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 = 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.