We leave the details to the reader. Then R,,, restricted to S2 is a differentiable map of S2 cf. Example 3 of Sec. In retrospect, what we have been doing up to now is extending the notions of differential calculus in R2 to regular surfaces. Since calculus is essentially a local theory, we defined an entity the regular surface which locally was a plane, up to diffeomorphisms, and this extension then became natural.
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The most important property of conformal maps is given by the fallowing theorem, which we shall not prove. Any two regular surfaces are locally conformal. Such a coordinate system is called isothermal. Once the existence of an isothermal coordinate system of a regular surface S is assumed, S is clearly locally conformal to a plane, and by composition locally conformal to any other surface.
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.
Remark 3. Isothermal pararnetrizations already appeared in Chap. Prove that F is a local diffeomorphism of U onto a cone C with the vertex at the origin and 2a as the angle of the vertex. Is F a local isometry? Use the stereographic projection cf. Exercise 16, Sec. Let be their regular tangent surfaces cf. Example 5, Sec. Prop, 2, Sec. Let x t, v be its tangent surface. If vl,. There exists an orthonormal basis v,,. If any of these conditions is satisfied, F is called a linear isometry of V into W.
Let G: R3 - R3 be a map such that that is, G is a distance-preserving map. Prove that there exists po E R3 and a linear isometry cf. Let S , , S,, and S, be regular surfaces. Prove that a. This implies that the isometries of a regular surface S constitute in a natural way a group, called the group of isometries of S.
Let S be a surface of revolution. Prove that the rotations about its axis are isornetries of S. Prove that the restric- tion of F to S is an isometry of S.
Construct an isometry p : C 3 C such that the set of fixed points of p, i. Let V and W be finite-dimensional vector spaces with inner products ,.
Let G: V --t W be a linear map. Prove that the following conditions are equivalent : a. If any of these conditions is satisfied, G is called a linear conformal map or a similitude.
We say that a differentiable map p : S1 --, S2 preserves angles when for every p E SI and every pair vl , v, E Tp S1 we have Prove that is locally conformal if and only if it preserves angles. Consider a triangle on the unit sphere so that its sides are made up of segments of Ioxodromes i. Example 4, Sec. Prove that the sum of the interior angles of such a triangle is n. Prove that if p is area-preserving and conformal, then q is an isometry. Let be a map defined as follows.
For each p E My the line passing through p and perpendicular to Oz meets Oz at the point q. Let 1 be the half-line starting from q and containingp Fig. Prove that p is an area-preserving diffeomorphism.
Manfredo do Carmo
Manfredo Do Carmo - Differential Geometry of Curves and Surfaces (1976)