General Relativity - General Tensors : Riemann - Christoffel Tensor or Curvature Tensor

The curvature tensor,also known as Riemann Curvature Tensor, is a central mathematical object in differential geometry and general relativity. It encodes the intrinsic curvature of a manifold and describes how vectors are twisted or rotated as they are parallel transported around infinitesimal loops. It’s foundational to the understanding of spacetime in general relativity. It connects geometry to physics,allowing Einstein to describe gravity not as force but as the curvature of spacetime shaped by energy and momentum. In flat spacetime ,i.e.,in special relativity its all components vanish. In curved spacetime,e.g.,around stars or black holes,it describes the tidal forces experienced by matter.