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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

2 votes

Lie groups admitting flat (bi)invariant metrics.

I'm a few years late with my answer, but if you're interested in the pseudo-Riemannian case as well, there is a structure theorem due to Baues for Lie algebras with invariant scalar products, see Theo …
HackVlix's user avatar
2 votes

Which Lie algebra admit symplectic forms

It's a bit late now, but there is also a more recent and very exhaustive paper by Cortés and Baues which adresses and summarizes many related questions: http://arxiv.org/abs/1307.1629
HackVlix's user avatar