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)

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

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)