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} .

Read more about this topic:  Ricci Curvature

Famous quotes containing the word definition:

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)

    “It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
    —Jane Adams (20th century)

    “I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)