# Differential Geometery: Images of Gauss Maps

## Homework Statement

this is a 2 part problem

let G: T-----> (Sigma) be the gaus map of the torus T derived from its outward unit normal U. What are the image curves under G of the meridians and parallels of T? What points of (Sigma) are the image of exactly two points of T?

And

Let G: M---->(Sigma) be the Gauss map of the saddle surface M: z=xy derived from the unit normal. What is the image under G of one of the straight lines, y= constant in M? How much of the sphere is covered by the entire image G(M)

## Homework Equations

Part 1:

Unit normal of z=xy

U= [-fxU1- fyU2+U3]/ (1+ (fx)^2 + (fy)^2)^1/2

## The Attempt at a Solution

I dont know how to do any rigorous proofs for these, but i am not sure we are suppose to.

Ok for part 1 here is what i think.

I think if you travel along the surface of the torus, along the meridian, you it will be mapped to a circle (in the xy plane). if you travel along the meridians you will get a circle in the xz plane.

And the points in sigma thare are exactly 2 points in T are the North and south poles of the unit sphere.

And for part 2 i am completley lost, am i even on the right track for part one?

any help is appreciated.