see also:
Riemannian geometry is the branch of Differential Geometry that studies Riemannian Manifolds, defined as smooth manifolds with a Riemannian Metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.