Curves on surfaces (differential geometry)

[Lee33]

A few topics we are covering in class are: Gauss map, Gauss curvature, normal curvature, shape operator, principal curvature. I am having difficulty understanding the concepts of curves on surfaces. For example, this problem:

Define the map ##\pi : (\mathbb{R}^3-\{(0,0,0)\})\to S^2## by ##\pi(p)=\frac{p}{||p||}.## Show that if ##\Sigma_R## is the sphere of radius ##R>0##, then the Gauss map of ##\Sigma_R## is ##\pi|_{\Sigma_R}## (which means the map ##\pi## restricted to the surface ##\Sigma_R##.) Compute the shape operator and the Gauss curvature of the sphere.

I don't even know where to start?

 

[hunt_mat]

It helps if you write down definitions, what is the Gauss map in question? Can you compute it?

 

[Lee33]

I know the Gauss maps a surface in ##\mathbb{R}^3## to the sphere ##S^2,## so ##\pi(p)## is a unit vector for all ##p\in \sum## such that ##\pi(p)## is orthogonal to the surface ##\mathbb{R}^3## at ##p##. Also, we defined the Gauss curvature as: ## K(p) = \kappa_1 \kappa_2 .##