A Course in Differential Geometry - download pdf or read online

By Wilhelm Klingenberg (auth.)

This English variation might function a textual content for a primary 12 months graduate path on differential geometry, as did for a very long time the Chicago Notes of Chern pointed out within the Preface to the German version. appropriate references for ordin­ ary differential equations are Hurewicz, W. Lectures on traditional differential equations. MIT Press, Cambridge, Mass., 1958, and for the topology of surfaces: Massey, Algebraic Topology, Springer-Verlag, big apple, 1977. Upon David Hoffman fell the tough activity of remodeling the tightly built German textual content into one that may mesh good with the extra comfy layout of the Graduate Texts in arithmetic sequence. There are a few e1aborations and several other new figures were further. I belief that the benefits of the German variation have survived while even as the efforts of David helped to explain the final belief of the path the place we attempted to place Geometry ahead of Formalism with no giving up mathematical rigour. 1 desire to thank David for his paintings and his enthusiasm through the entire interval of our collaboration. whilst i want to commend the editors of Springer-Verlag for his or her endurance and solid suggestion. Bonn Wilhelm Klingenberg June,1977 vii From the Preface to the German version This e-book has its origins in a one-semester direction in differential geometry which 1 have given again and again at Gottingen, Mainz, and Bonn.

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Fi, = O, thenfis called developable. 5 Proposidon. i) On a ruled surface, generators are asymptotic curves. Consequently, K ::; O.

3 Proposition. III is elliptic { parabolic hyperbolic 50 (with IIu. 6 Normal Form for a Surface, Special Coordinates at uo, then the surface represented by the second Taylor polynomial offis an elliPtic paraboloid, { parabolic cylinder, hyperbolic paraboloid. This representation gives us a geometric picture of what the sign of the Gauss curvature means, since its sign is the same as the sign of det II. 7 (a) Elliptic point; (b) hyperbolic point We now turn out attention to finding coordinates on a surface fitted to vector fields that are given in advance.

The third fundamenta! form is denoted by IIIu, III or dn· dn. 6 Proposition. 5)) under congruences of 1R3 and orientation-preserving changes of variables. 39 3 Surfaces: Local Theory i) LetBx = Rx + Xo be a congruence (det R = 1). < = dBfu' = Rfu' , ii = dBn. Therefore, if X, Y E Tuf. PROOF. ;l(dBX)·dBY = -dB(dn o df;l(X»·dBY = -dn odf;lX·Y = IIuCX, Y). ii) Let ep: V ~ U be an orientation-preserving change of variables and! = f o ep. ,. = (fu' x fu') det(~~:). Therefore ii = n o ep, since det(8u'/iJvk) > O.

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