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Differential geometry is a branch of mathematics which studies geometry using differential and integral calculus. In the elementary stages, differential geometry studies curves and surfaces in euclidean spaces. More generally, the objects studied are called differentiable manifolds. These are abstract spaces where the notion of tangent space is intrinsically attached.
It is interesting to see here how the notion of standard covariant derivative or connection of defined for vector fields are abstracted and how they are used to establish the central curvature's concept.