An Introduction to
                Riemannian Geometry
                Weincludeinthesenotesapresentationofthebasicsofdifferentialgeometry
                with a view to Riemannian geometry. We refer the reader to the classics on
                the subject for a more comprehensive and careful treatment [2, 3, 4, 5, 7].
                In addition to these texts, our exposition has benefited from the book [1]
                and lecture notes by Ben Andrews.
                Mat Langford
                Knoxville, December 2019       Last updated September, 2022
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                        Contents
                        An Introduction to Riemannian Geometry                             1
                           §1. Paracompactness, partitions of unity, and manifolds         5
                           §2. Differentiable manifolds                                    13
                           §3. The tangent space and tangent maps                         19
                           §4. Some differential topology                                  29
                           §5. The tensor algebra of a linear space                       35
                           §6. The tangent bundle and its tensor algebra                  47
                           §7. The Lie derivative and Lie algebras                        57
                           §8. Frobenius’ theorem                                         65
                           §9. Differential forms and the exterior calculus                67
                           §10. Orientability, integration, and Stokes’ Theorem           73
                           §11. Connections                                               77
                           §12. Geodesics and the exponential map                         89
                           §13. Torsion and curvature                                     93
                           §14. Riemannian metrics                                        99
                           §15. Convexity and completeness                               109
                           §16. Riemannian curvature                                     117
                           §17. Spaces of constant sectional curvature                   125
                           §18. Riemannian submanifolds                                  131
                           §19. First and second variations of arc-length                137
                           §20. Elementary comparison theorems                           147
                           §21. The cut locus and the injectivity radius                 157
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                                             ANINTRODUCTIONTORIEMANNIANGEOMETRY
                              §22.   Distance comparison                                             161
                              §23.   Integration on Riemannian manifolds                             165
                            Bibliography                                                             169
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