Thanks, Monique. I had a feeling the pronunciation would be something unintuitive.
MathematicalPhysicist said:
Which book are you using?
Hah! Well,... er,...
I'm actually trying to research possible Finslerian extensions of GR (i.e., generalization of Riemannian geometry where the fundamental metric tensor depends also on direction in the tangent bundle, not just position on the base manifold). So I'm really studying Finsler geometry, and also the associated concept of "Spray Spaces". There's a couple of books by Shen (and colleagues), also Xiaohuan & Mo. Then there's Finsler & Lagrange spaces by Bucatari & Miron. Also Sprays and Finsler material by Antonelli et al.
I must say that I'm not doing very well. Working with quantities on ##T(TM)## can be rather confusing. I haven't yet tried to ask questions over in the DG forum, since past (rare) Finsler questions over there tend not to get much response.
But the Nijenhuis stuff came up because of some (unpublished?) notes: "A Brief Introduction to Finsler Geometry" by Dahl (found via Google Scholar). He works through the usual horizontal--vertical decomposition of ##T(TM)##, and then uses this decomposition to define an almost--complex structure on ##T(TM)##. But I guess this will be gobbledegook to most people.
I took once a course on Symplectic Geometry and we discussed on this a little, there's the book by Dusa Mcduff which seems to be a standard on this topic.
So far I haven't approached my interest via symplectic spaces, since they seemed to be an afterthought here. But now I'm thinking I should try coming at it from that direction as well. I haven't looked at Mcduff before, but I'll take a look soon.
Cheers.