Structure on manifolds Thank you for your attention.
First, I would like to know why we see some different structures defined on manifolds: What is the necessity to have different structures, like Kähler structure, Sasakian Structures, Kenmotsu structures and so on.
Second, what is the application of shape operators since I saw in some papers the authors tried to find it in different ways and they found different values for it.
And last: can we define a complex structure on $M^{2n}$? Or Sasakian structure on $M^{2n-1}$?
Thank you for all your help in advance.
Best regards
 A: This became very long to leave as a comment, so I will provide a partial answer to #1 and an answer to #3.
For #1, do you know examples of manifolds with these features? Often we study the general theory because we seek to unify features of a class of examples that exhibit similar behaviour. In my experience, we rarely introduce a new structure to study unless we have a rich family of examples that cry out to be unified in one theory.
The answer to the last question is "no", not in general. Complex manifolds and Sasaki manifolds are quite special in the vast sea of manifolds. There are some obvious topological obstructions. For instance, the structure group of the tangent bundle to a complex manifold has to admit a reduction to $U(n)$. Stupid examples of non-complex manifolds are for instance non-orientable ones. One way to rule out Sasaki manifolds is to show that the manifold does not admit a contact structure. If, however, there is no topological obstruction, it is interesting to know if such structures exist. I haven't thought much about Sasaki manifolds, but I would guess that the manifold must fibre over a symplectic orbifold or something like this. In particular, there are many contact structures on $3$-manifolds that do not admit Sasaki structures. 
It is currently an interesting open problem in contact topology to know if an almost contact manifold (i.e. structure group in $\mathbb{1} \times U(n-1)$) admits a contact structure or not. I think the experts expect it to be true, but it remains open. To the best of my knowledge, it is also open as to whether $S^6$ admits a complex structure, and I don't believe we have a classification of Sasaki 5-manifolds either.
