PM 365 / AM 333: Elementary Dierential Geometry
Assignment 01; due Friday, 21 January 2011
[1]
Let : I Rn be a regular smooth parametrized curve in Rn .
(a) Suppose the curve does not pass through the origin, and let to be a parameter for which the point

PM 365 / AM 333: Elementary Dierential Geometry
Mid-Term test; SOLUTIONS
[1]
b
Using the fact that (s) is unit speed, we can compute T (s) = (s) = ( a sin( s ), a cos( s ), c ), and
c
c
c
c
s
a
s
a
T (s) = ( c2 cos( c ), c2 sin( c ), 0) = (s)N (s), from w

PMATH 365 Winter 2012
Assignment 6
DUE: at the beginning of class Wednesday 21 March 2012
1. Compute the Cristoel symbols k for the following surfaces and parametrizations:
ij
(a) the plane R2 with parametrization given by the identity;
(b) the cylinder C

PMATH 365 Winter 2012
1. Suppose that a space curve : I R3 is
parametrized by arc-length.
(a) Dene the Frenet frame, the curvature
and the torsion for the curve .
SOLUTION.
T = is the unit tangent vector,
T
is the principal normal vector,
P =
T
B = T P is

PMATH 365 Winter 2012
Assignment 7
DUE: at the beginning of class Wednesday 28 March 2012
1. Consider the surface M given by x2 + y 2 z 2 = 1. Given in terms of coordinates in R3 , we write
r(x, y, z) = x2 + y 2 + z 2 and can express the Gauss map as n(x,

PMATH 365 Winter 2012
Midterm 2
March 9, 2012.
x y
, and let S = det1 (1) be the surface
y z
2
consisting of the matrices with determinant 1. Given the parametrization X(u, v) = (u, v, 1v ) of
u
2
a neighbourhood of the identity matrix (1, 0, 1), we have

PMATH 365 Winter 2012
Assignment 1 (corrected version)
DUE: at the beginning of class Friday 13 January 2012
1. Let X be a vector eld on a curve : I Rn parametrized by arc-length (so | | = 1), that
is a map
X : I Rn
such that X(t) is thought as a vector w

PMATH 365 Winter 2012
Assignment 3
DUE: at the beginning of class Wednesday 25 January 2012
1. Consider the torus S obtained by rotating a circle of radius a in the xz-plane and centered at
(0, 0, b) around the x-axis.
(a) Find a parametrization of S.
(b)

PMATH 365 Winter 2012
Assignment 2
DUE: at the beginning of class Friday 20 January 2012
1. Let : I R3 be a regular parametrized curve. Suppose is smooth (for simplicity). Prove
that
(a) if the curvature function is identically 0, then the image of lies i

PMATH 365 Winter 2012
Assignment 4
DUE: at the beginning of class Monday 13 February 2012
1. Consider, as in assignment 3, the torus S obtained by rotating a circle of radius a in the xz-plane
and centered at (0, 0, b) around the x-axis. The parametrizati

PMATH 365 Winter 2012
Assignment 5
DUE: at the beginning of class Wednesday 7 March 2012
1. Given a map f : S1 S2 between two regular surfaces, we dene dfp : Tp S1 Tf (p) S2 in the
following way: recall that any vector v Tp S1 is the initial velocity (0)

PMATH 365 / AMATH 333 : Elementary Dierential Geometry
Instructor: S. Karigiannis
Mid-term Test: February 14, 2011; 1:30pm - 2:20pm
INSTRUCTIONS:
The test consists of 5 questions worth a total of 45 points.
ALL YOUR WORK MUST BE CONTAINED IN THE BOOKLETS.

PM 365 / AM 333: Elementary Dierential Geometry
Assignment 06; Solutions
[1]
Using the denition III(v1 , v2 ) = I(W (v1 ), W (v2 ) we see that in terms of matrices, we have
v1 T III v2 = (W v1 )T I (W v2 ) = v1 T W T I W v1 ,
so the matrix for III is W T

PM 365 / AM 333: Elementary Dierential Geometry
Assignment 01: SOLUTIONS
[1]
If (to ) is a point on the curve which is closest to the origin, then the distance from the origin
to the variable point (t) achieves a minimum at t = to . Hence the distance squ

PM 365 / AM 333: Elementary Dierential Geometry
Assignment 02; due Wednesday, 02 February 2011
[1]
Let (t) = (x(t), y (t) be a regular parametrized curve in R2 .
(a) Show that the curvature (t) is given by
(t) =
|x (t)y (t) y (t)x (t)|
3
(x (t)2 + y (t)2

PM 365 / AM 333: Elementary Dierential Geometry
Assignment 02: SOLUTIONS
[1]
We can think of (t) as being a space curve by taking the third coordinate to be z (t) = 0. We will use
the formula
| (t) (t)|
(t) =
| (t)|3
for the curvature of a parametrized sp

PM 365 / AM 333: Elementary Dierential Geometry
Assignment 03 (REVISED); due Friday, 18 February 2011
[1]
Let (s) be a unit-speed parametrized curve in R3 with curvature (s) > 0 for all s (s1 , s2 ). Let X (s)
be a unit vector eld dened along the parametr

PM 365 / AM 333: Elementary Dierential Geometry
Assignment 03; SOLUTIONS
[1]
We are given that (s) is a unit-speed parametrized space curve with curvature (s) > 0 for all s (s1 , s2 ).
We want to examine the regularity of the smooth parametrized surface

PM 365 / AM 333: Elementary Dierential Geometry
Assignment 04 (REVISED); due Friday, 11 March 2011
[1]
Let (s) for s (s1 , s2 ) be a unit-speed parametrized curve in R3 with curvature (s) > 0. Then the
ruled surface S dened as the image of the single coor

PM 365 / AM 333: Elementary Dierential Geometry
Assignment 04; Solutions
[1]
The map f : 1 : S S is a dieomorphism from S to S , because the map 1 f is
the identity map, which is a dieomorphism. So we just need to check that the rst fundamental form
of S

PM 365 / AM 333: Elementary Dierential Geometry
Assignment 05; due Wednesday, 23 March 2011
[1]
Let (s) be a unit-speed curve in R3 , and assume that there exists a positive constant C such that
0 < (s) < C everywhere on the curve. Given > 0, we dene the

PM 365 / AM 333: Elementary Dierential Geometry
Assignment 05; SOLUTIONS
[1]
We begin by computing the partial velocities:
s = + (cos tN + sin tB )
= T + cos t(T + B ) + sin t( N )
= (1 cos t)T sin tN + cos tB
and
t = sin tN + cos tB .
From these we can

PM 365 / AM 333: Elementary Dierential Geometry
Assignment 06; due Monday, 04 April 2011
[1]
Let be a regular surface in R3 . Recall that the rst fundamental form I, the second fundamental form
II, and the Weingarten map W at a point p on satisfy the foll