covariant differentiation; 3.3.2 Bianchi identities; 3.3.3 Other formulas; 3.4 Legendre transformation; 3.4.1 The dual norm in the dual space.
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3.4.2 Legendre transformation3.4.3 Example; Exercises; 4. S-Curvature; 4.1 Volume measures; 4.1.1 Busemann-Hausdor volume element; 4.1.2 The volume element induced from SM; 4.2 S-curvature; 4.2.1 Distortion; 4.2.2 S-curvature and E-curvature; 4.3 Isotropic S-curvature; 4.3.1 Isotropic S-curvature and isotropic E-curvature; 4.3.2 Randers metrics of isotropic S-curvature; 4.3.3 Geodesic flow; Exercises; 5. Riemann Curvature; 5.1 The second variation of arc length; 5.1.1 The second variation of length; 5.1.2 Elements of curvature and topology; 5.2 Scalar flag curvature; 5.2.1 Schur theorem.
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5.2.2 Constant flag curvature5.3 Global rigidity results; 5.3.1 Flag curvature with special conditions; 5.3.2 Manifolds with non-positive ag curvature; 5.4 Navigation; 5.4.1 Navigation problem; 5.4.2 Randers metrics and navigation; 5.4.3 Ricci curvature and Einstein metrics; Exercises; Further Studies; 6. Projective Changes; 6.1 The projective equivalence; 6.1.1 Projective equivalence; 6.1.2 Projective invariants; 6.2 Projectively at metrics; 6.2.1 Projectively at metrics; 6.2.2 Projectively at metrics with constant ag curvature; 6.3 Projectively at metrics with almost isotropic S-curvature.
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6.3.1 Randers metrics with almost isotropic S-curvature6.3.2 Projectively at metrics with almost isotropic S-curvature; 6.4 Some special projectively equivalent Finsler metrics; 6.4.1 Projectively equivalent Randers metrics; 6.4.2 The projective equivalence of ()-metrics; 6.4.3 The projective equivalence of quadratic ()-metrics; Exercises; 7. Comparison Theorems; 7.1 Volume comparison theorems for Finsler manifolds; 7.1.1 The Jacobian of the exponential map; 7.1.2 Distance function and comparison theorems; 7.1.3 Volume comparison theorems; 7.2 Berger-Kazdan comparison theorems.
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| Dewey Classification
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516.375
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| LC Classification
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QA689.S538 2016
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| Added Entry
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Shen, Zhongmin,1963-
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