Usage/Application of Raychaudhuri equation in Riemann geometry or pure maths

Question

While going through this paper by Witten and seeing a discussion about different aspects of Raychaudhari Equation and Einstein Field Equation. I want to ask if Raychaudhari Equation find any application in pure Maths specifically Riemann geometry since the only assumption in arriving it is to have a metric with signature (−+++) and geodesic equation. Hence it can be applied in the modified theory of gravity as well.

I searched the Raychaudhri equation in the A Panoramic View of Riemannian Geometry with no results while a google search gives no result as well. Finally searching on arXiv in Maths section only produces 7 results which are more or less about singularity theorem or the usual application of Raychaudari equation in GR.

This really bugs me since Raychaudhari equation is essentially about the evolution of geodesic equations therefore they should find at least some application in pure maths. Or does it happen that they are used but a more general version of them is used?

I have asked same question on maths SE but since there is no answer even after putting a bounty there I'm asking it here.

Answer 1

For any 1-form $\omega$ on a pseudo-Riemannian manifold, the product rule and commutation identity say that $$\omega^p\nabla_p\nabla_q\omega^q-\nabla^q(\omega^p\nabla_p\omega_q)=-\omega^p\omega^qR_{pq}-\nabla^q\omega^p\nabla_p\omega_q,$$ with (in the notation and language of the Raychaudhuri equation) the first term representing $\dot{\theta}$, the second term vanishing in the case of a geodesic congruence, the first term on the right being the Ricci term in the Raychaudhuri equation, and the last term being equivalent to the other terms in the Raychaudhuri equation. So, since the Raychaudhuri equation essentially amounts to a direct application of the commutation identity, you could say that it is used quite often in Riemannian geometry (eg as Bochner identity and Weitzenbock identity).

More specifically, you can view $\nabla_q\omega^q\equiv\theta$ as the Lie derivative of the volume form by the vector field $\theta^\sharp$, and then the Raychaudhuri equation is about how the volume form is deformed under the flow of a vector field. In this sense you could say that the Bishop volume comparison theorem is a Riemannian analogue to standard Raychaudhuri analysis (specifically 4.13-4.14 in Witten's notes).

Answer 2

The Bochner identity is a special case of the Raychaudhuri equation, obtained when the vector field is of gradient type.

Let $v$ be a vector field and let $a^\alpha=v^\alpha{}_{;\beta} v^\beta$ be its acceleration. The vector $v$ is neither normalized nor a gradient. The next identity \begin{align} \partial_v (v^\alpha{}_{;\alpha})&=v^\alpha{}_{;\alpha; \beta} v^\beta= v^\alpha{}_{;\beta; \alpha} v^\beta-R^\alpha{}_{\gamma \alpha \beta} v^\gamma v^\beta= (v^\alpha{}_{;\beta} v^\beta)_{; \alpha}-v^\alpha{}_{;\beta} v^\beta{}_{;\alpha}-Ric(v) \nonumber \\ &=a^\alpha{}_{;\alpha}-v_{\alpha;\beta} v^{\beta;\alpha}-Ric(v). \label{vod} \end{align} is used to derive both the Raychaudhuri equation (with acceleration and vorticity) and the Bochner equation. The connection with the former is easy, it is sufficient to assume $v$ normalized and split $v_{\alpha;\beta}$ in symmetric and antisymmetric parts. As for the connection with the latter we need to assume that $v$ is of gradient type. So let $v_\alpha=\partial_\alpha u$ where $u$ is a function. We have \begin{align*} \theta&= u_{;\alpha}{}^{;\alpha},\\ a_\alpha&=u_{;\alpha;\beta} v^\beta =u_{;\beta;\alpha} v^\beta=v_{\beta; \alpha} v^\beta=\tfrac{1}{2} (v^\beta v_\beta)_{; \alpha}=\tfrac{1}{2} (u^{;\beta} u_{;\beta})_{; \alpha},\\ u_{;\alpha ;\beta}&=u_{;\beta;\alpha}. \end{align*} Thus our first equation in display becomes the Bochner identity $$ (u_{;\alpha}{}^{;\alpha})_{;\beta} u^{;\beta}=\tfrac{1}{2} (u^{;\beta} u_{;\beta})_{; \alpha}{}^{;\alpha}-u_{;\alpha ;\beta} u^{;\alpha ;\beta} -Ric(u). $$ In this derivation the normalization of $v$ was not used so it holds in any signature. In the Riemannian case we get the usual form $$ g(\nabla u, \nabla \Delta u)= \tfrac{1}{2} \Delta( (\nabla u)^2)-(\textrm{Hess} \, u)^2-Ric(u). $$ The geodesic vorticity-free Raychaudhuri equation is just a particular case because the vector field can be seen as the gradient of the distance function from a hypersurface to which the congruence is orthogonal. In this case the first term on the right-hand side, corresponding to the divergence of the acceleration, vanishes.

The Raychaudhuri equation in general relativity is often conveniently replaced by a Bochner type equation while dealing with hypersurface-orthogonal timelike geodesic congruences, in which case the vector field is the gradient of the Lorentzian distance from the spacelike hypersurface. The Raychaudhuri equation for lightlike geodesic congruences (used e.g. in Penrose's singularity theorem) does not admit this type of reformulation so the Raychaudhuri equation rather than the Bochner equation remains the tool of choice in general relativity.

In Riemannian geometry you can use the Bochner equation instead of the Raychaudhuri equation because in most cases you can see the unit vector field of the geodesic congruence as the gradient of a distance function, i.e. you don't have the problem of lightlike geodesic congruences for which the analog of such a function does not exist.

Answer 3

The Raychaudhuri Equation is called the Raychaudhuri Equation because of a physist called Raychaudhuri. In mathematics the same equation occurs, for the reason you have pointed out, but it is called something else. In fact the same idea, studying the divergence of a flow of geodesics is so important that it has lots of names. Your issue is that you are searching math literature with physics terminology.

Beem, Ehrlich and Easley give a reasonable overview of the various faces of the mathematics have the same idea as the Raychaudhuri equation. For example, Chapter 10 covers Morse Index Theory, as well as the first and second variations of arc length; Section 12.1 gives a second introduction to Jacobi tensors and Section 12.2 discusses Jacobi tensors in the context of spacelike surfaces (a free boundary problem for the associated flow of geodesics). Definition 12.2 shows how Jacobi tensors are related to the Raychaudhuri equation. Appendix B provides even more of a connection and shows how a certain mathematically important Riccati equation is also related to Jacobi tensors.

Here's another example: The differential of the exponential equation is very closely related to Jacobi tensors, see Section 6.3.1 of Berger. So Jacobi tensors are very important. So important that Berger spends the rest of chapter 3 discussing them. Remember the differential equation that defines a Jacobi tensor is very closely related to Raychaudhuri's equation. I suggest that you also read chapter 7 of Berger because that stuff is connected as well.

The differential of the exponential map is closely related to Jacobi tensors so anything that involves the differential of exponential map involves Jacobi tensors and therefore is closely related to Raychaudhuri's equation.