Short answers, pulled from the story.
Bernhard Riemann published his groundbreaking work on geometry in 1854. Elwin Bruno Christoffel expanded upon these ideas shortly after to create the most common tool for expressing curvature in differential geometry.
The object measures the failure of second covariant derivatives to commute. It serves as a local invariant for Riemannian metrics by assigning a specific value to every point on a manifold.
The Riemann curvature tensor directly quantifies non-holonomy within a general manifold by measuring the difference between final and initial vectors after walking around a loop. This deflection identifies how lines appear straight only locally while failing to return to their original direction.
Skew symmetry applies to the first two indices of the object and a first algebraic Bianchi identity discovered by Ricci follows from primary rules. Simple calculations show such a tensor has n squared times n minus 1 divided by 2 independent components.
General relativity relies heavily on this central mathematical tool because the theory describes gravity through the curvature of spacetime. The tensor represents tidal forces experienced by a rigid body moving along a geodesic and allows physicists to model gravitational effects mathematically.