Just to have a record of some ideas regarding two (related?) problems.
1. The first has to do with my MSc thesis, the ambitious goal is to construct a link between Riemannian and Lorentzian geometry. This can be done by studying submanifolds of Minkowski space which inherit a changing-signature metric.
To begin with, we need to extend the classical concepts of curvature, parallel transport, geodesics, second fundamental form, etc. This can be done generically, at least in the transverse direction of the degeneracy of the metric (which is the interesting case).
2. Find new solutions of Einstein's equations and model galactic dynamics.
- Can we classify algebraically spaces with vanishing Ricci curvature? There is the Petrov, classification, but I am referring to a "Group-geometric" approach. For this, in each "class" we must have a preferred symmetric solution (eg. Minkowski), and all elements in this class can be generated by performing operations in the group.
- Following the idea that the inner product in Minkowski space is precisely the real part of the quaternionic group, is it possible to define a stronger version of Einstein equation, more manageable, by requiring some sort of quaternionic structure on the spacetime manifold? (spinors?)
- Variational principle. The Ricci flat condition translates into being a critical point of the total scalar curvature functional. So given a metric, is there a way to flow-deform it in order to prove existence of solutions.
- Weak solutions. Try to define a bilinear, 1st order, functional that correspond to the Ricci tensor.
- Foliations. Write the equations for $N \subset \Sigma \subset M$, where $\Sigma$ is a spacelike 3-manifold foliated by $N$. What about the case when M is foliated by N with fibres P, both 2-manifolds (null?)
3. Reformulate Einstein's equations.
- In terms of the light cone structure (S^3 bubdle over M?)
Warning: This is informal and vague. They are just baby ideas.