RicciKrümmung

Ricci curvature, or Ricci curvature tensor, is a fundamental concept in differential geometry, particularly within the study of Riemannian manifolds. It is a symmetric 2-tensor derived from the Riemann curvature tensor, representing the average curvature of a manifold in a pair of orthogonal directions.

Mathematically, for a Riemannian manifold \((M, g)\), the Ricci tensor \( \text{Ric} \) is obtained by contracting the Riemann curvature tensor \( R_{ijkl} \) over two indices: \( \text{Ric}_i^j = R_{i k l}^{\ \ j k} \). This operation reduces the rank of the curvature tensor from 4 to 2.

The Ricci tensor measures how the volume of a small geodesic ball changes under parallel transport, reflecting the average expansion or contraction of space in orthogonal directions. In contrast to the sectional curvature, which describes curvature in a specific 2-dimensional plane, the Ricci curvature provides a more global average measure.

In general relativity, the Einstein field equation relates the Ricci tensor to the matter content of spacetime, with the Ricci scalar (the trace of the Ricci tensor) appearing as the source of gravity. This connection links the geometric property of Ricci curvature to physical phenomena like gravitational attraction.

Ricci curvature is thus a key tool for analyzing the intrinsic geometry of manifolds and plays a central role in both pure mathematics and theoretical physics.