**Differential geometry** is basically the study of

geometry using

calculus. It has many applications in

physics, especially in the

theory of relativity. The central objects of study are Riemannian

manifolds, geometrical objects such as surfaces which locally look like

Euclidean space and therefore allow the definition of analytical concepts such as tangent vectors and

tangent space, differentiability, and vector and

tensor fields. The manifolds are equipped with a metric, which introduces geometry because it allows to measure distances and

angles locally and define concepts such as geodesics

?, curvature

? and torsion

?.