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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

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I never understand why people use the word "need" in this context. We don't need Kahler structure. We don't even need manifolds. We want them. – Qiaochu Yuan Apr 7 '13 at 6:54
So I change the word, if you know please explain why we want these structures on a manifold? – Merri Apr 7 '13 at 9:01
I don't understand what your second question means. Can you give some more details of what you are reading and/or looking to have clarified? – Sam Lisi Apr 8 '13 at 9:28

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.

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Thank you for the answers Now I am reading the book structures on Manifold which is written By Yano and Kon I saw some structures on manifolds which actually I could not Understand the application. Especially about shape operators. I sow some papers which tried to calculate value of the shape operator. What is this application Again Thanks and it could be so nice if you write any thing which you think could be useful for me. – Merri Apr 13 '13 at 16:54
@Merri: I don't know the book you mention, and I know very little about uses of shape operators. However, I would guess that the key feature of a "shape operator" is that it is the differential of a "Gauss map". There are many situations in which you have something that looks like a Gauss map (but isn't the classical one), so maybe there are many non-standard shape operators. I don't really know. – Sam Lisi Apr 13 '13 at 21:30
I should add the question: have you studied surfaces in $\mathbb{R}^3$? This is the easiest scenario with a shape operator, and it might be worthwhile to understand this first. If you already know about this use of the shape operator, I can't tell you anything further. – Sam Lisi Apr 13 '13 at 21:31

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