Ricci Curvature - Behavior Under Conformal Rescaling

Behavior Under Conformal Rescaling

If you change the metric g by multiplying it by a conformal factor, the Ricci tensor of the new, conformally related metric is given (Besse 1987, p. 59) by

where Δ = dd is the (positive spectrum) Hodge Laplacian, i.e., the opposite of the usual trace of the Hessian.

In particular, given a point p in a Riemannian manifold, it is always possible to find metrics conformal to the given metric g for which the Ricci tensor vanishes at p. Note, however, that this is only pointwise assertion; it is usually impossible to make the Ricci curvature vanish identically on the entire manifold by a conformal rescaling.

For two dimensional manifolds, the above formula shows that if f is a harmonic function, then the conformal scaling ge2ƒg does not change the Ricci curvature.

Read more about this topic:  Ricci Curvature

Famous quotes containing the word behavior:

    If you are willing to inconvenience yourself in the name of discipline, the battle is half over. Leave Grandma’s early if the children are acting impossible. Depart the ballpark in the sixth inning if you’ve warned the kids and their behavior is still poor. If we do something like this once, our kids will remember it for a long time.
    Fred G. Gosman (20th century)