By Wilhelm Klingenberg (auth.)

This English variation may perhaps 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 usual 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 tricky job of remodeling the tightly built German textual content into one that could mesh good with the extra comfortable layout of the Graduate Texts in arithmetic sequence. There are a few e1aborations and a number of other new figures were additional. I belief that the benefits of the German version have survived while even as the efforts of David helped to explain the final perception of the direction the place we attempted to place Geometry prior to Formalism with out giving up mathematical rigour. 1 desire to thank David for his paintings and his enthusiasm through the entire interval of our collaboration. whilst i need to commend the editors of Springer-Verlag for his or her persistence and strong recommendation. Bonn Wilhelm Klingenberg June,1977 vii From the Preface to the German variation This publication 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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P: V -+ U be a change of variables and let! p. Then lv(X, Y) = I,,(vlX, Y) for ali X, Y E Tv! = T,,(vli 37 3 Surfaces: Local Theory and l v(2, Y) = 14J(vMi>2, di>Y) for ali X, Ye T v1R2. i) Suppose Bx = Rx + xc, where R is an orthogonal map. Then dB = R. Therefore, if X, YeT,J, l u(RX,RY) = RX·RY = x·y = Iu(X, Y). "y = dfu o di>2'd/u o di>Y = Iu(di>2, di>Y), where u = i>(v). O PROOF. 6 Coro)]ary. Suppose the change of variables i> is given, in terms of coordinates, by ul = UI(V1 , v2 ), i = 1,2.

But c;(s) = X,(c,(s)), so these conditions are satisfied by hypothesis. AIso, s = v'(c;(s)), i #- j, implies that 1= Therefore dv' #- 0, i ~ (v'(c,(s)) = = dv'(C;(s)) = dv'(X,(c(s)). 1, 2. Remarks. i) The function vl(u) (resp. v2(u)) is an integral of the differential equation c(s) = Xl{c{s)) (resp. c(s) = Xz{c(s))). An integral of a differential equation (*) i(s) = I(x(s), s), XE U, is a differentiable function h: U -+ IR which is nonconstant on any open set and which is constant on integral curves of (*).

I) Letf: U -+ 1R3 be a surface. Let U E U. The inner product on 1R3 ;:;: T f (u)1R3 induces a quadratic form on Tuf c T/(U)1R3 ;:;: 1R3 by restriction. This form is called thefirstfundamentalform and is denoted sometimes by g or gu and sometimes by I or lu. ii) The inner product on Tf(U)1R3 ;:;: 1R3 composed with the linear map dfu: 1R2 ;:;: T uIR 2-+ T f (u)1R3 ;:;: 1R3 induces a quadratic form on T uIR 2 which is also called the first fundamental form. It is also denoted by g or 1, and it will sometimes be written "df· df" Remark.