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For an $n$-dimensional manifold $M$, almost complex and almost symplectic structures on $M$ correspond to reductions on the structure group of the tangent bundle, introducing a $GL(n/2,\mathbb{C})$ and $Sp(n)$ structure on $M$, respectively. With additional integrability (the vanishing of the Nijenhuis tensor or the closedness of the almost symplectic form) one has a complex or symplectic structure on $M$.

Since CR and Contact geometry are the odd-dimensional cousins of the above geometries, it seems as though it would make sense for CR and Contact structures to come from a reduction of the structure group of the manifold. However, these structures tend to be defined more in terms of their integrability conditions. I suppose what I am really hoping for is a more unified theory among the different geometric structures. The Riemannian, Almost Symplectic, and Almost Complex structures share an intimacy with their "2 out of 3" property arising from properties of the intersections of the reduced structure groups, and it would be nice if the relationship between Almost Contact and Almost CR structures stemmed from a similar place, and furthermore that their integrability conditions corresponded to some sort of "flatness" on the $G$-structure.

So I ask, do these $G$-structure correspondences exist for the above geometries, and do they share an analogous relationship with their even-dimensional counterparts?

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  • $\begingroup$ This might be relevant: eudml.org/doc/186418 $\endgroup$ – Qfwfq Sep 16 '17 at 0:13
  • $\begingroup$ Proposition 1.3, Chapter V of this book of K. Yano, M . Kon(books.google.com/…) , states that structure group of an almost contact manifold reduces to $U(n)\times 1$. $\endgroup$ – C.F.G Sep 16 '17 at 4:29
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Yes. You reduce the structure group of a contact $(2n+1)$-manifold to those bases of the tangent space for which the first $2n$ vectors form a conformal symplectic basis for the contact hyperplanes. Then you reduce the structure group by asking that the remaining tangent vector forms a basis in which the differential of some local choice of contact form is one on that vector, and its differential gives the symplectic structure on the hyerplane. It is an easy example of applying Cartan's method of equivalence. The same for CR geometries, which is a more famous example of the method of equivalence, worked out in detail in the lowest dimensional case by Cartan himself.

The relevant integrability is expressed in terms of the torsion. If you have a $G$-structure of the appropriate structure group $G$, it arises from such a contact or CR structure if and only if its torsion satisfies various requirements.

For the CR case, a very clear account is worked out in detail in:

MR1067341 (93h:32023) Jacobowitz, Howard An introduction to CR structures. Mathematical Surveys and Monographs, 32. American Mathematical Society, Providence, RI, 1990. x+237 pp. ISBN: 0-8218-1533-4

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I think that the relation of contact structures and CR structures to classical G-structures is a bit more complicated than suggested by Ben McKay's reply, although I certainly agree with his comments on the method of equivalence. Starting with contact structures, the corresponding G-structure corresponds to the stabilizer of a hyperplane in $\mathbb R^{2n+1}$. So such a structure is equivalent to a co-rank $1$ subbundle $H$ in the tangent bundle of a manifold $M$ of dimension $2n+1$. The structure function is the tensorial map $H\times H\to TM/H$ induced by the Lie bracket of vector fields. Contact structures then exactly correspond to the case that this structure function is non-degenerate in each point. For such structures, there is a further reduction of structure group of the associated graded vector bundle $H\oplus (TM/H)$ of the tangent bundle to the conformal symplectic group $CSp(2n,\mathbb R)$ (which is much smaller than the stabilizer mentioned above). The conceptual way to view this in my opinion is as a special case of the concept of a filtered manifold.

In the CR case, the story is a bit more complex. The underlying G-structure is given by a complex subbundle $H\subset TM$ of complex rank $n$ on a manifold of dimension $2n+1$. Note that the homogeneous model of this is $\mathbb C^n\times\mathbb R$, so this has infinite dimensional automorphism group. The structure function again is the tensorial map $H\times H\to TM/H$ induce by the Lie bracket of vector fields. Now as before, you can require this to be non-degenerate, but then there arises a compatibility condition with the complex structure on $H$. Namely you have to require that the above tensor is of type $(1,1)$ with respect to the complex structure, which in CR terms is sometimes called "partial integrability". If this condition is satisfied then you can view the partially integrable almost CR structure as the filtered analog of a G-structure: You have a filtered manifold (a contact manifold as described above) and a further reduction of the structure group of the associated graded vector bundle of $TM$ to the conformal unitary group $CSU(p,q)\subset CSp(2n,\mathbb R)$. Here $(p,q)$ is the signature of the CR structure, with $(n,0)$ corresponding to the strictly pseudoconvex case. For this, the homogeneous model is the unit sphere $S^{2n+1}\subset\mathbb C^{n+1}$, which has finite dimensional automorphism group (isomorphic to $PSU(n+1,1)$). This filtered G-structure now has an intrinsic torsion, namely the Nijenhuis tensor of the partially integrable almost CR structures, whose vanishing is equivalent to integrability of the CR structure.

Edit (in view of the comment by @FrancoisZiegler): I am not completely sure what is meant by an almost contact structure. The main point I wanted to make is that while contact forms, contact distributions, and almost CR structures do have an underlying G-structure, one has to impose an additional non-degeneracy condition on the intrinsic torsion (which I referred to as "structure function" in my reply). For contact distributions, this can be equivalently phrased as a filtered manifold structure modelled on a Heisenberg algebra (i.e. as a frame bundle for the associated graded vector bundle to the tangent bundle). In the almost CR case, there are further invariants related to integrability of the almost CR structure. These cannot be defined uniformly for all G-structures of the same type. However, if you impose the non-degeneracy condition as well as partial integrability, then you can view a partially integrable almost CR structure as a filtered analog of a G-structure. Then you can recover the Nijenhuis tensor, which is the natural obstruction to integrability of the CR-structure (and by the way does not show up in the 3-dimensionl case treated in the reference in the answer by @BenMcKay ) as a filtered analog of an intrinsic torsion.

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In addition to other excellent answers, I would like to approach from another angle.

Almost CR structures

A most general almost CR structure (i.e. of arbitrary codimension) on a real manifold $M$ is given by a complex subbundle

$$V\subset \mathbb C\otimes TM,$$

of the complexified tangent bundle of $M$, satisfying $V\cap \bar V=\{0\}$, where the conjugation on $V$ is induced by the standard conjugation on $\mathbb C\otimes TM$.

This definition is equivalent to its "real version" given by the pair $(H, J)$, where $H\subset TM$ is a real subbundle, and $J\colon H\to H$ is a complex structure on $H$. Indeed, given $(H, J)$ as above, the corresponding $V$ is the so-called $(0,1)$-subbundle given by

$$ V= H^{01} := \{ L\in \mathbb C\otimes H: JL = -iL \} = \{ X + iJX: X\in H\}.$$

Vice versa, every $V$ as above yields

$$H:= (V +\bar V)\cap TM,$$

and $J\colon H\to H$ is uniquely determined by the identity

$$X + iJX\in V, \quad X\in H.$$

(An equivalent description can be given via $(1,0)$ instead of $(0,1)$ vectors, but the former is preferred e.g. in the context of the $\bar\partial$ equation.)

CR codimension

An important invariant of an almost CR structure is its CR codimension, given by the complex codimension of $V\oplus \bar V$ in $\mathbb C\otimes TM$ or, equivalently, by the real codimension of $H$ in $TM$. The almost CR structures of codimension $0$ are precisely the almost complex structures, and those of codimension $1$ are the ones induced on real hypersurfaces in almost complex manifolds. Important examples of almost CR structures of higher codimension are given by real Lie group orbits, for instance, boundary components of bounded symmetric domains may have arbitrarily high CR codimension.

Almost CR structures as G-structures

To regard an almost CR structure $V$ as a G-structure, consider the subbundle of all frames in $TM$, consisting of all CR isomoprhisms from the flat model $\mathbb C^n\oplus \mathbb R^m$ onto $TM$. Then the group $P$ of all linear CR isomoprhisms of the flat model acts transitively on these frames at each point, and thus defines an $P$-structure on $M$. Vice versa, any such $P$-structure corresponds to an unique almost CR structure.

This construction works, in particular, for almost complex structures, since it is the special case of an almost CR structure corresponding to CR codimension 0.

Induced almost CR structures on real submanifolds of almost complex manifolds

For any real submanifold $M$ in an almost complex manifold $C$, there is canonical almost complex structure on each complex tangent space $H_p:= T_pM\cap J T_pM$ at a point $p\in M$, where $J$ is the complex structure of $C$. If $H_p$ is of constant dimension, $M$ is called CR submanifold of $C$ with the induced almost CR structure. If $M$ is a real hypersurface, it is always a CR submanifold.

Almost CR G-structure as a reduction of an almost complex G-structure

One way to see such reduction, is to consider a CR submanifold $M\subset C$ of an almost complex manifold $C$, such that $M$ is generic in $C$, which means $TM+JTM = TC|_M$, i.e. $TM$ needs to span the full tangent space to $C$ over $\mathbb C$. That genericity condition guarantees that any CR frame can be uniquely extended to a complex frame in $C$. Then the bundle of all CR frames on $M$ (defining the corresponding $G$-structure) is canonically identified with a subbundle of all complex frames on $C$ resticted to $M$, and hence can be seen as a reduction of the corresponding almost complex G-structure.

Integrability

An almost CR structure $V\subset \mathbb C\otimes TM$ is (formally) integrable if the subbundle $V$ is closed under Lie brackets. In contrast to almost complex structures, such formal integrability does not guarantee the existence of sufficiently many local CR functions (implying a local CR embedding into a complex manifold), which is a stronger property sometimes called integrability without the "formal" adjective. Most commonly CR structures are defined as formally integrable almost CR structures, whereas the latter ones are called locally embeddable.

Note that integrability played no role in the above discussion and comes as additional property. I suppose that was the confusion mentioned in the question.

Also note that, in contrast to complex structures, integrability of almost CR structures does not imply any ``flatness'' such as CR equivalences with their flat models.

Nondegeneracy conditions

The most well-known nondegeneracy condition is the one for the Levi form, which can be defined for any almost CR structure (or any codimension).

See e.g. my paper, Normal forms for almost non-integrable CR structures, Amer. J. of Math., 134 (2012), no. 4, 915-947, also available on the arxiv.org.

In particular, if $M$ is of CR codimension $1$ (i.e. of hypersurface type), adding a nondegenerate Levi tensor allows to reduce the almost CR G-structure to the one admitting a canonical Cartan connection, as more extensively described by Andreas Cap in his answer.

There are also many natural generalizations of the Levi-nondegeneracy condition in various directions, where many details can be found in this book:

MR1668103 (2000b:32066) Baouendi, M. Salah(1-UCSD); Ebenfelt, Peter(S-RIT); Rothschild, Linda Preiss(1-UCSD) Real submanifolds in complex space and their mappings.
Princeton Mathematical Series, 47. Princeton University Press, Princeton, NJ, 1999. xii+404 pp. ISBN: 0-691-00498-6

Contact and almost contact structures

In contrast to almost CR structures, contact structures already contain a nondegeneracy condition in their definition. The corresponding almost contact structure is then defined by adding a nondegenerate symplectic form to the hyperplane field, unrelated to the one given by the Lie bracket. That can be equivalently described by appropriate G-structure in a similar way as for almost CR structures described above.

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