By Wilhelm Klingenberg (auth.)
This English variation may perhaps function a textual content for a primary 12 months 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 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 tricky activity of remodeling the tightly built German textual content into one that could mesh good with the extra comfortable structure 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 overall notion of the direction 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 throughout the entire interval of our collaboration. even as i need to commend the editors of Springer-Verlag for his or her persistence and sturdy suggestion. 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 again and again at Gottingen, Mainz, and Bonn.
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Proof of corollary. 6). 1 Definition. A regular plane curve c: I ~ 1R2 is convex if, for all to EI, the curve lies entirely on one side of the tangent at c(to). 2 Theorem (A characterization of convex curves). Let c: I ~ 1R2 be a simple closed regular plane curve. Then c is convex if and only if one of the following conditions are true: Ie(t) ~ 0, alltEI or Ie(t) :s; 0, ali tEl. Remarks. i) If one of the above conditions hold then an orientation-reversing change of variables will produce the other.
Osserman, R. Isoperimetric and related inequalities. Proc. AMS Symp. in Pure and Applied Math. XXVII, Part 1, 207-215. ,. Dubins, L. E. On curves of minimal length with constraint an average curvature and prescribed initial and terminal positions and tangents. 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].
I«to) = O, to Ei. If I<(t) = const, tI :<;; t :<;; t 2, alI these tare vertices. 4 Theorem (Four vertex theorem). A convex, simple, c/osed smooth plane curve has at least four vertices. Remark. The theorem is true without the convexity hypothesis (although it is harder to prove). 4 Exercises and Some Further Results (due to G. Herglotz)2 Step 1. Since K(t) has a maximum and a minimum on 1, c(t) has at least two vertices. Without loss of generality, we may as sume that c is parameterized by arc length and that K(t) has a minimum at t = Oand a maximum at to, O < to < w, where I = [O, w].
A Course in Differential Geometry by Wilhelm Klingenberg (auth.)