# Curves on surfaces (differential geometry)

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?