Elementary Topics in Differential Geometry

In the past decade there has been a significant change in the freshman/ sophomore mathematics curriculum as taught at many, if not most, of our colleges. This has been brought about by the introduction of linear algebra into the curriculum at the sophomore level. The advantages of using linear algebra both in the teaching of differential equations and in the teaching of multivariate calculus are by now widely recognized. Several textbooks adopting this point of view are now available and have been widely adopted. Students completing the sophomore year now have a fair preliminary under­ standing of spaces of many dimensions. It should be apparent that courses on the junior level should draw upon and reinforce the concepts and skills learned during the previous year. Unfortunately, in differential geometry at least, this is usually not the case. Textbooks directed to students at this level generally restrict attention to 2-dimensional surfaces in 3-space rather than to surfaces of arbitrary dimension. Although most of the recent books do use linear algebra, it is only the algebra of ~3. The student's preliminary understanding of higher dimensions is not cultivated.

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Contenu

I Graphs and Level Sets.- 2 Vector Fields.- 3 The Tangent Space.- 4 Surfaces.- 5 Vector Fields on Surfaces; Orientation.- 6 The Gauss Map.- 7 Geodesics.- 8 Parallel Transport.- 9 The Weingarten Map.- 10 Curvature of Plane Curves.- 11 Arc Length and Line Integrals.- 12 Curvature of Surfaces.- 13 Convex Surfaces.- 14 Parametrized Surfaces.- 15 Local Equivalence of Surfaces and Parametrized Surfaces.- 16 Focal Points.- 17 Surface Area and Volume.- 18 Minimal Surfaces.- 19 The Exponential Map.- 20 Surfaces with Boundary.- 21 The Gauss-Bonnet Theorem.- 22 Rigid Motions and Congruence.- 23 Isometries.- 24 Riemannian Metrics.- Notational Index.