Differential Geometry Question

[Sistine]

Homework Statement

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 $$(\partial_1\sigma,\partial_2\sigma)$$?

Homework Equations

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

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

The Gauss map is defined as:

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

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?

 

[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.

 

[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.