University of Regina
Department of Mathematics and Statistics
MATH431/831 Dierential Geometry Winter 2014
Homework Assignment No. 1
Solutions
1. Find the curvature of the ellipse at an arbitrary point (see the notes, Section 1.2, Example
3) and check that
DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES
5. The Second Fundamental Form of a Surface
The main idea of this chapter is to try to measure to which extent a surface S is dierent from
a plane, in other words, how curved is a surface. The idea of doing thi
DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES
6. Gauss Theorema Egregium
Question. How can we decide if two given surfaces can be obtained from each other by
bending without stretching?
The simplest example is a at strip, say of paper, which can be rolled
DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES
2. Curves in Space
2.1. Curvature, Torsion, and the Frenet Frame. Curves in space are the natural
generalization of the curves in the plane which were discussed in Chapter 1 of the notes.
Namely, a parametrized
DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES
3. Regular Surfaces
3.1. The denition of a regular surface. Examples. The notion of surface we are
going to deal with in our course can be intuitively understood as the object obtained by a
potter full of phant
DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES
1. Curves in the Plane
1.1. Points, Vectors, and Their Coordinates. Points and vectors are fundamental
objects in Geometry. The notion of point is intuitive and clear to everyone. The notion
of vector is a bit
DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES
4. Lengths and Areas on a Surface
An important instrument in calculating distances and areas is the so called rst fundamental form of the surface S at a point P . This is nothing but the restriction of the scal
University of Regina
Department of Mathematics and Statistics
MATH431/831 Dierential Geometry Winter 2014
Homework Assignment No. 2 - Solutions
1. Consider the helix, as defined on page 2, Ch. 2 (Curves in Space) of the notes.
(a) Calculate the torsion of
University of Regina
Department of Mathematics and Statistics
MATH431/831 Dierential Geometry Winter 2014
Homework Assignment No. 3 - Solutions
1. Consider the map
(u, v ) = (cos u sin v, sin u sin v, cos v )
where 0 < u < 2 , 0 < v < .
(a) Show that (u,
DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES
7. Geodesics and the Theorem of Gauss-Bonnet
7.1. Geodesics on a Surface. The goal of this section is to give an answer to the following
question.
Question. What kind of curves on a given surface should be the