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 = d∗d 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 g ↦ e2ƒg does not change the Ricci curvature.
Read more about this topic: Ricci Curvature
Famous quotes containing the word behavior:
“Understanding child development takes the emphasis away from the child’s character—looking at the child as good or bad. The emphasis is put on behavior as communication. Discipline is thus seen as problem-solving. The child is helped to learn a more acceptable manner of communication.”
—Ellen Galinsky (20th century)