The Riemann Curvature Tensor is a mathematical object used in differential geometry to describe the curvature of a manifold. It provides a way to measure how much the geometry of a space deviates from being flat, which is essential in understanding the properties of curved spaces, such as those found in general relativity. This tensor is defined in terms of the Levi-Civita connection and involves the second derivatives of the metric tensor. It plays a crucial role in determining the geodesics, or the shortest paths, between points in a curved space, helping to explain how gravity affects the shape of the universe.