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.
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