# Differential Geometry Question

1. Nov 20, 2009

### Sistine

1. The problem statement, all variables and given/known data
Consider the following parametrization of a Torus:

$$\sigma(u,v)=((R+r\cos u)\cos v, (R+r\cos u)\sin v, r\sin u)$$

$$R>r,\quad (u,v)\in [0,2\pi)^2$$

1. Compute the Gauss map at a given point.

2. What are the eigenvalues of that map in the base $$(\partial_1\sigma,\partial_2\sigma)$$?

2. Relevant equations

$$\partial_1\sigma=\frac{\partial\sigma}{\partial u}$$

$$\partial_2\sigma=\frac{\partial\sigma}{\partial v}$$

The Gauss map is defined as:

$$N(u,v)=\frac{\partial_1\sigma\times\partial_2\sigma}{|\partial_1\sigma\times\partial_2\sigma|}$$

3. The attempt at a solution
Computing the Gauss map at a point $$p$$ is straightforward enough. But I'm not sure what part 2 of the question is asking me to do. How can I visualize the map as a matrix operator in a certain basis so that I can compute its eigenvalues?

2. Nov 20, 2009

### Dick

They have to mean the eigenvalues of the derivative of the Gauss map, dN, the mapping between the tangent spaces of the plane and the torus.

3. Nov 20, 2009

### Sistine

Perhaps your right. The image of the Gauss map at a point is perpendicular to the tangent space at that point , so that no linear combination of $$\partial_1\sigma$$, $$\partial_2\sigma$$ could ever represent N at that point. However is it possible to represent the map N:R2->R3 as a matrix? I'll try to find out if there is an error in the question.