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