By Wilhelm Klingenberg (auth.)

This English version may function a textual content for a primary yr graduate direction 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 usual differential equations. MIT Press, Cambridge, Mass., 1958, and for the topology of surfaces: Massey, Algebraic Topology, Springer-Verlag, long island, 1977. Upon David Hoffman fell the tricky job of reworking the tightly developed German textual content into one that might mesh good with the extra secure layout of the Graduate Texts in arithmetic sequence. There are a few e1aborations and a number of other new figures were further. I belief that the benefits of the German version have survived while whilst the efforts of David helped to explain the overall notion of the direction the place we attempted to place Geometry sooner than Formalism with no giving up mathematical rigour. 1 desire to thank David for his paintings and his enthusiasm throughout the entire interval of our collaboration. while i need to commend the editors of Springer-Verlag for his or her endurance and strong recommendation. Bonn Wilhelm Klingenberg June,1977 vii From the Preface to the German variation This e-book has its origins in a one-semester direction in differential geometry which 1 have given time and again at Gottingen, Mainz, and Bonn.

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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.

Amer. J. , 79, 497-516(1957). 11 Fenchel, W. Ober KrUmmung und Wendung geschlossener Raumkurven. Math. Ann. 101, 238-252 (1929). Ce. also Fenchel, W. On the differential geometry of c10sed space curves. Bull. Amer. Math. , 57, 44-54 (1951), ar Chem [A5]. 12 Fary, 1. Sur las courbure totale d'une courbe gauche faisant un noeud. Bull. Soc. Math. France, 77, 128-138 (1949). Milnor, J. On the differential geometry of closed space curves. Ann. , 52, 248-257 (1950). 10 A proof of Fenchel's theorem. , Ic(t)1 = r, tE [0, w].

F o ep. ,. = (fu' x fu') det(~~:). Therefore ii = n o ep, since det(8u'/iJvk) > O. dfdepY = IIuCdep2, depY) with u = ep(v). 7 Examples 1. R. ~ = - = g12 = O, (cos u cos v, cos u sin v, sin u) = -f(u, v) II = -dn·df= df·df= 1. 2. R. gu = b(- sin u cos v, - sin u sin v, cos u) g. = (a + b·cos u)(-sin v, cos v, O) g12 ;: F = gu· gv = O g22;: G = g~ = (a + b·COSU)2 n(u, v) = -(cos u cos v, cos u sin v, sin u). T/2[, n(u, v) = -f(u, v) wherefis as in (1) above. II = -dn·dg = dIdg hl l == L =fu·gu = h12 == M =fu·g.