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?



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