Riemanntensor

The Riemann tensor, or Riemann curvature tensor, is a central object in differential geometry. It encodes the curvature of a differentiable manifold equipped with an affine connection, most often the Levi-Civita connection derived from a metric. It measures how much the geometry deviates from being locally flat.

Intuitively, the Riemann tensor describes how parallel transporting a vector around a small closed loop fails to return the vector to its original direction. In a tangent space, it defines a curvature operator whose action depends on the plane of the loop, providing a local description of intrinsic curvature independent of any embedding.

In coordinates, the tensor with type (1,3) is written R^rho_{ sigma mu nu} = ∂_mu Γ^rho_{ sigma nu} - ∂_nu Γ^rho_{ sigma mu} + Γ^rho_{ lambda mu} Γ^lambda_{ sigma nu} - Γ^rho_{ lambda nu} Γ^lambda_{ sigma mu}, where Γ denotes the Christoffel symbols of the chosen connection. Contracting on indices gives the Ricci tensor R_{sigma nu} = R^rho_{ sigma rho nu}, and contracting further with the metric yields the scalar curvature R = g^{sigma nu} R_{sigma nu}.

The Riemann tensor has several symmetries: it is antisymmetric in the last two indices, R^rho_{ sigma mu nu} = - R^rho_{ sigma nu mu; the lowered form R_{rho sigma mu nu} obeys R_{rho sigma mu nu} = - R_{sigma rho mu nu} = - R_{rho sigma nu mu} = R_{mu nu rho sigma}. It also satisfies the first Bianchi identity: R_{rho sigma mu nu} + R_{rho mu nu sigma} + R_{rho nu sigma mu} = 0, and the second Bianchi identity in covariant form. The tensor vanishes if and only if the manifold is flat.

In physics, the Riemann tensor is foundational in general relativity, where its contractions yield the Ricci tensor and scalar curvature that enter the Einstein field equations.