A section on foundations reviews differential calculus and topology and explains tensor bundles, Riemannian curvature, and isometric immersions, while subsequent sections on curvature and topology and structure of the geodesic flow present information on symmetric spaces, the sphere theorem, Hamiltonian systems, and the theorem of the three closed geodesics. This second edition includes material on the author's Main Theorem for Surfaces of Genus 0. Annotation copyright by Book News, Inc., Portland, OR
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2-plane Alexandrov triangle angle assume atlas bundle chart called canonical Choose circle closed curves closed geodesic compact conjugate point consider constant curvature coordinates covariant derivation critical point defined Definition denote determined diffeomorphism differentiable mapping dimension dimensional distance eigenvalue element ellipse ellipsoid equation exists expp fibre finite flow line follows geodesic c(t geodesic flow given Hence Hilbert space homeomorphic homotopy hyperbolic integral isometry isomorphism Jacobi field Jacobi field Y(t Klingenberg Lemma linear minimizing geodesic morphism non-degenerate open neighborhood orbit orientation orthogonal parameter parameterized Proof Proposition prove Remark Riemannian manifold Riemannian metric scalar product sectional curvature sequence simple closed geodesics simply connected sphere submanifold subset subspace sufficiently small surface symmetric space symplectic tangent bundle tangent space tangent vectors tensor Theorem topological totally geodesic vector bundle