A Comprehensive Introduction to Differential Geometry, Vol. by Michael Spivak

By Michael Spivak

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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.

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