Ricci Curvature - Definition

Definition

Suppose that is an n-dimensional Riemannian manifold, equipped with its Levi-Civita connection . The Riemannian curvature tensor of is the tensor defined by

on vector fields . Let denote the tangent space of M at a point p. For any pair of tangent vectors at p, the Ricci tensor evaluated at is defined to be the trace of the linear map given by

In local coordinates (using the Einstein summation convention), one has

where

In terms of the Riemann curvature tensor and the Christoffel symbols, one has

R_{\alpha\beta} = {R^\rho}_{\alpha\rho\beta} = \partial_{\rho}{\Gamma^\rho_{\beta\alpha}} - \partial_{\beta}\Gamma^\rho_{\rho\alpha} + \Gamma^\rho_{\rho\lambda} \Gamma^\lambda_{\beta\alpha} - \Gamma^\rho_{\beta\lambda}\Gamma^\lambda_{\rho\alpha} =2 \Gamma^{\rho}_{{\alpha}} + 2 \Gamma^\rho_{\lambda \alpha} .

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