Abstract :

We shall talk about the idea of a ''connection'' on a Riemannian Manifold and how an appropriate connection allows us to introduce the ”curvature” of 'nice' manifolds. The techniques developed allows us to generalize the idea of ''straight lines'' in the real plane to the category of manifolds. To observe these ideas in motion, we shall compute these objects for Surfaces in space. We shall also talk about Euclid’s fifth axiom in the context of a general surface.