Curvature waves, harmonic curvature, and curvature flow

Question

Let me first say I'm better at physics than math and have big gaps in my understanding of Riemannian geometry. I do a lot better with intuitive explanations than identities and so forth.

I've been contemplating the idea of a Lorentzian manifold whose curvature is inherently wave-like; it could be described by the d'Alambert equation of the Riemann or Ricci tensor:

$$\Box R_{\mu\nu\sigma\rho} = 0$$

because the d'Alambertian is the wave operator, as we can see when we write it in an inertial frame:

$$\left(-\frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial z^2} + \frac{\partial^2}{\partial z^2}\right)R_{\mu\nu\sigma\rho} = 0.$$

Since the d'Alambertian is the generalization of the Laplacian, I would think that would also describe "harmonic curvature". But apparently harmonic curvature is given by the divergence, not the Laplacian:

$$\nabla^iR_{ijkl} = 0$$

Why is this so?

And the other thing is, when people describe flow of curvature, instead of the wave equation they use Ricci flow, which says the curvature determines how the metric changes:

$$\frac{\partial g_t}{\partial t} = -2\operatorname{Ric}_{g_t}$$

Now this apparently has soliton solutions, so I guess it's somehow a wave equation, but as I understand it was introduced more in the spirit of a diffusion equation, and I don't see why it should be preferable to the wave equation for the curvature tensor. Nor can I follow DeTurck's paper that supposedly generalizes Ricci flow to more than 3 dimensions, and even then is it really covariant?

Sorry if that's two questions in one post but it really boils down to: why is the Laplace equation of the curvature tensor apparently not studied, and these other equations are studied instead? Any insight would be appreciated!

Answer 1

1. The Ricci Flow (RF) equation $$ \frac{\partial g(t)}{\partial t}= -2 Ric(g(t)) $$ is a 2nd order PDE on the metric tensor $g(t)$. In contrast, your proposed d'Alambert equation equation is a 4-th order PDE (since the Riemann tensor itself is a 2nd order nonlinear differential operator). As a rule of thumb, differential equations of higher order are harder to analyze, which might be one answer to your question: Your proposed equation is simply harder than RF which itself is hard enough. (I have no idea if anybody looked at your equation since the differential geometry literature is so vast.)

2. More importantly, RF was introduced by Richard Hamilton in early 1980s as a potential tool for proving Thurston's Geometrization Conjecture (TGC). It is clear from the equations defining the RF that Ricci-flat metrics are stationary points of the flow. Moreover, whenever $g$ is an Einstein metric it is a critical point of the normalized Ricci flow (NRF), which is a modification of the RF. Since in dimension 3 a Riemannian metric is Einstein if and only if it has constant sectional curvature, Hamilton thought that RF might be useful in proving TGC (by establishing its convergence to a stationary point of the flow). This was quite a leap of faith, but, eventually, Perelman proved that Hamilton was right. Later on, RF was used to prove other conjectures of topological and geometrical flavor.

This explains popularity of the RF: (1) it has relatively low order (namely 2) and it is useful in proving something interesting.

I have no idea if the equation you propose can be used for proving some interesting geometric/topological results, but it is definitely more complex than the RF. I suggest you ask on MO if anybody (e.g. Robert Bryant, the one who introduced the Bryant Soliton in the context of the RF) saw any work in the literature on your equation.

3. About solitons. The notion of solitons in the context of RF (which, I think, is due to Hamilton himself), or of a more general geometric flow, an equation of the form, say, $$ \frac{\partial g(t)}{\partial t}= D(g(t)), $$ where $D$ is a differential operator, has only loose relation to classical solitons. One can define a geometric soliton as a family of Riemannian metrics $g(t)$ on a manifold $M$ such that there exists a family of diffeomorphisms $f_t: M\to M$ satisfying $$ f_t^*(g(t))=g(0), $$ i.e. we have a family of isometric metrics. More generally, one can allow metrics $g(t)$ such that $$ f_t^*(g(t))= a(t) g(0), $$ i.e. we have a family of metrics which are isometric up to a scalar. Depending on $a<1$ or $a>1$, one talks about shrinking or expanding solitons. Informally, "the shape of a metric does not change as the metric evolves." In this sense it is similar to a soliton/solitary wave: A solitary wave moves in space (this corresponds to having diffeomorphisms $f_t$ above) but preserves its shape. This terminological flexibility is useful since people consider both RF and its normalized version (NRF): A family of metrics which evolves by isometry under the NRF will evolve by isometry up to scaling under the RF.