Riemannian Manifolds and Homogeneous Geodesics (Springer Monographs in Mathematics)

This book is devoted to Killing vector fields and the one-parameter isometry groups of Riemannian manifolds generated by them. It also provides a detailed introduction to homogeneous geodesics, that is, geodesics that are integral curves of Killing vector fields, presenting both classical and modern results, some very recent, many of which are due to the authors. The main focus is on the class of Riemannian manifolds with homogeneous geodesics and on some of its important subclasses. To keep the exposition self-contained the book also includes useful general results not only on geodesic orbit manifolds, but also on smooth and Riemannian manifolds, Lie groups and Lie algebras, homogeneous Riemannian manifolds, and compact homogeneous Riemannian spaces. The intended audience is graduate students and researchers whose work involves differential geometry and transformation groups. Introduction Contents Chapter 1 Riemannian Manifolds 1.1 Fundamentals of the Theory of Smooth Manifolds 1.1.1 Smooth Manifolds 1.1.2 The Tangent Bundle over a Smooth Manifold 1.1.3 Smooth Vector Fields on M as Derivations of the Ring C^∞(M) 1.1.4 The Lie Commutator of Two Smooth Vector Fields 1.1.5 Tensor Fields on Smooth Manifolds 1.2 Manifolds with Connections and Riemannian Manifolds 1.2.1 Covariant Derivative 1.2.2 Riemannian Metrics 1.2.3 The Levi-Civita Connection on a Riemannian Manifold 1.3 Curvature of Manifolds with Connection and Riemannian Manifolds 1.3.1 The Curvature Tensor of a Manifold with a Covariant Derivative 1.3.2 The Curvature Tensor of a Riemannian Manifold 1.3.3 The Ricci Tensor of a Manifold with a Covariant Derivative 1.3.4 The Ricci Tensor of a Riemannian Manifold 1.3.5 Curvatures of Riemannian Manifolds 1.4 Fundamentals of the Geometry of Riemannian Manifolds 1.4.1 Parametrization of a Curve by the Arc Length 1.4.2 The First Variation of the Length of a Curve 1.4.3 Geodesics 1.4.4 The Geodesic Flow 1.4.5 The Exponential Map 1.4.6 The Shortest Curves 1.4.7 The Intrinsic Metric of a Riemannian Manifold 1.5 Some Global Properties of Riemannian Manifolds 1.6 Riemannian Submersions and the O'Neill Formulas 1.7 Miscellaneous Chapter 2 Lie Groups and Lie Algebras 2.1 Main Properties of Lie Groups 2.2 Smooth Actions of Lie Groups 2.3 Main Properties of Lie Algebras 2.3.1 Connections with Matrix Lie Algebras 2.3.2 Important Classes of Lie Algebras and Structural Results 2.3.3 Automorphisms and Derivations 2.4 Lie Groups with Left-invariant Riemannian Metrics 2.4.1 The Levi-Civita Connection on a Lie Group with a Left-invariant Riemannian Metric 2.4.2 Curvatures of a Lie Group with a Left-invariant Riemannian Metric 2.4.3 Curvature of Nilpotent Group-manifolds 2.4.4 Curvatures of Bi-invariant Riemannian Metrics on Lie Groups 2.5 Compact Lie Algebras and Lie Groups 2.5.1 Compact Lie Algebras 2.5.2 Cartan Subalgebras of Compact Lie Algebras 2.5.3 Root Systems and the Weyl Groups 2.5.4 Abstract Root Systems 2.6 Connections with Complex Lie Algebras 2.7 Invariant Norms and Inner Products on Lie Algebras Chapter 3 Isometric Flows and Killing Vector Fields on Riemannian Manifolds 3.1 Isometries and Killing Vector Fields on Riemannian Manifolds 3.2 The Curvature Tensor and Killing Vector Fields 3.3 Killing Vector Fields of Constant Length 3.4 Killing Vector Fields of Constant Length and Clifford–Wolf Translations 3.5 Killing Vector Fields and Curvatures 3.6 Isometric Flows and Points with Finite Period 3.7 Regular and Quasiregular Killing Vector Fields 3.7.1 Orbits of Isometric S1-actions on Riemannian Manifolds 3.7.2 Flows on Simply Connected Manifolds 3.7.3 Geodesic Flows and the Sasaki Metric on Tangent Bundles of Riemannian Manifolds 3.8 Discrete Subgroups of Lie Groups 3.9 Open Questions Chapter 4 Homogeneous Riemannian Manifolds 4.1 The Isometry Group of a Riemannian Manifold 4.2 Homogeneous Spaces 4.3 On the Topology of Compact Homogeneous Spaces 4.4 General Results on Reductive Decompositions 4.5 Invariant Affine Connections on Reductive Homogeneous Spaces 4.6 Main Properties of Homogeneous Riemannian Manifolds 4.7 Symmetric and Locally Symmetric Spaces 4.8 Killing Vector Fields of Constant Length on Locally Symmetric Manifolds 4.8.1 Symmetric Spaces 4.8.2 Non-simply Connected Manifolds 4.8.3 Locally Euclidean Spaces 4.8.4 Homogeneous Spherical Space Forms 4.9 Infinitesimal Structure of a Homogeneous Riemannian Space 4.10 Special Invariant Metrics 4.11 The Structure of the Set of Invariant Metrics 4.12 Compact Homogeneous Riemannian Manifolds of Positive Euler Characteristic 4.13 Some Special Compact Homogeneous Spaces 4.13.1 Aloff–Wallach Spaces 4.13.2 Generalized Wallach Spaces 4.13.3 Ledger–Obata Spaces 4.14 Curvatures of Homogeneous Riemannian Spaces 4.15 Homogeneous Riemannian Manifolds with Restrictions on Curvatures 4.15.1 Homogeneous Riemannian Spaces of Positive Sectional Curvature 4.15.2 Homogeneous Riemannian Spaces of Positive Ricci Curvature 4.15.3 Homogeneous Riemannian Spaces of Non-positive Sectional Curvature 4.15.4 Homogeneous Riemannian Spaces of Non-positive Ricci Curvature 4.16 Signature of the Ricci Curvature on Homogeneous Spaces 4.16.1 Signature of the Ricci Operator on Lie Groups 4.16.2 On Two Negative Eigenvalues of the Ricci Operator 4.16.3 Lie Groups with Metrics of Negative Ricci Curvature 4.17 Homogeneous Riemannian Spaces with Killing Vector Fields of Constant Length 4.17.1 Examples of Killing Vector Fields of Constant Length 4.17.2 Additional Properties of Killing Vector Fields of Constant Length 4.17.3 Unsolved Questions Chapter 5 Manifolds With Homogeneous Geodesics 5.1 Homogeneous Geodesics on Riemannian Manifolds 5.1.1 Homogeneity Criterion for a Geodesic 5.1.2 On the Closure of a Homogeneous Geodesic 5.1.3 Examples of Homogeneous Geodesics 5.1.4 A Special Quadratic Mapping 5.2 Geodesic Orbit Spaces 5.3 Important Subclasses of Geodesic Orbit Spaces 5.4 On the Radical and the Nilradical of the Lie Algebra of the Isometry Group of a Geodesic Orbit Space 5.5 Killing Vector Fields of Constant Length on GO-spaces 5.6 Geodesic Orbit Manifolds of Non-positive Ricci Curvature 5.7 Compact Geodesic Orbit Spaces 5.8 Applications of the Totally Geodesic Property 5.9 Geodesic Orbit Riemannian Metrics on Spheres 5.9.1 On Invariant Metrics on S^4n+3 for Transitive Actions of the Groups Sp(n+1), Sp(n+1)U(1), and Sp(n+1)Sp(1) 5.9.2 Sp(n+1)-invariant Geodesic Orbit Metrics on the Sphere S^4n+3 5.10 Compact GO-spaces of Positive Euler Characteristic 5.10.1 Basic Facts on Compact Homogeneous Manifolds of Positive Euler Characteristic 5.10.2 The Classification Theorem 5.10.3 Proof of the Classification Theorem 5.11 GO-spaces and Representations with Non-trivial Principal Isotropy Algebras 5.12 Miscellaneous 5.12.1 Compact GO-spaces with Two Isotropy Summands 5.12.2 Geodesic Orbit Metrics on Ledger–Obata Spaces 5.12.3 GO-spaces Fibered over Irreducible Symmetric Spaces 5.12.4 Geodesic Orbit Riemannian Structures on Rn 5.12.5 On Left-invariant Einstein Riemannian Metrics that are not Geodesic Orbit 5.12.6 Various Results on Homogeneous Geodesics and GO-spaces 5.12.7 Open Questions Chapter 6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics 6.1 δ-homogeneous and Clifford–Wolf Homogeneous Spaces 6.2 δ-homogeneous and Restrictively Clifford–Wolf Homogeneous Riemannian Manifolds 6.3 Some Structural Results 6.4 Generalized Normal Homogeneous and δ-homogeneous Riemannian Manifolds 6.5 Additional Symmetries of δ-homogeneous Metrics 6.6 Totally Geodesic Submanifolds 6.7 Properties of -vectors 6.7.1 The Case of Positive Euler Characteristic 6.8 Generalized Normal Homogeneous Manifolds of One Special Type 6.9 Classification of Generalized Normal Homogeneous Riemannian Manifolds of Positive Euler Characteristic 6.9.1 General Ideas and Constructions 6.9.2 The Spaces SO(2l+1)/U(l), l>=3 6.9.3 The Spaces Sp(l)/U(1).Sp(l-1), l>=2 6.10 Generalized Normal Homogeneous Metrics and 2-means in the Sense of W.J. Firey 6.11 Generalized Normal Homogeneous Riemannian Metrics on Spheres and Projective Spaces 6.11.1 Homogeneous Riemannian Manifolds Being Investigated 6.11.2 G-generalized Normal Homogeneous Metrics on Spheres 6.11.3 Spin(9)-generalized Normal Homogeneous Metrics on the Sphere S^15 6.12 The Chebyshev Norm on the Lie Algebra of the Isometry Group of a Compact Homogeneous Finsler Manifold 6.13 Calculations of the Chebyshev Norms for Some Manifolds 6.13.1 On the Löwner–John Ellipsoids 6.13.2 The Chebyshev Norm for Euclidean Spheres 6.13.3 The Chebyshev Norm for the Berger Spheres 6.13.4 The Chebyshev Norms for Invariant Metrics on SO(5)/U(2) 6.14 Almost Normal Homogeneous Riemannian Manifolds Chapter 7 Clifford–Wolf Homogeneous Riemannian Manifolds 7.1 Preliminary Results 7.2 Once More about Killing Vector Fields of Constant Length 7.3 The Proof of the Main Result 7.4 Clifford–Killing Spaces 7.5 Riemannian Manifolds with the Killing Property 7.6 Results by A. Hurwitz and J. Radon 7.7 Clifford–Killing Spaces on Spheres and Clifford Algebras and Modules 7.8 Clifford–Killing Spaces for S^2n-1 and Unit Radon Spheres in O(2n) 7.9 Triple Lie Systems in so(2n) and Totally Geodesic Spheres in SO(2n) 7.10 Lie Algebras in Clifford–Killing Spaces for S^2n-1 7.11 Killing Vector Fields of Constant Length and G-Clifford–Wolf Homogeneous Metrics on Round Spheres 7.11.1 Descriptions of Killing Vector Fields of Constant Length and G-Clifford–Wolf Homogeneous Metrics 7.11.2 The Spaces of Unit Killing Fields on Spheres 7.11.3 Unit Killing Fields on the Sphere S^15=Spin(9)/Spin(7) 7.12 Restrictively Clifford–Wolf Homogeneous Riemannian Manifolds References List of tables Symbol Index Subject Index