What is an example of a manifold such that:
(A) It is both a contact manifold and a CR manifold
(B) It is a contact manifold but not a CR manifold
(C) It is not a contact manifold but not a CR manifold?
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3$\begingroup$ The 3-sphere is both CR and contact. More generally, the boundary of any pseudoconvex domain is CR and contact, via the Levi form of the CR geometry. $\endgroup$– Ben McKayCommented Feb 21, 2016 at 10:05
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3$\begingroup$ A CR manifold can be seen as a contact manifold with CR structure, i.e., a complex structure J defined on the contact hyperplane. So the only nontrivial question is: does there exist a contact manifold which admits no CR structure? I think the answer is yes, but I do not have an example in mind. $\endgroup$– Chih-Wei ChenCommented Feb 29, 2016 at 16:39
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1$\begingroup$ I don't understand your last question. Are you looking for an orientable odd-dimensional manifold which does not carry any contact structure nor any CR structure ? $\endgroup$– M. DusCommented Jul 28, 2017 at 11:34
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2 Answers
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For a contact metric manifold $M$ we observe that $J = \varphi_{\vert D}$, i.e. the restriction of $\varphi$ to the contact distribution, defines an almost complex structure on $D=\ker\eta$. Then the associated almost CR-structure of $M$ is given by the holomorphic subbundle $$H = \{X − i JX\mid X \in D\}$$ of the complexification $TM^{\Bbb C}$ of the tangent bundle $TM$. We say that the associated almost CR-structure is integrable if $[H,H] \subset H$. This is equivalent to $[J, J ](X, Y ) = 0$, for any $X, Y \in D$, where $[J, J ]$ denotes the Nijenhuis torsion of $J$. It is known that the associated CR-structure of a three dimensional contact metric manifold is always integrable (S. Tanno 1989).
A contact metric manifold $M$ such that its associated almost CR-structure is integrable will be referred to as a contact strongly pseudo-convex CR-manifold.
It should be noted that every $(\kappa,\mu)$-manifold is a contact strongly pseudo convex integrable CR-manifold (D. E. Blair et al Contact metric manifolds satisfying a nullity condition doi:10.1007/BF02761646).
And answer of your questions:
A) By above comments, every $(\kappa,\mu)$-manifold is both a contact manifold and a CR manifold.
B) In Mitric [1991] and Tanno [1992] it was shown that the tangent sphere bundle with its standard contact metric structure is a CR-manifold if and only if the base manifold is of constant curvature. Thus if we choose $M$ with non-constant sectional curvature then the tangent sphere bundle $T_1M$ with its standard contact metric structure is a contact metric manifold that not a CR-manifold.
For (C) I do not have an example in mind.
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$\begingroup$ By your expression $H = \{X − i JX/X \in D\}$, do you mean $H = \{X − i JX\mid X \in D\}$? $\endgroup$– David Roberts ♦Commented Jul 27, 2017 at 23:10
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$\begingroup$ And by Blair et al, which paper do you mean? Generalization of Myers' theorem on a contact manifold.? (Searching MathSciNet gives four joint papers, but the only one with three authors only talks about the 3-dimensional case) I would like to provide links to your cited papers. $\endgroup$– David Roberts ♦Commented Jul 27, 2017 at 23:16
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1$\begingroup$ This paper: Contact metric manifolds satisfying a nullity condition $\endgroup$– C.F.GCommented Jul 28, 2017 at 5:28
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$\begingroup$ Ah, that paper was 1995. I added a link. $\endgroup$– David Roberts ♦Commented Jul 28, 2017 at 6:49
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The way I understand the questions, these may provide possible examples:
(A) These are precisely CR structures of CR codimension 1 with nondegenerate Levi-form. That is, the nondegeneracy of the Levi form implies the maximal nonintegrability of the corresponding hyperplane distribution.
(B) Take any example that is both contact and CR and perturb the CR structure to a non-integrable almost CR structure (any generic perturbation in dimension higher than 3).
(C) Take any hyperplane distribution that is not maximally non-integrable, then it is not contact. Then add any compatible non-integrable almost CR structure, which is locally always possible.
