# 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
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.

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(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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