A Course in Differential GeometryThis English edition could serve as a text for a first year graduate course on differential geometry, as did for a long time the Chicago Notes of Chern mentioned in the Preface to the German Edition. Suitable references for ordin ary differential equations are Hurewicz, W. Lectures on ordinary differential equations. MIT Press, Cambridge, Mass., 1958, and for the topology of surfaces: Massey, Algebraic Topology, Springer-Verlag, New York, 1977. Upon David Hoffman fell the difficult task of transforming the tightly constructed German text into one which would mesh well with the more relaxed format of the Graduate Texts in Mathematics series. There are some e1aborations and several new figures have been added. I trust that the merits of the German edition have survived whereas at the same time the efforts of David helped to elucidate the general conception of the Course where we tried to put Geometry before Formalism without giving up mathematical rigour. 1 wish to thank David for his work and his enthusiasm during the whole period of our collaboration. At the same time I would like to commend the editors of Springer-Verlag for their patience and good advice. Bonn Wilhelm Klingenberg June,1977 vii From the Preface to the German Edition This book has its origins in a one-semester course in differential geometry which 1 have given many times at Gottingen, Mainz, and Bonn. |
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bijective c(to change of variables circle closed geodesic compact surface conjugate points consider convex coordinate system curvature functions curve c(t defined denote developable surface diffeomorphism differentiable function differentiable mapping differential equations du² e₁(t equal equivalent Euclidean example exists follows Frenet frame Gauss curvature geodesic coordinates given GL(n implies inner product integral isometry Jacobi field K₁ Lemma Let c(t line element linear map linearly independent manifold Math matrix minimal surfaces neighborhood open set oriented surface orthogonal osculating p₁ parallel translation parameterized plane curve polygon principal curvature principal direction PROOF Proposition prove Remark Riemannian metric satisfies second fundamental form sphere Suppose surface f surface of revolution surface with Riemannian t₁ tangent space tangent vector tangential vector field theorem topology torus unique vector field X₁