ARCHIVUM MATHEMATICUM (BRNO)
Tomus 44 (2008), 491–510
REMARKS ON SPECIAL SYMPLECTIC CONNECTIONS
Martin Panák and Vojtěch Žádník
Abstract. The notion of special symplectic connections is closely related to
parabolic contact geometries due to the work of M. Cahen and L. Schwachhöfer.
We remind their characterization and reinterpret the result in terms of
generalized Weyl connections. The aim of this paper is to provide an alternative
and more explicit construction of special symplectic connections of three
types from the list. This is done by pulling back an ambient linear connection
from the total space of a natural scale bundle over the homogeneous model of
the corresponding parabolic contact structure.
1. Introduction
Special symplectic connection on a symplectic manifold (M, ω) is a torsion-free
linear connection preserving ω which is special in the sense of definitions in 1.1.
The definition of special symplectic connection is rather wide, however, there is a
nice link between special symplectic connections and parabolic contact geometries,
which was established in the profound paper [3]. The main result of that paper
states that, locally, any special symplectic connection on M comes via a symplectic
reduction from a specific linear connection on a one-dimension bigger contact
manifold C, the homogeneous model of some parabolic contact geometry. All the
necessary background on parabolic contact geometries is collected in section 2. The
construction and the characterization from [3] is quickly reminded in section 3,
culminating in Theorem 3.2.
In the next section we provide an alternative and rather direct approach to
special symplectic connections. Firstly we reinterpret the previous results in terms
of parabolic geometries so that the specific linear connections on C are exactly the
exact Weyl connections corresponding to specific choices of scales. A choice of scale
further defines a bundle projection from TC to the contact distribution D ⊂ TC
and this gives rise to a partial contact connection on D. By the very construction,
the only ingredient which yields the special symplectic connection on M is just the
partial contact connection associated to the choice of scale, Proposition 4.2.
Finally, the direct construction of special symplectic connections works via a
pull-back of an ambient symplectic connection on the total space of a canonical
2000 Mathematics Subject Classification: primary 53D15; secondary 53C15, 53B15.
Key words and phrases: special symplectic connections, parabolic contact geometries, Weyl
structures and connections.
The first author was supported by the grant nr. 201/05/P088, the second author by the grant
nr. 201/06/P379, both grants of the Grant Agency of the Czech Republic.
492 M. PANÁK AND V. ŽÁDNÍK
scale bundle ˆC → C. Namely for several specified cases we can find a convenient
ambient connection on ˆC and then compare the exact Weyl connection and the
pull-back connection on C corresponding to the choice of scale so that they coincide
on the contact distribution D, Theorem 4.3. By the previous results, they give
rise the same symplectic connection on M after the reduction. This construction
applies to the projective contact structures, CR structures of hypersurface type,
and Lagrangean contact structures, which are dealt in subsections 4.4, 4.5, and 4.6,
respectively.
1.1. Special symplectic connections. Given a smooth manifold M with a symplectic
structure ω ∈ Ω2
(M), linear connection on M is said to be symplectic if
it is torsion free and ω is parallel with respect to . There is a lot of symplectic
connections to a given symplectic structure, hence studying this subject, further
restrictive conditions appear. Following the article [3], we consider the special
symplectic connections defined as symplectic connections belonging to some of the
following classes:
(i) Connections of Ricci type. The curvature tensor of a symplectic connection
decomposes under the action of the symplectic group into two irreducible components.
One of them corresponds to the Ricci curvature and the other one is the
Ricci-flat part. If the curvature tensor consists only of the Ricci curvature part,
then the connection is said to be of Ricci type.
(ii) Bochner–Kähler connections. Let the symplectic form be the Kähler form
of a (pseudo-)Kähler metric and let the connection preserve this (pseudo-)Kähler
structure. The curvature tensor decomposes similarly as in the previous case into
two parts but this time under the action of the (pseudo-)unitary group. These are
called Ricci curvature and Bochner curvature. If the Bochner curvature vanishes,
the connection is called Bochner–Kähler.
(iii) Bochner–bi-Lagrangean connections. A bi-Lagrangean structure on a symplectic
manifold consists of two complementary Lagrangean distributions. If a symplectic
connection preserves such structure, i.e. both the Lagrangean distributions
are parallel, then again the curvature tensor decomposes into the Ricci and Bochner
part. If the Bochner curvature vanishes, we speak about Bochner–bi-Lagrangean
connections.
(iv) Connections with special symplectic holonomies. We say that a symplectic
connection has special symplectic holonomy if its holonomy is contained in a
proper absolutely irreducible subgroup of the symplectic group. Special symplectic
holonomies are completely classified and studied by various people.
Connections of Ricci type are characterized in the interesting article [2], see
remark 4.4(a) for some detail. The Bochner–Kähler metrics (marginally also the
Bochner–bi-Lagrangean structures) have been thoroughly studied in the deep article
[1]. See also [11] for further investigation of the subject which is more relevant
to our recent interests. For more remarks and references on special symplectic
connections we generally refer to [3].
Note that all the previous definitions admit an analogy in complex/holomorphic
setting but we are dealing only with the real structures in this paper.
REMARKS ON SPECIAL SYMPLECTIC CONNECTIONS 493
Acknowledgement. We would like to thank in the first place to Andreas Čap
for the fruitful discussions and suggestions concerning mostly the Weyl structures,
especially in the technical part of 4.4. Among others we would like to mention
Lorenz Schwachhöfer, Jan Slovák and Jiří Vanžura, who were willing to discuss
some aspects of the geometries in this article.
2. Parabolic contact geometries and Weyl connections
In this section we provide the necessary background from parabolic geometries
and generalized Weyl structures as can be found in [13], [7] or, the most
comprehensively, in [6].
2.1. Parabolic contact geometries. Semisimple Lie algebra admits a contact
grading if there is a grading g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 such that g−2 is one
dimensional and the Lie bracket [ , ] : g−1 × g−1 → g−2 is non-degenerate. If g
admits a contact grading, then g has to be simple. Any complex simple Lie algebra,
except sl(2, C), admits a unique contact grading, but this is not guaranteed generally
in real case. However, the split real form of complex simple Lie algebra and most
of non-compact non-complex real Lie algebras admit a contact grading.
Let g be a real simple Lie algebra admitting a contact grading, let p := g0⊕g1⊕g2
be the corresponding parabolic subalgebra, and let p+ := g1 ⊕ g2. Let further z(g0)
be the center of g0. Let E ∈ z(g0) be the grading element of g and let g0 ⊂ g0 be
the orthogonal complement of E with respect to the Killing form on g. From the
invariance of the Killing form and the fact that [g−2, g2] = E , the subalgebra
g0 ⊂ g0 is equivalently characterized by the fact that [g0, g2] = 0. For later use let
us denote p := g0 ⊕ p+.1
For a semisimple Lie group G and a parabolic subgroup P ⊂ G, parabolic
geometry of type (G, P) on a smooth manifold M consists of a principal P-bundle
G → M and a Cartan connection η ∈ Ω1
(G, g), where g is the Lie algebra of G.
If g is simple Lie algebra admitting a contact grading and the Lie subalgebra
p ⊂ g of P corresponds to this grading, then we speak about parabolic contact
geometry. The contact grading of g gives rise to a contact structure on M as follows.
Under the usual identification TM ∼= G ×P g/p via η, the P-invariant subspace
(g−1 ⊕ p)/p ⊂ g/p, defines a distribution D ⊂ TM, namely
(1) D ∼= G ×P (g−1 ⊕ p)/p .
For regular parabolic geometries of these types, the distribution D ⊂ TM defined
by (1) is a contact distribution. The Lie bracket of vector fields induces the so-called
Levi bracket on the associated graded bundle gr(TM) = D ⊕ TM/D, which is an
algebraic bracket of the form L : D ∧ D → TM/D. The regularity means the Levi
bracket corresponds to the Lie bracket on g− = g−1 ⊕ g−2.
Any contact distribution can be always given as the kernel of a contact form
θ ∈ Ω1
(M), i.e. a one-form such that θ∧(dθ)n
is a volume form on M. In particular,
the restriction of dθ to D ∧ D is non-degenerate. For any choice of contact form θ,
let rθ ∈ X(M) be the corresponding Reeb vector field, i.e. the unique vector field
1Note that in [3] the essential subalgebras g0 and p are denoted by h and p0, respectively.
494 M. PANÁK AND V. ŽÁDNÍK
on M satisfying rθ dθ = 0 and θ(rθ) = 1. This further provides a trivialization of
the quotient bundle TM/D so that TM ∼= D ⊕ R. Next, if X and Y are sections
of D = ker θ then dθ(X, Y ) = −θ([X, Y ]) by the definition of exterior differential.
Altogether, under the trivialization above, the restriction of dθ to D ∧ D coincides
with the Levi bracket L up to the sign.
2.2. Weyl structures. Let (G → M, η) be a parabolic geometry of type (G, P).
Let p ⊂ g be the Lie algebras of the Lie groups P ⊂ G and let g = g−k ⊕ · · · ⊕ g0 ⊕
· · · ⊕ gk be the corresponding grading of g. Let G0 ⊂ P be the Lie group with Lie
algebra g0 and let P+ := exp p+ so that P = G0 P+. Let further G0 := G/P+ → M
be the underlying G0-bundle and let π0 : G → G0 be the canonical projection. The
filtration of the Lie algebra g gives rise to a filtration of TM and the principal
G0-bundle G0 → M plays the role of the frame bundle of the associated graded
gr(TM). The reduction of the structure group of TM to G0 often corresponds to
an additional geometric structure on M and this collection of data we call the
underlying structure on M (see e.g. [7] for more precise formulations).
A Weyl structure for the parabolic geometry (G → M, η) is a global smooth
G0-equivariant section σ : G0 → G of the projection π0. In particular, any Weyl
structure provides a reduction of the P-principal bundle G → M to the subgroup
G0 ⊂ P. Denote by ηi the gi-component of the Cartan connection η ∈ Ω1
(G, g).
For a Weyl structure σ : G0 → G, the pull-back σ∗
η0 defines a principal connection
on the principal bundle G0; this is called the Weyl connection of the Weyl structure
σ. Next, the form σ∗
η− ∈ Ω1
(G0, g−) provides an identification of the tangent
bundle TM with the associated graded tangent bundle gr(TM) and the form
σ∗
η+ ∈ Ω1
(G0, p+) is called the Rho-tensor, denoted by Pσ
. The Rho-tensor is
used to compare the Cartan connection η on G and the principal connection on G
extending the Weyl connection σ∗
η0 from the image of σ : G0 → G.
Any Weyl connection induces connections on all bundles associated to G0, in
particular, there is an induced linear connection on TM. By definition, any Weyl
connection preserves the underlying structure on M. On the other hand, there
are particularly convenient bundles such that the induced connection from σ∗
η0 is
sufficient to determine whole the Weyl structure σ. These are the so-called bundles
of scales, the oriented line bundles over M defined as follows.
2.3. Scales and exact Weyl connections. Let L → M be a principal R+-bundle
associated to G0. This is determined by a group homomorphism λ : G0 → R+
whose derivative is denoted by λ : g0 → R. The Lie algebra g0 is reductive, i.e. g0
splits into a direct sum of the center z(g0) and the semisimple part, hence the only
elements that can act non-trivially by λ are from z(g0). Next, the restriction of
the Killing form B to g0 and further to z(g0) is non-degenerate. Altogether, for
any representation λ : g0 → R there is a unique element Eλ ∈ z(g0) such that
(2) λ (A) = B(Eλ, A)
for all A ∈ g0. By Schur’s lemma, Eλ acts by a real scalar on any irreducible
representation of G0. An element Eλ ∈ z(g0) is called a scaling element if it acts
by a non-zero real scalar on each G0-irreducible component of p+. (In general,
REMARKS ON SPECIAL SYMPLECTIC CONNECTIONS 495
the grading element of g is a scaling element.) A bundle of scales is a principal
R+-bundle associated to G0 via a homomorphism λ : G0 → R+, whose derivative is
given by (2) for some scaling element Eλ. Bundle of scales Lλ
→ M corresponding
to λ is naturally identified with G0/ ker λ, the orbit space of the action of the
normal subgroup ker λ ⊂ G0 on G0.
Let Lλ
→ M be a fixed bundle of scales and let σ : G0 → G be a Weyl structure
of a parabolic geometry (G → M, η). Then the Weyl connection σ∗
η0 on G0 induces
a principal connection on Lλ
and [7, Theorem 3.12] shows that this mapping
establishes a bijective correspondence between the set of Weyl structures and the
set of principal connections on Lλ
. Note that the surjectivity part of the statement
is rather implicit, however there is a distinguished subclass of Weyl structures which
allow more satisfactory interpretation, namely the exact Weyl strucures defined
as follows. Any bundle of scales is trivial and so it admits global smooth sections,
which we usually refer to as choices of scale. Any choice of scale gives rise to a flat
principal connection on Lλ
and the corresponding Weyl structure is then called
exact.
Furthermore, due to the identification Lλ
= G0/ ker λ, the sections of Lλ
→ M
are in a bijective correspondence with reductions of the principal bundle G0 → M to
the structure group ker λ ⊂ G0. Altogether for any choice of scale, the composition
of the two reductions above is a reduction of the principal P-bundle G → M to the
structure group ker λ ⊂ G0 ⊂ P; let us denote the resulting bundle by G0. Hence
the corresponding exact Weyl connection has holonomy in ker λ and by general
principles from the theory of G-structures, it preserves the geometric quantity
corresponding to the choice of scale.
In the cases of parabolic contact geometries, the canonical candidate for the
bundle of scales is the bundle of positive contact one-forms. Note that this is the
bundle of scales corresponding to (a non-zero multiple of) the grading element
E ∈ z(g0), hence the Lie subalgebra ker λ ⊂ g0 is identified with g0 from 2.1. Let
G0 be the connected subgroup in G corresponding to g0 ⊂ g. Reinterpreting the
general principles above: the choice of a contact one-form θ ∈ Ω1
(M) yields a
reduction G0 ⊂ G of the principal bundle G → M to the subgroup G0 ⊂ P and a
principal connection on G0, which preserves not only the underlying structure on
M (so in particular the contact distribution D = ker θ), but moreover the form θ
itself. In other words, θ is parallel with respect to the induced linear connection on
TM.
3. Characterization of special symplectic connections
In this section the quick review of the construction of the special symplectic
connections from the article [3] is described. Consult e.g. [8] for details on invariant
symplectic structures on homogeneous spaces.
3.1. Adjoint orbit and its projectivization. Let g be a real simple Lie algebra
admitting a contact grading and let e2
+ ∈ g be a maximal root element, i.e. a
generator of g2. Let G be a connected Lie group with Lie algebra g. Consider the
496 M. PANÁK AND V. ŽÁDNÍK
adjoint orbit of e2
+ and its oriented projectivization:
(3) ˆC := AdG(e2
+) ⊂ g , C := Po
( ˆC) ⊂ Po
(g) .
The restriction of the natural projection p : g\{0} → Po
(g) to ˆC yields the principal
R+-bundle p : ˆC → C, which we call the cone. The right action of R+ is just the
multiplication by positive real scalars. The fundamental vector field of this action
is the Euler vector field ˆE defined as ˆE(x) := x, for any x ∈ ˆC ⊂ g.
Since ˆC is an adjoint orbit of G in g, and g can be identified with g∗
via the
Killing form, there is a canonical G-invariant symplectic form ˆΩ on ˆC. For any
X, Y ∈ g and α ∈ ˆC ⊂ g∗
, the value of ˆΩ is given by the formula
ˆΩ(ad∗
X(α), ad∗
Y (α)) := α([X, Y ]) ,
where ad∗
: g → gl(g∗
) is the infinitesimal coadjoint representation and ad∗
X(α) =
−α ◦ adX is viewed as an element of Tα
ˆC. Under the identification g ∼= g∗
the
previous formula reads as
(4) ˆΩ(adX(a), adY (a)) = B(a, [X, Y ]) ,
for any X, Y ∈ g and a ∈ ˆC ⊂ g, where B : g×g → R is the Killing form. The Euler
vector field and the canonical symplectic form defines a (canonical) G-invariant
one-form ˆα on ˆC by
(5) ˆα :=
1
2
ˆE ˆΩ .
Immediately from definitions it follows that L ˆE
ˆΩ = 2ˆΩ and consequently dˆα = ˆΩ.
Lemma. Let p : ˆC → C be the cone defined by (3) and let P ⊂ P be the connected
subgroups in G corresponding to the subalgebras p ⊂ p ⊂ g from 2.1. Then
ˆC ∼= G/P and C ∼= G/P so that the contact distribution D ⊂ T(G/P) is identified
with Tp· ker ˆα ⊂ TC.
Proof. By definition, the group G acts transitively both on ˆC and C = ˆC/R+.
Since [A, e2
+] = 0 if and only if A ∈ p and we assume the Lie subgroup P ⊂ G
corresponding to p ⊂ g is connected, the stabilizer of e2
+ is precisely P . Hence
the orbit ˆC is identified with the homogeneous space G/P . Since P ⊃ P is also
connected, P/P is identified with the subgroup {exp tE : t ∈ R} ∼= R+ in P. Hence
P preserves the ray of positive multiples of e2
+ so that C = ˆC/R+ is identified with
G/P.
For the last part of the statement, note that the Euler vector field is generated by
(a non-zero multiple of) the grading element E ∈ z(g0). The canonical one-form ˆα on
ˆC is G-invariant, so it is determined by its value in the origin o ∈ G/P , i.e. e2
+ ∈ ˆC,
which is a P -invariant one-form φ on g/p . By (4) and (5), φ is explicitly given as
φ(X) = B(e2
+, [E, X]), possibly up to a non-zero scalar multiple. The formula is
obviously independent of the representative of X in g/p and the kernel of φ is just
(g−1⊕p)/p . The tangent map of the projection p : ˆC → C corresponds to the natural
projection g/p → g/p, hence Tp· ker ˆα ⊂ TC corresponds to (g−1 ⊕ p)/p ⊂ g/p
which defines the contact distribution D ⊂ T(G/P) in (1).
REMARKS ON SPECIAL SYMPLECTIC CONNECTIONS 497
Remarks. (a) Note that in contrast to the definition of the cone in [3] we do not
assume the center of G is trivial. Hence the two approaches differ by a (usually
finite) covering. Because of the very local character of all the constructions that
follow, this causes no problem and we will not mention the difference below.
(b) The homogeneous space ˆC ∼= G/P is an example of a symplectic homogeneous
space, i.e. a homogeneous space with an invariant symplectic structure. According
to [8, Corollary 1], for G being semisimple, any simply connected symplectic
homogeneous space of a Lie group G is isomorphic to a covering of some G-orbit
in g, which is thought with the (restriction of the) canonical symplectic form.
Moreover the covering map and hence the orbit are unique.
(c) According to [3, Prop. 3.2], the bundle ˆC → C can be identified with the
bundle of positive contact forms on C so that ˆΩ = dˆα corresponds to the restriction
of the canonical symplectic form on the cotangent bundle T∗
C. In detail, a section
s: C → ˆC yields the contact one-form θs := s∗
ˆα and, by the naturality of the
exterior differential, dθs = s∗ ˆΩ.
3.2. General construction. Let a be an element of a real simple Lie algebra g
admitting a contact grading. With the notation as before, let ξa be the fundamental
vector field of the left action of G on C ∼= G/P corresponding to a ∈ g. Let us denote
by Ca the (open) subset in C where ξa is transverse to the contact distribution
D ⊂ TC and oriented in accordance with a fixed orientation of TC/D. The vector
field ξa gives rise to a unique contact one-form θa on Ca such that ξa is its Reeb
field. In other words, θa ∈ Ω1
(Ca) is uniquely determined by the conditions
(6) ker θa = D and θa(ξa) = 1 .
Since ξa is a contact symmetry, i.e. Lξa
D ⊂ D, it easily follows that Lξa
θa = 0
and consequently ξa dθa = 0. Let Ta ⊂ G denote the one-parameter subgroup
corresponding to the fixed element a ∈ g. We say that an open subset U ⊂ Ca is
regular if the local leaf space MU := Ta \ U is a manifold. Since ξa dθa = 0 and
dθa has maximal rank, it descends to a symplectic form ωa on MU , for any regular
U ⊂ Ca.
Next, let π: G → G/P ∼= C be the canonical P-principal bundle and consider
its restriction to Ca. If Ca is non-empty, then [3, Theorem 3.4] describes explicitly
a subset Γa in π−1
(Ca) ⊂ G, which forms a G0-principal bundle over Ca where
G0 is the subgroup of P as in 2.3. For a regular open subset U ⊂ Ca, denote
ΓU := π−1
(U) ⊂ Γa. Note that ΓU is invariant under the action of Ta. Denoting
BU := Ta \ ΓU , BU → MU is a G0-principal bundle and [3, Theorem 3.5] shows
that the restriction of the (g−2 ⊕ g−1 ⊕ g0)-component of the Maurer–Cartan form
µ ∈ Ω1
(G, g) to ΓU descends to a (g−1 ⊕ g0)-valued coframe on BU . Altogether,
the bundle BU → MU is interpreted as a classical G0-structure and the g0-part of
the coframe above induces a linear connection on MU . It turns out this connection
is special symplectic connection with respect to the symplectic form ωa.
Surprisingly, [3, Theorem B] proves that any special symplectic connection can
be at least locally obtained by the previous construction. With an assumption on
498 M. PANÁK AND V. ŽÁDNÍK
dim g ≥ 14, which is equivalent to dim MU ≥ 4, we reformulate the main result of
[3] as follows.
Theorem ([3]). Let g be a simple Lie algebra of dimension ≥ 14 admitting a
contact grading. With the same notation as above, let a ∈ g be such that Ca ⊂ C is
non-empty and let U ⊂ Ca be regular. Then
(a) the local quotient MU carries a special symplectic connection,
(b) locally, connections from (a) exhaust all the special symplectic connections.
An instance of the correspondence between the various classes of special symplectic
connections and contact gradings of simple Lie algebras is as follows. For
dim MU = 2n, special symplectic connections of type (i), (ii) and (iii), according
to the definitions in 1.1, corresponds to the contact grading of simple Lie algebras
sp(2n + 2, R), su(p + 1, q + 1) with p + q = n, and sl(n + 2, R), respectively. The
corresponding parabolic contact structure on C ∼= G/P is the projective contact
structure, CR structure of hypersurface type, and Lagrangean contact structure,
respectively. Details on each of these structures are treated in the next section in
details.
4. Alternative realization of special symplectic connections
Below we describe parabolic contact structures corresponding to special symplectic
connections of type (i), (ii) and (iii) as mentioned above. The aim of this section
is, for each of the listed cases, to provide the characterization of Theorem 3.2, and
so the realization of special symplectic connections, in more explicit and satisfactory
way. For this purpose we interpret the model cone p: ˆC ∼= G/P → G/P ∼= C in
each particular case and look for a natural ambient connection ˆ on ˆC which is
good enough to give rise the easier interpretation. We start with a reinterpretation
of the construction from 3.2 in terms of Weyl structures and conenctions.
4.1. Partial contact connections. In order to formulate the next results we
need the notion of partial contact connections. For a general distribution D ⊂
TM on a smooth manifold M, a partial linear connection on M is an operator
Γ(D) × X(M) → X(M) satisfying the usual conditions for linear connections.
In other words, we modify the notion of linear connection on TM just by the
requirement to differentiate only in the directions lying in D. If a partial linear
connection preserves D, then restricting also the second argument to D yields an
operator of the type Γ(D) × Γ(D) → Γ(D); in the case the distribution D ⊂ TM
is contact, we speak about the partial contact connection.
Given a contact distribution D ⊂ TM and a classical linear connection on M,
any choice of a contact one-form induces a partial contact connection D
as follows.
Let θ ∈ Ω1
(M) be a contact one-form with the contact subbundle D and let rθ
be the corresponding Reeb vector field as in 2.1. Let us denote by πθ : TM → D
the bundle projection induced by θ, namely the projection to D in the direction of
rθ ⊂ TM. Now for any X, Y ∈ Γ(D), the formula
(7) D
XY := πθ( XY )
REMARKS ON SPECIAL SYMPLECTIC CONNECTIONS 499
defines a partial contact connection and we say that D
is induced from by θ.
The contact torsion of the partial contact connection D
is a tensor field of type
D ∧ D → D defined as the projection to D of the classical torsion. More precisely,
if D
is induced from by θ, TD
denotes the contact torsion of D
and T is the
torsion of , then TD
(X, Y ) = πθ(T(X, Y )) = D
XY − D
Y X − πθ([X, Y ]) for any
X, Y ∈ Γ(D).
4.2. General construction revisited. In the construction of special symplectic
connection in 3.2, we started with a choice of an element a ∈ g which in particular
induced a contact one-form θa on Ca. Then we described the G0-principal bundle
Γa → Ca which is actually a reduction of the P-principal bundle π−1
(Ca) → Ca to the
structure group G0 ⊂ P. In terms of subsection 2.3, the couple (π−1
(Ca) → Ca, µ)
forms a flat parabolic geometry of type (G, P) and the contact form θa represents
a choice of scale. In this vein, the reduction Γa ⊂ π−1
(Ca) above is interpreted
as (the image of) an exact Weyl structure and below we show this is exactly
the one corresponding to θa. In particular, the restriction of the g0-part of the
Maurer–Cartan form µ to Γa defines the exact Weyl connection preserving θa.
Further restriction to a regular subset U ⊂ Ca and the factorization by Ta finally
yielded a special symplectic connection on MU = Ta \ U. In the current setting
together with the definitions in 4.1, it is obvious that the resulting connection on
MU is fully determined by the partial contact connection induced by θa from the
exact Weyl connection on U ⊂ Ca corresponding to θa. Since any Weyl connection
preserves the contact distribution D, the induced partial contact connection is just
the restriction to the directions in D. Altogether, we can recapitulate the results
in 3.2 as follows.
Proposition. Let a ∈ g be so that Ca ⊂ C is non-empty and let U ⊂ Ca be regular.
Let θa be the contact one-form on U ⊂ Ca determined by a ∈ g as in (6). Then the
special symplectic connection on MU constructed in 3.2 is fully determined by the
partial contact connection induced from the exact Weyl connection corresponding
to θa.
Proof. According to the discussion above, we only need to show that the exact
Weyl structure represented by Γa ⊂ G is the one corresponding to the scale θa.
This easily follows from the definitions of θa and Γa:
The contact one-form θa is defined in (6) by ξa, the fundamental vector field
corresponding to the element a ∈ g. The vector field ξa on Ca ⊂ G/P is the
projection of the right invariant vector field on π−1
(Ca) ⊂ G generated by a. Using
the identification TC ∼= G ×P (g/p), the frame form corresponding to ξa is the
equivariant map G → g/p given by g → Adg−1 (a) + p.
On the other hand, the subset Γa ⊂ π−1
(Ca) is explicitly described in the proof
of [3, Theorem 3.4] as
Γa = g ∈ G : Adg−1 (a) = 1
2 e2
− + p ,
where e2
− is the unique element of g−2 such that B(e2
−, e2
+) = 1. Obviously, the
restriction of the frame form of ξa to Γa is constant, which just means that the
vector field ξa is parallel with respect to the exact Weyl connection corresponding
500 M. PANÁK AND V. ŽÁDNÍK
to Γa. Since ξa is the Reeb vector field of the contact one-form θa, the latter is
parallel if and only if the former is, which completes the proof.
4.3. Pull-back connections. Let p: ˆC → C be the cone as in 3.1. Any smooth
section s: C → ˆC determines a principal connection on ˆC; the corresponding
horizontal lift of vector fields is denoted as X → Xhor
. An ambient linear connection
ˆ on ˆC defines a linear connection s
on C by the formula
(8) s
XY := Tp( ˆ Xhor Y hor
) .
We call s
the pull-back connection corresponding to s. On the other hand, for
any section s, which we call a choice of scale by 2.3, let θs = s∗
ˆα be the contact
form and let ¯ s
be the corresponding exact Weyl connection on C. In the rest of
this section, we are looking for an ambient connection ˆ on ˆC so that both s
and
¯ s
induce the same partial contact connection on D ⊂ TC. For this reason it turns
out that ˆ has to be symplectic, i.e. ˆ ˆΩ = 0.
The following statement provides together with Theorem 3.2 and Proposition
4.2 the desired simple realization of special symplectic connections of type (i), (ii),
and (iii) according to the list in 1.1. The point is that in all these cases the ambient
connection ˆ is very natural and easy to describe.
Theorem. Let ˆC → C be the model cone for g = sp(2n + 2, R), su(p + 1, q + 1) or
sl(n + 2, R). Then there is an ambient symplectic connection ˆ on the total space
of ˆC so that, for any section s: C → ˆC, the induced partial contact connections of
the exact Weyl connection and the pull-back connection corresponding to s coincide.
Although the definition of the cone ˆC → C is pretty general, its convenient
interpretation necessary to find a natural candidate for ˆ is no more universal.
In order to prove the Theorem, we deal in following three subsections with each
case individually. It follows that the reasonable interpretation of the cone in any
discussed case is more or less standard and we refer primarily to [6] for a lot of
details. The candidate for an ambient connection ˆ is almost canonical, therefore
in the proofs of subsequent Propositions we focus only in the justification of the
choices.
Note that a natural guess for ˆ to be a G-invariant symplectic connection
on ˆC = G/P does help only for contact projective structures, i.e. the structures
corresponding to the contact grading of g = sp(2n + 2, R). This is due to the
following statement, which is an immediate corollary of [12, Theorem 3]: For a
connected real simple Lie group G with Lie algebra g, the nilpotent adjoint orbit
C = AdG(e2
+) admits a G-invariant linear connection if and only if g ∼= sp(m, R).
For a reader’s convenience we assume the dimension of C = G/P to be always
m = 2n + 1. Consequently, dim ˆC = 2n + 2 and we further continue the convention
that all important objects on ˆC are denoted with the hat.
4.4. Contact projective structures. Contact projective structures correspond
to the contact grading of the Lie algebra g = sp(2n + 2, R), the only real form of
sp(2n + 2, C) admitting the contact grading. These structures are studied in [9]
REMARKS ON SPECIAL SYMPLECTIC CONNECTIONS 501
in whole generality: contact projective structure on a contact manifold (M, D) is
defined as a contact path geometry such that the paths are among geodesics of a
linear connection on M; the paths are then called contact geodesics. In analogy
to classical projective structures, a contact projective structure is given by a class
of linear connections [ ] on TM having the same contact torsion and the same
non-parametrized geodesics such that the following property is satisfied: if a geodesic
is tangent to D in one point then it remains tangent to D everywhere.
The model contact projective structure is observed on the projectivization of
symplectic vector space (R2n+2
, ˆΩ) with ˆΩ being a standard symplectic form. Let G
be the group of linear automorphisms of R2n+2
preserving ˆΩ, i.e. G := Sp(2n+2, R).
In order to represent conveniently the contact grading of the corresponding Lie
algebra, let ˆΩ be given by the matrix


0 0 1
0 J 0
−1 0 0

, with respect to the standard
basis of R2n+2
, where J =
0 In
−In 0
and In is the identity matrix of rank n. For
Jt
= −J, the Lie algebra g = sp(2n + 2, R) is represented by block matrices of the
form
g =





a Z z
X A JZt
x −Xt
J −a

 : A ∈ sp(2n, R)



,
where the non-specified entries are arbitrary, i.e. x, a, z ∈ R, X ∈ R2n
and Z ∈ R2n∗
,
and the fact A ∈ sp(2n, R) means that At
J + JA = 0. Particular subspaces of the
contact grading g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 of g is read along the diagonals so that
g−2 is represented by x ∈ R, g−1 by X ∈ R2n
, etc. In particular, g0 is represented
by the pairs (a, A) ∈ R × sp(2n, R) so that sp(2n, R) is the semisimple part gss
0
and the center z(g0) is generated by the grading element E corresponding to the
pair (1, 0). Following the general setup in 2.1, p = g0 ⊕ g1 ⊕ g2, p = gss
0 ⊕ g1 ⊕ g2,
and P, P are the corresponding connected Lie subgroups in G. Schematically, the
parabolic subgroup P ⊂ G is given as
P =





r ∗ ∗
0 ∗ ∗
0 0 r−1

 : r ∈ R+



and P ⊂ P corresponds to r = 1. Easily, G acts transitively on R2n+2
\ {0}, P is
the stabilizer of the first vector of the standard basis, and P is the stabilizer of the
corresponding ray. Hence ˆC ∼= G/P is identified with R2n+2
\ {0} and its oriented
projectivization C ∼= G/P is further identified with the sphere S2n+1
⊂ R2n+2
.
Altogether, we have interpreted the model cone for contact projective structures as
ˆC ∼= R2n+2
\ {0} → S2n+1 ∼= C .
It is easy to check that the canonical symplectic form on ˆC corresponds to the
standard symplectic form on R2n+2
, which is G-invariant by definition. As a
particular interpretation of the general definition in 3.1, the contact distribution
502 M. PANÁK AND V. ŽÁDNÍK
D ⊂ TS2n+1
is given by Dv = v⊥
∩ TvS2n+1
, where v ∈ S2n+1
and v⊥
= {x ∈
R2n+2
: ˆΩ(v, x) = 0}.
Next, let ˆ be the canonical flat connection on R2n+2
. Then the connections
on S2n+1
defined by (8) form projectively equivalent connections having the great
circles as common non-parametrized geodesics. Any great circle is the intersection
of S2n+1
with a plane passing through 0. If the plane is isotropic with respect to ˆΩ,
we end up with contact geodesics. Note that no connection in the class preserves the
contact distribution, since it is obviously torsion-free, however the induced partial
contact connection coincides with the restriction of an exact Weyl connection to D:
Proposition. Let ˆC → C be the model cone for g = sp(2n+2, R). Then C ∼= S2n+1
,
ˆC ∼= R2n+2
\ {0}, ˆΩ corresponds to the standard symplectic form on R2n+2
, and the
ambient symplectic connection ˆ from Theorem 4.3 is the canonical flat connection
on R2n+2
.
Proof. Since ˆC ∼= G/P , the tangent bundle T ˆC is identified with the associated
bundle G×P (g/p ) via the Maurer–Cartan form µ on G, where the action of P on
g/p is induced from the adjoint representation. On the other hand, ˆC ∼= R2n+2
\{0},
so g/p ∼= R2n+2
as vector spaces. R2n+2
is the standard representation of G and
the essential observation for the next development is that its restriction to P ⊂ G
is isomorphic to the representation of P on g/p . Explicitly, the isomorphism
R2n+2
→ g/p is given by
(9)


a
X
x

 →


a 0 0
X 0 0
x −Xt
J −a

 + p .
Altogether, T ˆC ∼= G×P R2n+2
and since the representation of P is the restriction of
a representation of G, the Maurer–Cartan form µ induces a linear connection on T ˆC
by general arguments as in [5]. More precisely, G×P R2n+2 ∼= (G×P G)×G R2n+2
,
where the (homogeneous) principal bundle G ×P G → ˆC represents the symplectic
frame bundle of ˆC. The Maurer–Cartan form µ on G extends to a G-invariant
principal connection on G ×P G. The latter connection induces connections on
all associated bundles, in particular, this gives rise to a flat invariant symplectic
connection on T ˆC, i.e. the canonical flat connection ˆ on R2n+2
. Due to this
interpretation of ˆ , we are going to describe the covariant derivative with respect
to ˆ in an alternative way which will provide a comparison of the pull-back and
exact Weyl connections.
For a vector field ˆX ∈ X( ˆC) let us denote by f ˆX the corresponding frame form,
i.e. the P -equivariant map from G to g/p ∼= R2n+2
. As ˆ is an instance of tractor
connection, the frame form of the covariant derivative of ˆY in the direction of ˆX
turns out to be expressed as
(10) fˆ ˆX
ˆY = ˆξ ·fˆY + µ(ˆξ) ◦ fˆY ,
where ˆξ ∈ X(G) is a lift of ˆX ∈ X( ˆC) and ◦ denotes the standard representation of
g on R2n+2
; see [5, section 2] or [13, section 2.15].
REMARKS ON SPECIAL SYMPLECTIC CONNECTIONS 503
From now on, let s : C → ˆC be a fixed section of the model cone, i.e. a choice of
scale, and let σs
: G0 → G be the corresponding exact Weyl structure, where G0 is
the principal G0-bundle as in 2.3. Since R2n+2 ∼= g−⊕ E as G0-modules, the section
s provides the identification T ˆC ∼= G0 ×G0
(g− ⊕ E ) (similarly, TC ∼= G0 ×G0
g−).
If ˆX is a vector field on ˆC and f ˆX the corresponding frame form as above, than the
frame form corresponding to the identification T ˆC ∼= G0 ×G0
(g− ⊕ E ) is given by
f ˆX ◦ σs
. (Similarly for vector fields on C.) Restricting to the image of σs
within G,
we do not distinguish between these two interpretations.
In the definition of the pull-back connection, Xhor
∈ X( ˆC) denotes the horizontal
lift of vector field X ∈ X(C) with respect to the principal connection on ˆC determined
by s. According to the identifications above, the horizontality in terms of the frame
forms is expressed as fXhor = (0, fX)t
∈ R2n+2 ∼= E ⊕ g−. Hence the formula (10)
yield
(11) fˆ
Xhor Y hor =
0
ˆξ ·fY
+ µ(ˆξ) ◦
0
fY
.
The tangent map of the projection p: ˆC → C corresponds to the projection π: g− ⊕
E → g− in the direction of E , hence the result of the covariant derivative s
XY
with respect to the pull-back connection defined by (8) corresponds to the g− part
of (11).
On the other hand, the covariant derivative ¯ s
with respect to the exact Weyl
connection corresponding to s is given by f¯ s
X
Y = ξs
· (fY ◦ σs
), where ξs
∈ X(G0)
is the horizontal lift of X ∈ X(C) with respect to the principal connection on G0.
This is characterized by µ0(Tσs
·ξs
) = 0, i.e. Tσs
·ξs
= ξ + ζPs
(ξ)
where ξ is the lift
such that µ(ξ) ∈ g− and Ps
is the Rho-tensor. Since ξs
· (fY ◦ σs
) = (Tσs
·ξs
)·fY ,
we conclude by the formula
(12) f¯ s
X
Y = ξ ·fY − ad Ps
(ξ) (fY ) .
Altogether, considering Tσs
·ξs
instead of ˆξ in (11), the desired comparison of
the pull-back connection and the exact Weyl connection determined by s is given
by
(13) f s
X
Y − ¯ s
X
Y = π µ(ξ) + Ps
(ξ) ◦
0
fY
,
where π denotes the projection g− ⊕ E → g− as before. In particular, expressing
the standard action on the right hand side of (13) for X, Y ∈ Γ(D), i.e. for µ(ξ)
and fY having values in g−1, the difference tensor turns out to be of the form
(14) s
XY − ¯ s
XY = −dθs(X, Y )rs
where θs and rs is the contact form and the Reeb vector field, respectively, corresponding
to the scale s: C → ˆC. This shows that the induced partial contact
connections of the pull-back connection and the exact Weyl connection determined
by s coincide.
504 M. PANÁK AND V. ŽÁDNÍK
Remarks. (a) The paper [2] provides a characterization of symplectic connections
of Ricci type with specific symplectic connections obtained by a reduction procedure
from a hypersurface in a symplectic vector space. More specifically, for a ∈ g =
sp(2n + 2, R) the hypersurface in R2n+2
is defined by
Σa := x ∈ R2n+2
: ˆΩ(x, ax) = 1 ,
where ˆΩ is the standard symplectic form, and all the connections are induced
from the flat ambient connection on R2n+2
. Basically, this is just another view on
the description of pull-back connections which is conceivable whenever C can be
interpreted as a hypersurface in ˆC; the section s : C → ˆC is then understood as a
deformation of the hypersurface. In the current case, C ∼= S2n+1
⊂ R2n+2
\ {0} ∼= ˆC
and one easily shows that for the section sa corresponding to an element a ∈ g,
the image of sa really coincides with the hypersurface Σa above.
(b) Note that the argument in the proof of Proposition above can be directly
generalized in at least two ways: First, the homogeneous model and the flat connection
ˆ can be replaced by a general manifold M with contact projective structure
and the unique ambient connection on the total space of a scale bundle over M,
respectively, which is established in [9, Theorem B]. The general ambient connection
is induced by a canonical Cartan connection in the very same manner as above.
Second, the comparison of pull-back connections and exact Weyl connections can
be extended to general Weyl connections. Indeed, any Weyl connection corresponds
by 2.3 to a principal connection on a scale bundle, which actually is the important
ingredient in the definition of pull-back connections in (8). The fact that the
principal connection on the bundle of scales is given by a section plays no role in
this context.
(c) By [9, Theorem A], any choice of scale determines a unique linear connection
on M so that it preserves the corresponding contact form and its differential,
represents the contact projective structure, and has a normalized torsion. Note that
this is neither the pull-back connection nor the exact Weyl connection, however
the induced partial contact connection is still the same. Connections of this type
are close analogies of Webster–Tanaka connections well known in CR geometry.
4.5. CR structures of hypersurface type. These structures correspond to the
contact grading of the Lie algebra g = su(p + 1, q + 1), a real form of sl(n + 2, C),
where p + q = n once for all. In fact the correct full name of the general geometric
structure of this type is non-degenerate partially integrable almost CR structure of
hypersurface type. This structure on a smooth manifold M is given by a contact
distribution D ⊂ TM with a complex structure J : D → D so that the Levi
bracket L : D ∧ D → TM/D is compatible with the complex structure, i.e.
L(J−, J−) = L(−, −) for any −, − ∈ Γ(D). A choice of contact form provides an
identification of TxM/Dx with R, for any x ∈ M, and the latter condition on the
Levi bracket says that L(−, J−) is a non-degenerate symmetric bilinear form on
D, that is a pseudo-metric. Hence L(−, J−) + iL(−, −) is a Hermitean form on D
whose signature (p, q) is the signature of the CR structure.
REMARKS ON SPECIAL SYMPLECTIC CONNECTIONS 505
The classical examples of CR structures of the above type are induced on
non-degenerate real hypersurfaces in Cn+1
. In general, for a real submanifold
M ⊂ Cn+1
, the CR structure on M is induced from the ambient complex space
Cn+1
so that the distribution D is the maximal complex subbundle in TM, and the
complex structure J is the restriction to D of the multiplication by i. The model
CR structures of hypersurface type are induced on the so-called hyperquadrics, cf.
[10]. A typical hyperquadric of signature (p, q) is described as a graph
(15) Q := (z, w) ∈ Cn
× C : (w) = h(z, z) ,
or as
(16) S := {(z, w) ∈ Cn
× C : h(z, z) + |w|2
= 1},
where h is a Hermitean form of signature (p, q). It turns out that the induced
CR structures on Q and S are equivalent and the equivalence is established
by the restriction of the biholomorphism (z, w) → z
w−i , 1−iw
w−i . Note that this
identification is almost global (only the point (0, i) ∈ S is mapped to infinity) and
projective. In particular, Q and S are different affine realizations of a projective
hyperquadric in CPn+1
which is identified with the homogeneous space G/P as
follows.
Let G be the group of complex linear automorphisms of Cn+2
preserving a
Hermitean form H of signature (p + 1, q + 1), i.e. G := SU(p + 1, q + 1). Let
the Hermitean form H be given by the matrix


0 0 − i
2
0 I 0
i
2 0 0

, with respect to
the standard basis (e0, e1, . . . , en, en+1), where I =
Ip 0
0 −Iq
represents the
Hermitean form h of signature (p, q) on e1, . . . , en ⊂ Cn+2
. According to this
choice, the Lie algebra g = su(p+1, q+1) is represented by matrices of the following
form with blocks of sizes 1, n, and 1
g =





c 2iZ v
X A I ¯Zt
u −2i ¯Xt
I −¯c

 : u, v ∈ R, A ∈ u(p, q), tr(A) + 2i (c) = 0



,
where the non-specified entries are arbitrary, i.e. X ∈ Cn
, Z ∈ Cn∗
, and c ∈ C.
(Note that A ∈ u(p, q) means ¯At
I + IA = 0, so in particular tr(A) is purely
imaginary complex number.) The contact grading of g is read along the diagonals
as in 4.4. In particular, g0 is represented by the pairs (c, A) ∈ C × u(p, q) with
the constrain tr(A) + 2i (c) = 0. The center z(g0) is two-dimensional, where the
grading element E corresponds to the pair (1, 0), and the semisimple part gss
0
is isomorphic to su(p, q). The subalgebra g0
∼= u(p, q) corresponds to the pairs
of the form (−1
2 tr(A), A). Note that the compatibility of the Levi bracket with
the complex structure on D is reflected here by the fact that [iX, iY ] = [X, Y ]
for any X, Y ∈ g−1. Subalgebras p ⊂ p ⊂ g are defined as in 2.1, P ⊂ P are
the corresponding connected subgroups in G. The parabolic subgroup P ⊂ G is
506 M. PANÁK AND V. ŽÁDNÍK
schematically indicated as
P =





reiθ
∗ ∗
0 ∗ ∗
0 0 1
r eiθ

 : r ∈ R+



and P ⊂ P corresponds to r = 1.
Let N be the set of non-zero null-vectors in Cn+2
with respect to the Hermitean
form H. Clearly, G preserves and acts transitively on N. If Q ⊂ G denotes the
stabilizer of the first vector of the standard basis then N is identified with the
homogeneous space G/Q. Obviously Q ⊂ P ⊂ P corresponds to r = 1 and θ = 0
according to the description of P above. Since P /Q ∼= U(1), the group of complex
numbers of unit length, the homogeneous space G/P is identified with N/U(1).
Next P ⊃ P is the stabilizer of the complex line generated by the first vector of
the standard basis, so the homogeneous space G/P is identified with N/C∗
, the
complex projectivization of N. Altogether a natural interpretation of the model
cone in this case is
ˆC ∼= N/U(1) → N/C∗ ∼= C.
A direct substitution shows that the hyperquadric Q from (15) is the intersection
of N with the complex hyperplane z0 = 1. According to the new basis (e0 +
ien+1, e1, . . . , en, e0 − ien+1) of Cn+2
, the Hermitean metric H is in the diagonal
form so that the hyperquadric S from (16) is the intersection of N with the complex
hyperplane z0 = 1 (where the dash refers to coordinates with respect to the new
basis). This recovers the identification above, in particular, both Q and S are
identified with N/C∗ ∼= C.
From now on, let C be the hyperquadric S in the hyperplane z0 = 1 which we
naturally identify with Cn+1
. This hyperplane without the origin is further identified
with N/U(1) ∼= ˆC under the map (z , w ) → ( |h(z , z ) + |w |2|, z , w ). Denote
by ˆh the induced Hermitean metric (of signature (p + 1, q)) on this hyperplane and
let ˆΩ be its imaginary part. Obviously, both ˆh and ˆΩ are G-invariant, and an easy
calculation shows that ˆΩ corresponds to the canonical symplectic form on ˆC up to
non-zero constant multiple. Altogether, the defining equation (16) for S ⊂ Cn+1
reads as
(17) S = z ∈ Cn+1
: ˆh(z, z) = 1
and the most satisfactory interpretation of the model cone is
ˆC ∼= Cn+1
\ {0} → S ∼= C .
Proposition. Let ˆC → C be the model cone for g = su(p + 1, q + 1). Then
ˆC ∼= Cn+1
\ {0} and C ∼= S, the hyperquadric in Cn+1
\ {0} given by (17), where
ˆh is the Hermitean metric of signature (p + 1, q). Further, ˆΩ corresponds to the
imaginary part of ˆh and the ambient symplectic connection ˆ from Theorem 4.3 is
the canonical flat connection on Cn+1
.
Proof. The connection ˆ is obviously symplectic, i.e. ˆ is torsion-free and ˆ ˆΩ = 0.
By definition, ˆΩ is the imaginary part of the Hermitean metric ˆh on Cn+1
. Its
REMARKS ON SPECIAL SYMPLECTIC CONNECTIONS 507
real part ˆg is then expressed in terms of ˆΩ and the standard complex structure on
Cn+1
as ˆg = ˆΩ(−, i−). This is a real pseudo-metric on Cn+1 ∼= R2n+2
of signature
(2p + 2, 2q) and ˆ can be seen as the Levi–Civita connection of ˆg.
As in general, let ˆα := ˆE ˆΩ. Let s: S → Cn+1
\ {0} be a section of the cone
and let θ := s∗
ˆα be the corresponding contact one-form on S. Then g := dθ(−, i−)
is a non-degenerate symmetric bilinear form on the contact distribution D which
has to be preserved by the Weyl connection ¯ s
. Next, since we deal with the
homogeneous model, the contact torsion of ¯ s
vanishes. In fact, the corresponding
partial contact connection on D is uniquely determined by the fact that (i) it leaves
g to be parallel and (ii) its contact torsion vanishes.
In order to prove the statement, it suffices to show that (i) and (ii) is satisfied
also by the partial contact connection induced by the pull-back connection s
corresponding to s. However, since ˆ is torsion-free, the pull-back connection s
is torsion-free as well, hence the condition (ii) is satisfied trivially. The condition
(i) follows as follows: For X, Y, Z ∈ Γ(D), expand
( s
Xg)(Y, Z) = X ·dθ(Y, iZ) − dθ( s
XY, iZ) − dθ(Y, i s
XZ) .
Since θ = s∗
ˆα and dˆα = ˆΩ, by the naturality of exterior differential we have got
dθ = s∗ ˆΩ. Next easily, Ts·X = Xhor
◦ s and, by the definition of the pull-back
connection in 4.3, Ts· s
XY = ˆ Xhor Y hor
◦ s mod ˆE . Since ˆα = 1
2
ˆE ˆΩ and
D = Tp· ker ˆα, the previous formula is rewritten as
X · ˆΩ(Ts·Y, Ts·iZ) − ˆΩ(Ts· s
XY, Ts·iZ) − ˆΩ(Ts·Y, Ts·i s
XZ)
= Xhor
· ˆΩ(Y hor
, iZhor
) − ˆΩ( ˆ Xhor Y hor
, iZhor
) − ˆΩ(Y hor
, i ˆ Xhor Zhor
) .
However, the very last expression is just ( ˆ Xhor ˆg)(Y hor
, Zhor
), which vanishes
trivially by definitions.
4.6. Lagrangean contact structures. Lagrangean contact structures correspond
to the contact grading of g = sl(n + 2, R), another real form of sl(n + 2, R).
Lagrangean contact strucure on a smooth manifold M consists of the contact
distribution D ⊂ TM and a fixed decomposition D = L⊕R so that the subbundles
L and R are Lagrangean, i.e. isotropic with respect to the Levi bracket L: D ∧D →
TM/D. These structures was profoundly studied in [14] where we refer for a lot of
details. The model Lagrangean contact structure appears on the projectivization
of the cotangent bundle of real projective space; let us present the algebraic
background first.
The contact grading of g = sl(n + 2, R) is read diagonally as in 4.4 and 4.5 from
the following block decomposition
g =





a Z1 z
X1 B Z2
x X2 c

 : a + tr(B) + c = 0



,
where as usual the non-specified entries are arbitrary, i.e. x, a, c, z ∈ R, X1, Z2 ∈
Rn
, X2, Z1 ∈ Rn∗
, and B ∈ gl(n, R). The subalgebra g0 is represented by the
triples (a, B, c) ∈ R × gl(n, R) × R so that a + tr(B) + c = 0. The center z(g0)
508 M. PANÁK AND V. ŽÁDNÍK
is two-dimensional and the grading element E corresponds to (1, 0, −1). The
semisimple part gss
0 is isomorphic to sl(n, R) and the subalgebra g0
∼= gl(n, R)
is represented by all triples of the form −1
2 tr(B), B, −1
2 tr(B) . The subspace
g−1 defining the contact distribution is split as g−1 = gL
−1 ⊕ gR
−1, where gL
−1 is
represented by X1 ∈ Rn
and gR
−1 by X2 ∈ Rn∗
, so that this splitting is invariant
under the adjoint action of g0. Furthermore, the subspaces gL
−1 and gR
−1 are isotropic
with respect to the bracket [ , ]: g−1 × g−1 → g−2, which reflects the geometric
definition of the structure in terms of Levi bracket. Similarly, g1 splits as gL
1 ⊕ gR
1 .
The subalgebras p ⊂ p ⊂ g are given as before. Let G be the group SL(n + 2, R).
The connected parabolic subgroup P ⊂ G corresponding to p ⊂ g is schematically
indicated as
P =





pq ∗ ∗
0 ∗ ∗
0 0 p
q

 : p, q ∈ R+



and P ⊂ P corresponds to q = 1.
The homogeneous space G/P is naturally identified with the set of flags of
half-lines in hyperplanes in Rn+2
. Indeed, the standard action of G on Rn+2
descends to a transitive action both on rays and hyperplanes in Rn+2
, so G acts
transitively on the set of flags of above type. The subgroup P is the stabilizer of the
flag ⊂ ρ where and ρ is the ray and the hyperplane generated by the first and
the first n + 1 vectors from the standard basis, respectively. Obviously, P = ˜P ∩ ¯P
where ˜P is the stabilizer of and ¯P stabilizes ρ. Note that both ˜P and ¯P are also
parabolic.
We claim that G/P ∼= Po
(T∗
Sn+1
) which is the oriented projectivization of the
cotangent bundle of projective sphere, the oriented projectivization of Rn+2
. This
can be clarified as follows: The projective sphere Sn+1 ∼= Po
(Rn+2
) is identified
with G/ ˜P, where ˜P ⊂ G is the stabilizer of the ray as above. Let ˜p ⊂ g be
the Lie algebra of ˜P and let g = ˜g−1 ⊕ ˜g0 ⊕ ˜g1 be the corresponding grading
of g. As usual, (g/˜p)∗ ∼= ˜g∗
−1
∼= ˜g1, hence T∗
Sn+1 ∼= T∗
(G/ ˜P) is identified with
G × ˜P ˜g1 via the Maurer–Cartan form on G. Now, the adjoint action of ˜P on ˜g1 is
transitive and an easy direct calculation shows that the stabilizer of a convenient
element of ˜g1 is precisely P ⊂ P ⊂ ˜P; the subgroup P ⊂ ˜P is the stabilizer of the
corresponding ray. Altogether, ˜g1
∼= ˜P/P and Po
(˜g1) ∼= ˜P/P, so T∗
Sn+1 ∼= G/P
and Po
(T∗
Sn+1
) ∼= G/P. Hence the interpretation of the model cone for Lagrangean
contact structures is
ˆC ∼= T∗
Sn+1
→ Po
(T∗
Sn+1
) ∼= C
so that the canonical G-invariant symplectic form on ˆC corresponds to the canonical
symplectic form on the cotangent bundle T∗
Sn+1
, cf. remark 3.1(c).
Now we are going to expose a general construction following [14]; it turns out this
will be useful to find a candidate for the ambient connection ˆ on ˆC ∼= T∗
Sn+1
. Let
M be a manifold with linear torsion-free connection and let H ⊂ TT∗
M be the
corresponding horizontal distributions on the cotangent bundle over M. Together
with the vertical subbundle V of the projection p : T∗
M → M we have got an
REMARKS ON SPECIAL SYMPLECTIC CONNECTIONS 509
almost product structure on T∗
M. Let ˆα be the canonical one-form and ˆΩ = dˆα
the canonical symplectic form on T∗
M. By definition of ˆΩ, the subbundle V is
isotropic with respect to ˆΩ. The complementary subbundle H determined by the
connection is isotropic if and only if is torsion-free. After the projectivization,
the decomposition V ⊕ H = TT∗
M yields a Lagrangean contact structure on
P(T∗
M). Moreover, the almost product structure on T∗
M and so the Lagrangean
contact structure on P(T∗
M) are independent on the choice of connection from the
projectively equivalent class [ ]. Altogether, starting with a projective structure
on a smooth manifold M, this gives rise to a Lagrangean contact structure on the
projectivized cotangent bundle of M. Note that in terms of parabolic geometries,
this construction is an instance of the so-called correspondence space construction
[4, section 4] which is formally powered by the inclusion P ⊂ ˜P of parabolic
subgroups in G. As a particular implementation of a general principle, locally flat
projective structure on M gives rise to a locally flat Lagrangean contact structure
on P(T∗
M). This is actually observed elementarily in the previous paragraph
provided we consider oriented projectivization instead of the usual one.
Proposition. Let ˆC → C be the model cone for g = sl(n+2, R). Then ˆC ∼= T∗
Sn+1
,
C ∼= Po
(T∗
Sn+1
), and ˆΩ corresponds to the canonical symplectic form on cotangent
bundle. Let further J : TT∗
Sn+1
→ TT∗
Sn+1
be the almost product structure given
by the projective structure on Sn+1
as above. Then the bilinear form ˆg := ˆΩ(−, J−)
on T∗
Sn+1
is symmetric and non-degenerate and the ambient symplectic connection
ˆ from Theorem 4.3 is the Levi–Civita connection of ˆg.
Proof. Let Sn+1
⊂ Rn+2
be the standard projective sphere. The projective structure
[ ] is induced from the canonical flat connection in Rn+2
, in particular, any
connection in the class is torsion-free. As before, this ensures that both subbundles
V and H from the corresponding decomposition of TT∗
Sn+1
are isotropic with
respect to the canonical symplectic form ˆΩ. The decomposition V ⊕ H = TT∗
Sn+1
determines the product structure J so that V and H is the eigenspace of J corresponding
to the eigenvalue 1 and −1, respectively. Since both ˆΩ and J are
non-degenerate, the same holds true also for ˆg := ˆΩ(−, J−). Since both V and
H are isotropic with respect to ˆΩ, the bilinear form ˆg turns out to be symmetric,
hence it is a pseudo-metric on T∗
Sn+1
.
The rest of the proof is completely parallel to that in 4.5 up to the interchange
between the almost complex and almost product structure on ˆC and D ⊂ TC,
respectively.
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Department of Mathematics and Statistics
Masaryk University
611 37 Brno, Czech Republic
E-mail: naca@math.muni.cz zadnik@math.muni.cz