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:

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

    “... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
    —Adrienne Rich (b. 1929)

    “One definition of man is “an intelligence served by organs.””
    —Ralph Waldo Emerson (1803–1882)