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
?.