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
Read more about this topic: Ricci Curvature
Famous quotes containing the word definition:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possessafter many mysterieswhat one loves.”
—François, Duc De La Rochefoucauld (16131680)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (18421910)
“... 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 lensif we are unaware that women even have a historywe live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)