Question

The proof that there exist isothermal coordinate systems on any regular surface is delicate and will not be taken up here. The interested reader may consult L. Bers, Riemann Surfaces, New York University, Institute of Mathe matical Sciences, New York, 1957-1958, pp. 15-35. Remark 3. Isothermal parametrizations already appeared in Chap. 3 in the context of minimal surfaces; cf. Prop. 2 and Exercise 13 of Sec. 3-5 EXERCISES 1. Let F:U R2R3 be given by F(u, v) = (u sin c cos v, u sin a sin v, u cos a), (u, v) E U=[(u, ) e R2; u> 0, aconst. 228 Intrinsic Geometry of Surfaces a. Prove that Fis a local diffeomorphism of U onto a cone C with the vertex at the origin and 2c as the angle of the vertex. b. Is Fa local isometry?

Answer 1

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