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MTH 255 : Diﬀerential Geometry I Fall 2009 Instructor: Ibrahim Unal Final Exam 12/11/2009, Friday Name : I.D. Problems 1 2 3 4 5 6 Total Score 1 MTH 255 : Diﬀerential Geometry I Fall 2009 Take-Home Final Exam Due 12/11/2009, Friday 12:00pm You should do it BY YOURSELF and write your answers clearly. Please write your name and ID number and staple your exam papers with the cover page otherwise it will not be accepted. Problem 1.(20 pts) Consider the set of points S in R3 given by x2 y 2 z= + 2 a2 b (a) Prove that S is a surface. (Hint : Use Implicit Function Theorem) (b) Show that x(u, v) = (au cos v, bu sin v, u2 ), u > 0 is a local chart for S. (c) Set a = 1 and b = 2 and sketch the surface. Sketch on the surface some lines where u = const and v = const, i.e. local coordinate curves on S induced by x (d) Show that S is diﬀeomorphic to the sphere without north pole for any a > 0 and b > 0. Problem 2.(25 pts) Let M be the surface deﬁned by z = x2 + 3xy − 5y 2 (a) Show that at p = (0, 0, 0) the unit vectors u1 = (1, 0, 0) and u2 = (0, 1, 0) deﬁne a basis for the tangent space to M (b)Show that α1 (t) = (t, 0, t2 ) α2 = (0, t, −5t2 ) deﬁne paths on M , passing through p and with tangents u1 and u2 respectively at p. 2 (c)Find a unit normal vector ﬁeld U deﬁned on a neighborhood of p on M . d) Compute the covariant derivatives u1 U and u2 U where U is the unit normal vector computed in part (c) and u1 and u2 are the tangent vectors in part(a). (e) Compute the Shape operator at p with respect to the base given in (a). (f ) Compute the Gaussian curvature and the mean curvature at p. Problem 3.(10 pts) (a) For any compact surface S in R3 , prove that there must exist a point x0 on S such that K(x0 ) > 0 Hint: Choose the smallest r such that the sphere S 2 (r) of radius r centered at the origin contains the surface S, and let x0 be the point where S 2 (r) touches S. Then argue that at this point K(x0 ) > 0 (b)(5 pts) Prove that there is no compact minimal surface in R3 . Problem 4.(15 pts) Let M be a surface in R3 oriented by a unit normal vector ﬁeld U = g1 U1 + g2 U2 + g3 U3 Then the Gauss map G : M −→ Σ of M sends each point p of M to the point (g1 (p), g2 (p), g3 (p)) of the unit sphere Σ. (Pictorially : Move U (p) to the origin by parallel motion; there it points to G(p). Thus G completely describes the turning of U as it traverses M . (a) For each of the following surfaces, describe the image G(M ) of the Gauss map in the sphere Σ (i) Cylinder, x2 + y 2 = r2 (ii) Cone, z = x2 + y 2 (iii) Plane, x + y + z = 0 (iv) Sphere, (x − 1)2 + y 2 + (z + 2)2 = 1 (b) Show that the shape operator of M is (minus) the tangent map of its Gauss map: If S and G : M −→ Σ are both derived from U , then S(v) and −G∗ (v) are parallel for every tangent vector v to M . 3 Problem 5.(20 pts) A mapping of surfaces F : M −→ N is conformal provided that there exists a real valued function λ > 0 on M such that ||F∗ (vp )|| = λ(p)||vp || for all vp ∈ Tp M .If λ(p) = 1, it is called a local isometry. Let F : M −→ N be a conformal mapping. (a) Show that for all vp , wp ∈ Tp M , F∗ (vp ) · F∗ (wp ) = λ(p)vp · wp . (b) Prove that a conformal mapping preserves angles in the following sense: If θ is an angle between v and w at p ∈ M then θ is an angle of F∗ (v) and F∗ (w) at F (p). (c) Show that the stereographic projection P : Σ0 −→ R2 is conformal with scale factor P (p) λ(p) = 1 + 4 . (d) Prove that a regular mapping F = (f, g) : R2 −→ R2 is conformal and det(J(F )) > 0 if and only if the Cauchy-Riemann equations ∂f ∂g ∂f ∂g ∂u = ∂v and ∂v = − ∂u hold. Problem 6.(15 pts) Let α be a unit-speed curve in M ⊂ R3 . If E1 , E2 , E3 is an adapted frame ﬁeld such that E1 restricted to α is the unit tangent T . (a)α is a geodesic of M if and only if ω12 (T ) = 0 b) If F : M −→ N is a local isometry, F will map geodesics to geodesics. (Hint : Use Lemma 5.3 in Section 6.5 without proving.) c) Show that unit-speed parametrization of a meridian on the sphere is a geodesic. Is this true for parallels, too ? 4