By Wilhelm Klingenberg (auth.)

ISBN-10: 1461299233

ISBN-13: 9781461299233

ISBN-10: 146129925X

ISBN-13: 9781461299257

This English variation may 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 variation. compatible 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, ny, 1977. Upon David Hoffman fell the tough job of reworking the tightly built German textual content into one that could mesh good with the extra cozy layout of the Graduate Texts in arithmetic sequence. There are a few e1aborations and several other new figures were extra. I belief that the advantages of the German version have survived while even as the efforts of David helped to explain the overall belief of the direction the place we attempted to place Geometry prior to 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. whilst i need to commend the editors of Springer-Verlag for his or her persistence and sturdy recommendation. Bonn Wilhelm Klingenberg June,1977 vii From the Preface to the German variation This publication has its origins in a one-semester path in differential geometry which 1 have given repeatedly at Gottingen, Mainz, and Bonn.

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**Example text**

1976, p. B. This is in general not an orthonormal frame. 1 A quick review of quadratic forms 1. Let T be a real vector-space. A symmetric bilinear form or a quadratic form is a map {J: T x T --+ IR satisfying (J(X, Y) = (J(Y, X) (J(aX + bY,Z) = a{J(X,Z) + b{J(Y,Z) Here a, b E IR and X, Y, Z E (symmetry) (bilinearity). T. {J is positive definite if X =1- O => (J(X, X) > O. Example: The standard inner product in Euclidean space IRn. 35 3 Surfaces: Local Theory 2. The matrix representation of {:J with respect to a basis e" 1 is the matrix S; i s; n, of T (glj) : = (f:J(eh ei»~' If X = Lf ~ef' Y = L' 'TJ'e" then (:J(X, Y) = Lf" ~'TJfgf" Suppose fk, 1 s; k s; n, is another basis of T.

Then there is a change olvariables : Vo -+ Uo c U near Uo with (0) = Uo with the lollowing properties: il/ = 1 , 0 /(v) - /(0) = V1 Jf Xl = h(uo), then Vi = PROOF. + V 2X 2 + r(v)no, X1 ul uh - + v = (vi, v2). o(lu - uol) and rvv(O) = hlluo). Since {Xl> X 2 , n} forms a basis in T f (u)1R3, we may write I(u) - I(uo) + V 2(U)X2 + q(u)no = V 1(U)Xl for some functions vl(u), q(u) with vl(uo) = q(uo) = o. The first order of business is to find an inverse for v = (v 1(u), v2(u». I(uo) ov L OUl (uo)X k = k, k «ovkjoUI)(UO» is an invertible matrix.

13 Horn, R. A. On Fenchel's theorem. Amer. Math. Monthly, 78, 380-381 (1971). 1 Definitions. i) U will always denote an open set in 1R2. Points of U will be denoted by u E 1R2, or by (u l , u2 ) E IR x IR or (u, v) E IR x IR. ii) A differentiable mapping f: U ~ 1R3 such that dfu: Tu IR2 ~ T f (u)1R3 is injective for alI u E U is a (parameterized) surface patch, or simply a surface. A mapping f satisfying this condition is called regular. The u E U are called parameters of f iii) The two-dimensionallinear subspace dfu(IR~) c T f (u)1R3 is called the tangent space of f al u, and will be denoted by Tuf Elements of Tuf are called tangent vectors (of/at u).

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