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: $K(p) = \kappa_1 \kappa_2 .$