Riemannian Manifolds: An Introduction to Curvature

What Is Curvature?.- Review of Tensors, Manifolds, and Vector Bundles.- Definitions and Examples of Riemannian Metrics.- Connections.- Riemannian Geodesics.- Geodesics and Distance.- Curvature.- Riemannian Submanifolds.- The Gauss-Bonnet Theorem.- Jacobi Fields.- Curvature and Topology.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  C. Gauss,et al.  General investigations of curved surfaces , 1902 .

[3]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[4]  H. Hopf,et al.  Über die Drehung der Tangenten und Sehnen ebener Kurven , 1935 .

[5]  H. Rauch.  On Differential Geometry in the Large. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[6]  J. Nash.  The imbedding problem for Riemannian manifolds , 1956 .

[7]  P. Hartman.  Ordinary Differential Equations , 1965 .

[8]  J. C. Burkill,et al.  Ordinary Differential Equations , 1964 .

[9]  K. Nomizu,et al.  Foundations of Differential Geometry, Volume I. , 1965 .

[10]  George F. R. Ellis,et al.  The Large Scale Structure of Space-Time , 2023 .

[11]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[12]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[13]  William S. Massey,et al.  Algebraic Topology: An Introduction , 1977 .

[14]  H. Lawson.  Lectures On Minimal Submanifolds , 1980 .

[15]  R. Hamilton.  Three-manifolds with positive Ricci curvature , 1982 .

[16]  Joseph A. Wolf.  Spaces of Constant Curvature , 1984 .

[17]  J. Kazdan,et al.  Prescribing the Curvature of a Riemannian Manifold , 1985 .

[18]  Thomas H. Parker,et al.  The Yamabe problem , 1987 .

[19]  A. J. Sieradski.  An Introduction to Topology and Homotopy , 1991 .

[20]  B. M. Fulk.  MATH , 1992 .

[21]  P. Pflug,et al.  Invariant Distances and Metrics in Complex Analysis , 1993 .

[22]  Albert Schwarz,et al.  Elements of Homotopy Theory , 1993 .

[23]  J. Eschenburg.  Comparison Theorems in Riemannian Geometry , 1994 .

[24]  M. Abate,et al.  Finsler Metrics - A Global Approach: with Applications to Geometric Function Theory , 1994 .

[25]  I. Chavel.  Riemannian Geometry: Subject Index , 2006 .

[26]  Shiing-Shen Chern,et al.  Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction , 1997 .

[27]  Frank Morgan,et al.  Riemannian geometry : a beginner's guide , 1998 .

[28]  Frédéric Jean,et al.  Sub-Riemannian Geometry , 2022 .