MTH 255 Differential Geometry I Fall 2009 by sparkunder16

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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




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                   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 .




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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 ?




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