I have been studying Symplectic Geometry. Previously I studied Riemannian Geometry.
In Symplectic Geometry I learned the existence of an almost complex structure and how some special almost complex manifolds are integrable thereby becoming a complex manifold.
Now I am seeing that we want to focus on a manifold which has all 3 of these structures (symplectic,riemannian and complex) and in some sense are compatible.
My question is : All we did was define special types of 2 forms on a manifold and then started looking at compatibility between them. So there should be no reason to give these forms extra privilege, I could just as well define some weird structure like maybe a closed k-form which satisfies blah-blah properties and that would lead to a new geometry etc.
How does a Mathematician know which structures are "important" and worth our attention?
Best Answer
Perhaps, you could remember that Kähler geometry was invented before symplectic geometry. It is "natural", in the sense that it is the natural geometry of complex projective manifolds. As a submanifold of $\bf CP^N$ a complex algebraic manifold is endowed with a Kähler structure, and this structure is the main tool to prove basic results : Hodge Theorems, Hard Lefschetz Theorem , for instance.