UPC: 9783319917542 | Graduate Texts in Mathematics: Introduction to Riemannian Manifolds (Hardcover)
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UPC lookup results for: 9783319917542 | Graduate Texts in Mathematics: Introduction to Riemannian Manifolds (Hardcover)
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.
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