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AuthorLee, John M. author
TitleRiemannian Manifolds [electronic resource] : An Introduction to Curvature / by John M. Lee
ImprintNew York, NY : Springer New York : Imprint: Springer, 1997
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Descript XV, 226 p. online resource


Thisbookisdesignedasatextbookforaone-quarterorone-semestergr- uate course on Riemannian geometry, for students who are familiar with topological and di?erentiable manifolds. It focuses on developing an in- mate acquaintance with the geometric meaning of curvature. In so doing, it introduces and demonstrates the uses of all the main technical tools needed for a careful study of Riemannian manifolds. I have selected a set of topics that can reasonably be covered in ten to ?fteen weeks, instead of making any attempt to provide an encyclopedic treatment of the subject. The book begins with a careful treatment of the machineryofmetrics,connections,andgeodesics,withoutwhichonecannot claim to be doing Riemannian geometry. It then introduces the Riemann curvature tensor, and quickly moves on to submanifold theory in order to give the curvature tensor a concrete quantitative interpretation. From then on, all e?orts are bent toward proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet theorem (expressing thetotalcurvatureofasurfaceintermsofitstopologicaltype),theCartan- Hadamard theorem (restricting the topology of manifolds of nonpositive curvature), Bonnet's theorem (giving analogous restrictions on manifolds of strictly positive curvature), and a special case of the Cartan-Ambrose- Hicks theorem (characterizing manifolds of constant curvature). Many other results and techniques might reasonably claim a place in an introductory Riemannian geometry course, but could not be included due to time constraints


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

Mathematics Differential geometry Mathematics Differential Geometry


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