What are the research areas in differential geometry involving Lie Groups action?

Question

I love Lie groups actions! My contact with this topic occurred in the context of differentiable manifolds and riemannian geometry courses, and while studying Klein geometries. My knowledge is still very limited, though.

I would like to specialize on a research area in which I could work a lot with Lie groups action from a geometric point of view. It would be very nice if I could find some suggestions/possibilities here. I'm not connected to the university now and I couldn't find pithy information on the internet.

Thank you all, in advance.

Answer 1

One example: the representation theory of Lie groups is a large and classic area of mathematics.

Answer 2

I'm by no means an expert, but I think ergodic theory and dynamical systems might be an interesting general direction to consider.

Answer 3

The geometry of "highly symmetric" spaces can frequently be framed almost entirely in terms of Lie groups, by studying the symmetry groups rather than the spaces themselves. See for instance the study of Riemannian space forms or symmetric spaces, which have been (partially) classified using the theory of Lie groups and their representations.

I'm not sure where the cutting edge of these fields lie, but there are certainly open problems which have a very Lie-theoretic flavor.