Riemannian connection on a surface

A Riemannian connection on a surface (or more generally on any Riemannian manifold) is a way to define how to differentiate vector fields along the surface, while keeping the geometric structure provided by the Riemannian metric in mind. ### Key Concepts 1. **Riemannian Metric**: A Riemannian manifold has an inner product defined on the tangent space at each point, called the Riemannian metric.