Geom Dedicata
DOI 10.1007/s10711-009-9426-6
ORIGINAL PAPER
Contact projective structures and chains
Andreas ˇCap · Vojtˇech Žádník
Received: 12 February 2009 / Accepted: 23 September 2009
© Springer Science+Business Media B.V. 2009
Abstract Contact projective structures have been thoroughly studied by D. Fox. He associated
to a contact projective structure a canonical projective structure on the same manifold.
We interpret Fox’s construction in terms of the equivalent parabolic (Cartan) geometries,
showing that it is an analog of Fefferman’s construction of a conformal structure associated
to a CR structure. We show that, on the level of Cartan connections, this Fefferman-type
construction is compatible with normality if and only if the initial structure has vanishing
contact torsion. This leads to a geometric description of the paths that have to be added to the
contact geodesics of a contact projective structure in order to obtain the subordinate projective
structure. They are exactly the chains associated to the contact projective structure, which are
analogs of the Chern–Moser chains in CR geometry. Finally, we analyze the consequences
for the geometry of chains and prove that a chain–preserving contactomorphism must be a
morphism of contact projective structures.
Keywords Projective structure · Contact projective structure · Path geometry ·
Fefferman construction · Chains · Cartan connection · Parabolic geometry
Mathematics Subject Classification (2000) 53A20 · 53B10 · 53B15 · 53C15 · 53D10
A. ˇCap (B)
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Vienna, Austria
e-mail: Andreas.Cap@univie.ac.at
A. ˇCap
International Erwin Schrödinger Institute for Mathematical Physics,
Boltzmanngasse 9, 1090 Vienna, Austria
V. Žádník
Faculty of Education, Masaryk University, Poˇríˇcí 31, 60300 Brno, Czech Republic
e-mail: zadnik@math.muni.cz
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1 Introduction
Classical projective structures can be viewed as describing the geometry of geodesics of
affine connections, viewed as unparametrized curves (paths). The study of these structures
was a very active part of differential geometry in the first decades of the 20th century. After
some time of less activity, the interest in these geometries has been revived during the last
years. Much of this recent interest is related to the fact that they form a simple instance of
the large class of so-called parabolic geometries.
Among the parabolic geometries there is also a contact analog of classical projective structures,
called contact projective structures. Such a structure is given by a contact structure and
a family of paths in directions tangent to the contact distribution, which can be realized as
geodesics of some affine connection. While basic ideas on these structures can be traced
back to the classical era, they have been formally introduced and thoroughly studied by Fox
in [10]. One of the main results in that article is that any contact projective structure can be
canonically extended to a projective structure on the same manifold.
Studying Fox’s canonical projective structure is the main purpose of this article. We first
review some fundamental facts on projective and contact projective structures in Sect.2. In
Sect.3, we give a geometric description of the paths in directions transverse to the contact
distribution that have to be added to the given paths in contact directions in order to obtain
the canonical projective structure. To describe these curves, recall that for CR manifolds
of hypersurface type, there are the so-called Chern–Moser chains introduced in [9]. They
form a family of canonical unparametrized curves available in all directions transverse to the
contact distribution. The description of Chern–Moser chains via the canonical Cartan connection
associated to a CR structure easily generalizes to all parabolic contact structures, see
[7]. In this way, one obtains a family of chains associated to any contact projective structure,
and these are the curves to be added in order to get the canonical projective structure, see
Corollary3.3.
This description is obtained via another result, which is of independent interest. In [10],
the canonical projective structure was obtained via the so-called ambient descriptions or
cone descriptions of contact projective and projective structures. We give a description in
terms of the canonical Cartan connections, which shows that it is an analog of the Fefferman
construction as described in [2], see Proposition3.3. We also show that this Fefferman type
construction produces not only the canonical projective structure but also its canonical Cartan
connection if and only if the initial contact projective structure has vanishing contact torsion,
see Theorem3.2.
The fact that the chains of a contact projective structure can be realized as geodesics of an
affine connection is in sharp contrast to the cases of CR structures and Lagrangean contact
structures, see [8]. In the latter article, we have studied chains via the path geometry they
determine. In Sect.4 we discuss these issues in the contact projective case, where they are
rather easy. In spite of the fact that the contact projective structure cannot be recovered from
the path geometry of chains, we are able to prove that contactomorphisms which preserve
chains actually are morphisms of contact projective structures, see Theorem4.3.
2 Projective and contact projective structures
2.1 Projective structures
A projective structure on a smooth manifold M is given by a class of projectively equivalent
linear connections [∇] on T M. Two connections are called projectively equivalent if their
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difference tensor is of the form A(ξ, η) = ϒ(ξ)η + ϒ(η)ξ for some one-form ϒ ∈ Ω1(M).
Note that such a tensor is symmetric, so by definition, projectively equivalent connections
have the same torsion. It is a well known classical result that two connections which have
the same torsion are projectively equivalent if and only if they have the same geodesics up
to parametrization.
This means that a projective structure on M is given by a class of linear connections on
T M which have the same torsion and the same unparametrized geodesics. Since symmetrizing
a connection does not change its geodesics, it is usually assumed the connections in the
class are torsion-free, which is a natural normalization of the structure.
We can also interpret this description as saying that a projective structure on M is given
by the smooth family of paths (unparametrized curves) formed by the geodesics. This is an
example of a path geometry, i.e. a smooth family of paths with exactly one path through each
point in each direction, see Sect.4.1 for the precise definition. We will return to this point of
view there.
The model projective structure is given by the real projective space RPm = PRm+1 with
the class of connections induced from the canonical flat connection on Rm. The geodesics
of these connections are the projective lines. The group of diffeomorphisms of RPm which
preserve this structure is PGL(m + 1, R), the quotient of GL(m + 1, R) by its center. For
our purposes it is better to work with oriented projective structures. This means replacing
RPm by the sphere Sm, viewed as the space of rays in Rm+1. Then the appropriate group is
˜G := SL(m + 1, R), and the distinguished paths are the great circles on Sm. If m is even,
then ˜G is isomorphic to PGL(m + 1, R), while for odd m it is a two-fold covering. In any
case, ˜G acts transitively both on Rm+1 \ {0} and on Sm. Let ˜P ⊂ ˜G be the stabilizer of
the ray generated by the first vector of the standard basis of Rm+1 and let ˜Q ⊂ ˜P be the
stabilizer of the vector itself, so Sm ∼= ˜G/ ˜P and Rm+1 \ {0} ∼= ˜G/ ˜Q. In terms of matrices,
˜P is represented by block matrices
˜P =
det(A)−1 Z
0 A
: A ∈ GL+
(m, R), Z ∈ Rm∗
,
where GL+(m, R) = {A ∈ GL(m, R) : det(A) > 0}. The subgroup ˜Q ⊂ ˜P is given by
those matrices in ˜P for which A ∈ SL(m, R). ˜P is a parabolic subgroup of the simple Lie
group ˜G and the corresponding grading of the Lie algebra ˜g = sl(m + 1, R) is given by the
block decomposition
˜g0 ˜g1
˜g−1 ˜g0
with blocks of sizes 1 and m. Hence ˜g−1
∼= Rm, ˜g0
∼= gl(m, R), and ˜g1
∼= Rm∗. The Lie
algebras of ˜P and ˜Q are ˜p = ˜g0 ⊕ ˜g1 and ˜q = ˜gss
0 ⊕ ˜g1, respectively. Here ˜gss
0 denotes the
semisimple part of ˜g0, which is isomorphic to sl(m, R).
General oriented projective structures admit an equivalent description as Cartan geometries
of type ( ˜G, ˜P). Projecting to the lower right block defines a homomorphism from ˜P
onto GL+(m, R), so we can view the latter group as a quotient of ˜P. Then with notation as
above, the following holds, see e.g. [3]:
Theorem 2.1 Let M be a smooth manifold of dimension ≥ 2 which is endowed with an
oriented projective structure. Then the oriented linear frame bundle of M can be canonically
extended to a principal ˜P–bundle ˜G → M, which can be endowed with a Cartan connection
˜ω ∈ Ω1( ˜G, ˜g). The pair ( ˜G, ˜ω) is uniquely determined up to isomorphism if one in addition
requires ˜ω to be normal in the sense recalled in Sect.3.2 below.
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The normalization condition on the Cartan connection in the theorem is the usual condition
for parabolic geometries, namely that the curvature has values in the kernel of the Kostant
codifferential.
The relation between the Cartan geometry ( ˜G → M, ˜ω) and the projective structure can
be described in several ways. On the one hand, the quotient ˜G0 → M of ˜G by the connected
subgroup with Lie algebra g1 can be naturally identified with the oriented linear frame bundle
of M. One can then take the component ˜ω0 of ˜ω in g0 and pull it back via smooth sections
σ : ˜G0 → ˜G satisfying an equivariancy condition. This leads to principal connections on ˜G0
which exactly correspond to the connections in the projective class.
More easily, the unparametrized geodesics of the projective structure are given by projections
to M of flow lines of the constant vector fields ˜ω−1(X) ∈ X( ˜G) with X ∈ ˜g−1.
2.2 Contact projective structures
Recall that a contact manifold (M, H) is a smooth manifold M of odd dimension
2n + 1 together with a smooth subbundle H ⊂ T M of corank one which is maximally
non-integrable. This means that the bilinear bundle map L : H × H → T M/H induced by
the Lie bracket of vector fields, which is called the Levi–bracket, is non-degenerate. The contact
analog of projective structures was formally introduced and thoroughly studied in [10].
Similarly to classical projective structures this contact analog can be described in several
equivalent ways.
The simplest description is via the analog of path geometries, for which one only considers
paths which are everywhere tangent to the contact distribution H. Then a contact projective
structure can be defined as a smooth family of such contact paths with one path through
each point in each direction in H, which are among the geodesics of some linear connection
on T M. This is the definition used in [10]. While this implicitly also provides a definition
as an equivalence class of linear connections on T M, more work is needed to obtain a nice
description of this type.
Indeed, Theorem A of [10] provides a special class of such connections. One starts with
a contact form θ ∈ Ω1(M), i.e. a one-form whose kernel in each point x ∈ M is the contact
distribution Hx . We will always assume that the contact structure in question admits global
contact forms. This amounts to the fact that the line bundle T M/H (or equivalently its dual)
admit global nonzero sections. Equivalently, these line bundles have to be orientable, and we
further assume that an orientation has been chosen. Then we can talk about positive contact
forms, and given one such form θ, any other is obtained by multiplication by a positive
smooth function. Given a positive contact form θ, Theorem A of [10] shows the existence
of a linear connection ∇ on T M which has the given paths among its geodesics, satisfies
∇θ = 0 and ∇dθ = 0 as well as normalization conditions on its torsion.
Since ∇θ = 0, the connection ∇ preserves the subbundle H ⊂ T M, and of course the
geodesics in contact directions depend only on the restriction of ∇ to a linear connection on
the vector bundle H → M. Further, it turns out that all of ∇ is determined by the restriction
of ∇ to an operator (H) × (H) → (H), a so-called partial connection. Finally,
viewing the Levi–bracket L as a bundle map Λ2 H → T M/H, its kernel defines a corank
one subbundle Λ2
0 H ⊂ Λ2 H. Since ∇dθ = 0, the linear connection on Λ2 H induced by
∇ preserves this subbundle, so ∇ (respectively, its restriction) is a (partial) contact connection.
Now parallel to the projective case, one can define contact projective equivalence of
partial contact connections, and characterize this in terms of the difference tensor. This leads
to a formula in terms of a smooth section H∗ which is similar to the one for projectively
equivalent connections, see formula (2.8) of [10].
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There is a significant difference to the case of projective structures, which concerns torsion.
For linear connections on the tangent bundle, one can always remove the torsion without
changing the geodesics. This is no more true in the contact setting. By Theorem 2.1 of [10],
the restrictions of the torsions of all the representative connections ∇ associated to contact
forms as above to Λ2 H∗ ⊗ H ⊂ Λ2T ∗ M ⊗ T M coincide. This is called the contact torsion
of the contact projective structure. (In the setting of partial contact connections, one has to
further restrict to (Λ2
0 H)∗ ⊗ H, but this needs only minor adaptions.)
The model contact projective structure is given by the space of rays in a symplectic vector
space. Consider R2n+2 with the standard linear symplectic form Ω. Then Ω induces a contact
structure on the space S2n+1 of rays, and the great circles tangent to the contact subbundle
(which can locally be realized as geodesics for the standard flat connection on R2n+1) define a
contact projective structure. One of the main results of [10] is the construction of a canonical
projective structure from a contact projective structure. For the homogeneous model, it is
obvious how to do this: One simply adds that great circles which are transverse to the contact
distribution, to obtain the homogeneous model of oriented projective structures.
The contact projective structure on S2n+1 constructed above is evidently homogeneous
under the symplectic group G := Sp(2n +2, R). It is easy to see that the actions of elements
of G are exactly those diffeomorphisms of S2n+1 which preserve both the contact structure
and the projective structure. Generalizing this result to the curved case will be the main aim
of Sect.4. Now G acts transitively both on R2n+2 \ {0} and on S2n+1, so as homogeneous
spaces S2n+1 ∼= G/P and R2n+2 \{0} ∼= G/Q, where P is the stabilizer of the ray generated
by the first vector of the standard basis of R2n+2, and Q is the stabilizer of that vector. For
the obvious inclusion G → ˜G (with m = 2n + 1) we get P = G ∩ ˜P and Q = G ∩ ˜Q. As
in 2.1, P is a parabolic subgroup in the simple Lie group G. To obtain a nice presentation of
the Lie algebra g of G, it is best to choose Ω to be represented by the matrix
⎛
⎝
0 0 1
0 J 0
−1 0 0
⎞
⎠ ,
with J = 0 In
−In 0 and In denoting identity matrix of rank n. Using this form, the Lie algebra
g = sp(2n + 2, R) has the form
g =
⎧
⎨
⎩
⎛
⎝
a U w
X A JUt
z −Xt J −a
⎞
⎠
⎫
⎬
⎭
,
with blocks of sizes 1, 2n and 1, a, z, w ∈ R, X ∈ R2n, U ∈ R2n∗, and A ∈ sp(2n, R) (with
respect to J). We obtain a grading g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 with g−2 corresponding to
z, g−1 to X, g0 to a and A, and so on. By construction, p is formed by the matrices which are
block upper triangular, i.e. satisfy z = 0 and X = 0, so p = g0 ⊕ g1 ⊕ g2. The subalgebra
q ⊂ p corresponds to those matrices, which in addition satisfy a = 0. For the algebra ˜g from
2.1, we simply obtain all matrices of the same size, and the comparison with the description
in 2.1 shows the various grading components and subalgebras.
In Theorem C of [10], the author proves existence of a canonical Cartan connection associated
to a contact projective structure, which reads as follows:
Theorem 2.2 Let (M, H) be a contact manifold which admits a global contact form and is
endowed with a contact projective structure. Then there exists a principal P–bundle p : G →
M endowed with a Cartan connection ω ∈ Ω1(G, g) such that H = T p(ω−1(g−1 ⊕ p)) and
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the contact geodesics are projections to M of flow lines of constant vector fields ω−1(X) ∈
X(G) with X ∈ g−1. The pair (G, ω) is uniquely determined up to isomorphism provided that
one in addition requires ω to be normal in the sense discussed in Sect.3.2 below.
Remark 2.2 (1) The normalization condition in the theorem is a generalization of the uniform
normalization condition for parabolic geometries. As we shall discuss in more
detail in Sect.3.2 below, it reduces to the standard condition if and only if the projective
contact structure has vanishing contact torsion, see Proposition 4.1 of [10]. In this
special case, the theorem follows from general results on parabolic geometries, see [6].
(2) The description of the relation between the Cartan geometry and the underlying contact
projective structure in the theorem is different from the original one in [10]. The
characterization in [10] uses the ambient connection to be discussed in Sect.2.3 below.
Section 4.3 of [10] discusses the characterization of contact geodesics via the development
of curves (induced by the Cartan connection ω) into the homogeneous model
G/P = S2n+1. Contact geodesics on M are exactly the curves which develop to the
contact geodesics in the model. In [7] it is shown how the description in terms of development
is equivalent to being a projection of an integral curve of a certain type of
constant vector fields (of the Cartan connection ω). Then it suffices to observe that the
contact geodesics in the model G/P are precisely the orbits of one-parameter subgroups
of G generated by elements of g−1.
(3) There is another distinguished family of curves in the model space. As in (2), they may
be either characterized via development or as projections of integral curves of constant
vector fields, but this time with generator in g−2. In view of the similarity to the concept
in CR geometry introduced by Chern–Moser, these are called chains. In particular, a
chain is uniquely determined by its initial direction as an unparametrized curve. For
the homogeneous model S2n+1, the chains are exactly those great circles which are
transverse to the contact distribution.
2.3 Ambient descriptions
The basis of the construction of a projective structure subordinate to a contact projective
structure is the so-called ambient description or cone description of projective and contact
projective structures. In the projective case, this goes back to the work of Tracy Thomas in
the 1930s, in the contact projective case it is due to Fox. In [10] the ambient connection
is constructed first (in Theorem B) and then used to construct a Cartan connection. Here
we take the opposite point of view, and use the Cartan connection to construct the ambient
connection.
The starting point for the ambient description of both types of structure is a principal
bundle L → M with structure group R+, namely the frame bundle of the bundle of ( −1
m+1 )–
densities. In the contact projective case, it is easy to see that one may also view this density
bundle as a square root of the bundle of positive contact forms. In the (oriented) projective
case, L can be constructed from the Cartan bundle ˜G → M via a homomorphism ˜P → R+
with kernel ˜Q ⊂ ˜P. Hence L ∼= ˜G/ ˜Q, and ˜G → L is a principal bundle with structure group
˜Q, on which ˜ω is a Cartan connection. In particular, T L ∼= ˜G × ˜Q (˜g/˜q) with the action of ˜Q
coming from the adjoint representation. In the contact projective case, there is a completely
analogous description in terms of the canonical Cartan geometry (G → M, ω) induced by
the contact projective structure. In particular, T L ∼= G ×Q (g/q) in the contact projective
case.
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Proposition 2.3 Consider ˜G := SL(m + 1, R) and let ˜Q ⊂ ˜G be the stabilizer of the first
vector in the standard basis of Rm+1. Then, as a representation of ˜Q, ˜g/˜q is isomorphic to
the restriction to ˜Q of the standard representation Rm+1 of ˜G.
If m is odd, say m = 2n + 1, then the analogous statement holds for G = Sp(2n + 2, R)
and the stabilizer Q ⊂ G of the first basis vector.
Proof The Lie subalgebra ˜q ⊂ ˜g consists of all matrices for which all entries in the first column
are zero. To describe the ˜Q-representation ˜g/˜q, we may thus simply look at the action of
the adjoint representation of ˜Q to the first column of matrices. Since ˜G is a matrix group, the
adjoint representation is given by conjugation. By definition, the first column of any matrix
in ˜Q equals the first unit vector, so multiplying any matrix from the right by an element of
˜Q leaves the first column unchanged. But this implies that for A ∈ ˜Q and X ∈ ˜g, the first
column of AX A−1 equals the first column of AX, which implies the claim for ˜Q. But then
the same statement is true for any subgroup of ˜Q, hence in particular for Q = ˜Q ∩ G in the
case of odd m.
Using this, we may view the bundle T L → L as the associated bundle ˜G × ˜Q Rm+1,
respectively, G ×Q R2n+2, and since the inducing representations are restrictions of representations
of ˜G, respectively G, we can invoke the general construction of [4]. This shows
that the Cartan connection ˜ω, respectively, ω induces a linear connection on T L.
Theorem 2.3 (1) The linear connection on T L induced by a projective structure on M as
described above coincides with the ambient connection from Theorem 3.1 of [10].
(2) The linear connection on T L induced by a contact projective structure on M as
described above coincides with the ambient connection from Theorem B of [10].
Proof There are various ways to prove this, which all boil down to rather straightforward
verifications. On the one hand, one may simply verify that the linear connections we have constructed
satisfy the properties listed in the theorems of [10], and then invoke the uniqueness
parts of these theorems.
Even easier, one may follow the construction of the Cartan connection in [10] backwards.
First, one lifts the principal R+–action on L to a free right action by vector bundle homomorphisms
on T L in such a way that the orbit space T L/R+ (which evidently is a vector
bundle over L/R+ = M) can be identified with the standard tractor bundle of the structure
in question. Then one shows that the ambient connection induces a tractor connection on
that bundle, which by the general methods of [4] gives rise to a Cartan connection. This
corresponds to the fact that ˜P/ ˜Q ∼= R+ acts on ˜G/ ˜Q = L with orbit space ˜G/ ˜P = M (and
the analogous statement for P/Q). Now one immediately verifies that the lift of the action
is exactly defined in such a way that the linear connection on T L induces the usual tractor
connection on the tractor bundles, which completes the proof.
3 The subordinate projective structure
Having collected the background, we can now move to proving the first main results of
this article. We show that the construction of a projective structure subordinate to a contact
projective structure in [10] can be interpreted as a generalized Fefferman construction.
This interpretation leads to immediate payoff, since it implies a geometric description of the
subordinate projective structure in terms of chains.
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3.1 The Fefferman–type construction
The scheme for generalized Fefferman constructions is by now fairly familiar, see [2], where
also the application to contact projective structures was suggested. As before, consider G =
Sp(2n + 2, R) and ˜G := SL(2n + 2, R), let ψ : G → ˜G be the obvious inclusion. Then
put i := ψ|P : P → ˜P and α := ψ : g → ˜g. Now suppose that we have given a contact
manifold (M, H) of dimension m = 2n + 1, which is endowed with a contact projective
structure, and let (G → M, ω) be the canonical Cartan geometry of type (G, P) determined
by this structure as in 2.2. Then the homomorphism i : P → ˜P defines a left action of P
on ˜P, so we can form the associated bundle ˜G := G ×P ˜P → M. This clearly is a principal
bundle with structure group ˜P, and we have a natural map j : G → ˜G induced by mapping
u ∈ G to the class of (u, e). It is easy to prove (compare with 3.1 and 3.2 of [8]) that there is
a unique Cartan connection ˜ω ∈ Ω1( ˜G, ˜g) such that j∗ ˜ω = α ◦ ω.
This construction actually defines a functor from Cartan geometries of type (G, P) to
Cartan geometries of type ( ˜G, ˜P), both defined on the same manifold. (The fact that we
obtain a geometry on the same manifold is due to the fact that ˜P ∩ G is already a parabolic
subgroup of G. For other Fefferman–type constructions, this is not the case. Then one has
to pass to a parabolic subgroup containing this intersection, and the new geometry will be
defined on the total space of a natural bundle.) Since any Cartan geometry of type ( ˜G, ˜P) on
a manifold M gives rise to an underlying projective structure, we obtain a functor mapping
contact projective structures to projective structures on the same manifold. It is not clear,
however, whether the Cartan connection ˜ω is normal and hence coincides with the canonical
Cartan connection associated to the projective structure in general. This is a familiar
phenomenon for generalized Fefferman constructions.
Before we discuss the question of normality of ˜ω, we give a geometric description of the
projective structure produced by the generalized Fefferman construction.
Proposition 3.1 Let (G → M, ω) be a Cartan geometry of type (G, P) and let ( ˜G → M, ˜ω)
be the Cartan geometry of type ( ˜G, ˜P) obtained by the generalized Fefferman construction.
Then the paths of the projective structure determined by ( ˜G → M, ˜ω) are the projections
of the flow lines of the constant vector fields ω−1(X) ∈ X(G) generated by elements
X ∈ g−1 ∪ g−2.
Proof It is well known that the paths of the projective structure can be obtained as the projections
of the flow lines of the constant vector fields ˜ω−1( ˜X) for all elements ˜X ∈ ˜g−1.
Moreover, it is well known that there is exactly one such path through each point of M in
each direction. As we have seen above, viewing g as a subalgebra of ˜g, the Cartan connection
˜ω is characterized by j∗ ˜ω = ω. In particular, for X ∈ g ⊂ ˜g, the constant vector fields
ω−1(X) ∈ X(G) and ˜ω−1(X) ∈ X( ˜G) are j-related. Hence their flows are j-related and in
particular have the same projection to M.
From the description of the Lie algebras g ⊂ ˜g in 2.1 and 2.2, we first see that g−2 ⊂ ˜g−1.
Hence for X ∈ g−2, the projection of the flow line of ω−1(X) is among the paths of the
projective structure. The tangent directions of these paths exhaust all directions which are
transverse to the contact distribution.
On the other hand, g−1 ⊂ ˜g−1 ⊕ ˜g0 (with a nontrivial component in ˜g0 for any nonzero
element of g−1). For X ∈ g−1 let ˜X be the ˜g−1-component of X (i.e. the matrix with the
same first column as X and all other columns zero), and put ˜A = X − ˜X ∈ ˜g0. From
the explicit presentations of g and ˜g one immediately verifies that [ ˜A, ˜X] = 0, and hence
Ad(exp(−t ˜A))( ˜X) = ˜X for all t. Now for u ∈ G, let ˜c(t) be the flow line of the constant
vector field ˜ω−1( ˜X). Then the curve c(t) := ˜c(t) · exp(t ˜A) has the same projection to M
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as ˜c(t), so this projection is among the paths determined by the projective structure. But
denoting by r the principal right action and by ζ ˜A the fundamental vector field generated by
˜A, one computes that
c (t) = Trexp(−t ˜A)
· ˜c(t) + ζ ˜A(c(t)),
and so ˜ω(c (t)) = Ad(exp(−t ˜A))( ˜X) + ˜A = X for all t. This shows that the flow line of
ω−1(X) is also among the paths of the induced projective structure. Since the tangents of
such paths exhaust all directions in the contact distribution, this completes the proof.
Remark 3.1 (1) A nice alternative argument for the last part of the proof is as follows:
Since [ ˜A, ˜X] = 0, we get exp(t X) = exp(t ˜X) exp(t ˜A), and hence the exponential
curves generated by X and ˜X have the same projection to ˜G/ ˜P. Via development, this
implies the same result for the flow lines of the constant vector fields.
(2) A regular Cartan geometry (G → M, ω) of type (G, P) as in the proposition gives rise
to a contact projective structure on M. The distinguished paths (in contact directions) of
this structure are the flow lines of the vector fields ω−1(X) for X ∈ g−1. The proposition
in particular says, that these are among the paths of the projective structure obtained via
the generalized Fefferman construction. Hence the projective structure obtained from
the generalized Fefferman construction is subordinate to the initial contact projective
structures in the sense of Definition 3.1 of [10].
3.2 Normality
As mentioned above, there are few general results on the compatibility of Fefferman type
constructions with normality of Cartan connections, except for the fact that the result of a
generalized Fefferman construction is locally flat if and only if the original geometry is locally
flat. To discuss normality in our case, let us first recall the normalization condition used for
parabolic geometries. Consider a semisimple Lie algebra g with a parabolic subalgebra p
and the corresponding grading g = g− ⊕ g0 ⊕ p+ (with p = g0 ⊕ p+). Then the Killing
form induces a duality between g/p and p+, which is equivariant for the natural action of
any parabolic subgroup P ⊂ G with Lie algebra p. Now there is a standard complex for
computing the Lie algebra homology of p+ with coefficients in g. The differential in this
complex is often denoted by ∂∗ and referred to as the Kostant codifferential since it can
also be obtained by dualizing a Lie algebra cohomology differential. For the normalization
condition, we need the map
∂∗
: Λ2
p+ ⊗ g → p+ ⊗ g,
which on decomposable elements is given by
∂∗
(Z ∧ W ⊗ A) = −W ⊗ [Z, A] + Z ⊗ [W, A] − [Z, W] ⊗ A.
Now the curvature of a Cartan geometry (G, ω) of type (G, P) can be described by the
curvature function κ : G → L(Λ2(g/p), g), which is characterized by
κ(u)(X + p, Y + p) = dω(ω−1
(X)(u), ω−1
(Y)(u)) + [X, Y].
As noted above, the target space of κ can be identified with Λ2p+ ⊗ g, and the geometry is
called normal if ∂∗ ◦ κ = 0.
This is the normalization condition used for projective structures in Theorem2.1. For
contact projective structures, this normalization condition is not general enough, however.
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Geom Dedicata
The reason for this can be seen from one of the nice properties of the normalization condition
given by the Kostant codifferential. Namely, there is an operator : Λ2p+ ⊗g → Λ2p+ ⊗g
called the Kostant Laplacian. This is not equivariant for the action of the parabolic subgroup
P but only for its Levi component, a subgroup G0 ⊂ P with Lie algebra g0. Now due to the
gradings on p+ and g, the space Λ2p+ ⊗ g is naturally graded, so one may split the curvature
function κ into homogeneous components with respect to these gradings. One has to assume
throughout that the geometry is regular, so all homogeneous components of degree less or
equal to zero vanish identically. If this is the case, then it turns out that the lowest nonzero
homogeneous component of κ always has values in ker( ). This is extremely useful, since
ker( ) can be computed explicitly as a G0–representation (which is the main step towards
the proof of Kostant’s version of the Bott–Borel–Weil theorem in [11]).
For the parabolic pairs (g, p) and (˜g, ˜p) considered in Sect.2, the description of ker( ) is
particularly easy. In each case, this is an irreducible representation of G0, contained in one
homogeneity. The result is listed in the tables below, and ker( ) is always the component of
highest weight in the indicated subrepresentation.
(g, p), n = 1
homog. contained in
3 g1 ∧ g2 ⊗ g0
(g, p), n > 1
homog. contained in
2 g1 ∧ g1 ⊗ g0
(˜g, ˜p), m = 2
homog. contained in
3 ˜g1 ∧ ˜g1 ⊗ ˜g1
(˜g, ˜p), m > 2
homog. contained in
2 ˜g1 ∧ ˜g1 ⊗ ˜g0
In particular, we see that in all cases the maps in ker( ) have values in p ⊂ g, respectively,
in ˜p ⊂ ˜g. Since the same is evidently true for all maps of higher homogeneous degree, we see
that in both cases the curvature function of a regular normal parabolic geometry always has
values in Λ2p+ ⊗ p, i.e. such geometries are always torsion free. Now it is easy to see that a
Cartan geometry of type (G, P) is torsion free if and only if the induced contact projective
structure has vanishing contact torsion.
Therefore, to deal with contact projective structures with non-vanishing contact torsion,
one has to generalize the normalization condition, and this has been done in [10]. In Definition
4.1 of that article, the author explicitly describes a P–submodule K ⊂ Λ2(g/p)∗ ⊗ g,
which consists of maps of positive homogeneity and contains ker(∂∗). The normalization
condition used in Theorem2.2 then is that the curvature function has values in K.
Having the necessary background at hand, we can now clarify compatibility of the generalized
Fefferman construction with normality.
Theorem 3.2 Let (G → M, ω) be a Cartan geometry of type (G, P) whose curvature function
has values in the P–mdoule K from above and let ( ˜G → M, ˜ω) be the result of the
generalized Fefferman construction from 3.1. Then ˜ω is normal if and only if ω is torsion
free. Moreover, ˜ω is locally flat if and only if ω is locally flat.
Proof The description of the generalized Fefferman construction in 3.1 immediately leads
to the relation between the curvatures of the two geometries. Let us denote by ˜κ and κ the
curvature functions of ˜ω and ω. Noting that α = ψ : g → ˜g is a homomorphism of Lie
algebras, we obtain (compare with Proposition 3.3 of [8])
˜κ( j(u))( ˜X + ˜p, ˜Y + ˜p) = α(κ(u)(X + p, Y + p)), (1)
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Geom Dedicata
for all u ∈ G, ˜X, ˜Y ∈ ˜g and X, Y ∈ g such that α(X) + ˜p = ˜X + ˜p and likewise for Y and ˜Y.
Note that for given ˜X, we can always find an element X with this property, since α induces
a linear isomorphism g/p → ˜g/˜p. Note also that by equivariancy (1) uniquely determines ˜κ.
Since α is injective, we see that ˜κ vanishes identically if and only if κ vanishes identically,
so the statement about local flatness follows readily.
Second, ˜ω by definition is torsion free if and only if ˜κ has values in Λ2(˜g/˜p)∗ ⊗ ˜p, and
since ˜p ⊂ ˜g is ˜P–invariant this is equivalent to the same statement for ˜κ ◦ j. Since α−1(˜p) = p
by construction, the latter statement via (1) is equivalent to κ having values in Λ2(g/p) ⊗ p
and hence to torsion freeness of ω. As we have seen above, normal Cartan connections of
type ( ˜G, ˜P) are always torsion free, so we see that normality of ˜ω implies torsion freeness
of ω.
Let us conversely assume that ω is torsion free and that its curvature function has values in
K. Then by Proposition 4.1 of [10] the curvature function ˜κ has values in ker(∂∗), so we may
apply general results for parabolic geometries. The isomorphism α : g/p → ˜g/˜p induced by
α is equivariant over the inclusion P → ˜P, so the same is true for ϕ := (α−1)∗ : (g/p)∗ →
(˜g/˜p)∗. Hence also the map Φ := Λ2ϕ ⊗ α is equivariant in the same sense, and in terms
of this map we can write (1) as ˜κ ◦ j = Φ ◦ κ. Now let ˜∂∗ be the Kostant codifferential
associated to (˜g, ˜p). Then equivariancy of Φ implies that Φ−1(ker(˜∂∗)) ⊂ Λ2(g/p)∗ ⊗ g is
a P–submodule. Clearly, normality of ˜ω is equivalent to the fact that κ has values in this
P–submodule. By Corollary 3.2 of [1], this is equivalent to the fact that the harmonic part
κH of the curvature function has values in it.
As discussed above, κH (u) has values in ker( ) ⊂ ∧2(g/p)∗ ⊗ g, which is an irreducible
representation of G0. Denoting by w ∈ ker( ) a highest weight vector in this representation,
it is therefore, sufficient to prove that Φ(w) ∈ ker(˜∂∗). This can be verified by a simple
direct computation. Alternatively, it is easy to verify that the G0–representation (˜g/˜p)∗ ⊗ ˜g
in which ˜∂∗ has values does not contain an irreducible component isomorphic to ker( ).
Remark 3.2 The proof of this theorem is significantly easier then the proofs of normality for
the classical Fefferman construction (see [5]) or other generalized Fefferman constructions.
This is due to the fact that G ∩ ˜P = P and hence g ∩ ˜p = p in our case. The second property
directly shows that torsion freeness of ω implies torsion freeness of ˜ω, which otherwise needs
more involved proofs. On the other hand, the first property implies equivariancy of the map
Φ, which together with the general results obtained using BGG sequences allow a reduction
of the problem to harmonic curvature.
3.3 Comparing to the construction by Fox
For a contact projective structure on a contact manifold (M, H), there is the canonical
Cartan geometry (G → M, ω) from Theorem 2.2. Applying to this geometry the generalized
Fefferman construction from 3.1, we obtain a canonical projective structure on M,
which is subordinate to the contact projective structure in the sense of Remark 3.1(2). Our
final aim in this section is to prove that the result coincides with the subordinate projective
structure constructed in Sect.3.3 of [10].
Fox’s construction is based on the ambient description of contact projective and projective
structures as discussed in 2.3. There we already noticed that the spaces on which the ambient
connection is defined are the same for both types of structures. Moreover, the ambient
connection associated to a contact projective structure in Theorem B of [10] satisfies all
the properties of a projective ambient connection from Theorem 3.1 of [10], except for torsion
freeness. Symmetrizing the contact projective ambient connection, one obtains a torsion
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Geom Dedicata
free connection, which then is the canonical connection associated to a projective structure.
This is the canonical projective structure as defined by Fox. Notice that the ambient connection
associated to a contact projective structure is torsion free if and only if the structure
has vanishing contact torsion. This in turn is equivalent to the fact that this ambient connection
coincides with the ambient connection of the canonical subordinate projective structure
defined by Fox, which is the analog of Theorem3.2 in this setting.
Proposition 3.3 For a contact projective structure on a contact manifold (M, H), the subordinate
projective structure obtained via the generalized Fefferman construction coincides
with the subordinate projective structure constructed in Sect.3.3 of [10].
Proof Let (G → M, ω) be the Cartan geometry associated to the contact projective structure
as in Theorem2.2 and let ( ˜G → M, ˜ω) be the result of the generalized Fefferman construction
from 3.1. We want to show that, via the procedure from 2.3, these two Cartan geometries
lead to the same ambient connection. From property 4 of an ambient connection in Theorem
3.1 of [10] one easily concludes that the paths of a projective structure can be realized as projections
of geodesics of the ambient connection. Since symmetrizing the projective ambient
connection does not change its geodesics, this will complete the proof.
To compute the ambient connections, recall that we can realize the space L on which the
ambient connection is defined as G/Q or ˜G/ ˜Q. Further, g/q ∼= R2n+2 as a representation
of Q and ˜g/˜q ∼= R2n+2 as a representation of ˜Q. These identifications are compatible with
the isomorphism g/q ∼= ˜g/˜q induced by the inclusion g → ˜g. Using T L ∼= G ×Q (g/q),
vector fields on L are in bijective correspondence with Q–equivariant smooth functions
G → R2n+2. Likewise, T L ∼= ˜G × ˜Q (˜g/˜q) identifies such vector fields with ˜Q–equivariant
smooth functions ˜G → R2n+2. The correspondence between functions and vector fields is
given by taking preimages under the Cartan connections, and then projecting to the base. For
the canonical inclusion j : G → ˜G, we by definition have j∗ ˜ω = ω (identifying g with a
subset of ˜g). This immediately implies that if ˜f : ˜G → R2n+2 is the equivariant function
corresponding to η ∈ X(L), then the equivariant function f : G → R2n+2 corresponding to
η is simply given by f = ˜f ◦ j.
As we know from 2.3, the ambient connections are special cases of tractor connections,
so their actions are described in terms of equivariant functions in the proof of Theorem 2.7
in [4]. We first look at the contact projective ambient connection. Given another vector field
ξ ∈ X(L), we first have to choose a lift ˆξ ∈ X(G). Then the function G → R2n+2 corresponding
to the covariant derivative of η in direction ξ is given by ˆξ · f + ω(ˆξ) ◦ f . (In
the first summand the vector field differentiates the function, while in the second, ω(ˆξ) acts
algebraically on the values of f .) Now we can extend T j ◦ ˆξ to a lift ˜ξ ∈ X( ˜G) of ξ. From
j∗ ˜ω = ω we conclude that ˜ω(˜ξ) ◦ j = ω(ˆξ), and by construction ˆξ · f = (˜ξ · ˜f ) ◦ j. But
this says that the function describing the covariant derivative with respect to the projective
ambient connection is just the equivariant extension of the function describing the covariant
derivative with respect to the contact projective ambient connection. This shows that the two
connections actually coincide, which completes the proof.
Of course, the nice geometric interpretation of the subordinate projective structure provided
by the generalized Fefferman construction in Proposition3.1 now carries over to the
construction by Fox.
Corollary 3.3 In the language of paths, the canonical subordinate projective structure
defined in [10] is obtained by adding the chains of a contact projective structure to the
contact geodesics.
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Geom Dedicata
4 The path geometry of chains
The chains in a contact projective structure determine a generalized path geometry. For
Lagrangean contact structures and partially integrable almost CR structures, this path geometry
and its relation to the parabolic geometry associated to the original structure has been
discussed in [8]. For contact projective structures, this relation is much easier, since the path
geometry of chains is obtained as a restriction of the path geometry induced by the subordinate
projective structure. This is a simple instance of the general construction of correspondence
spaces from [1]. Therefore, the analogs of the results form [8] on the path geometry of chains
are rather easy to deduce. Still this path geometry turns out to be very useful, since it allows
us to prove that a contactomorphism between two contact projective structures which maps
chains to chains actually is a morphism of the contact projective structures.
4.1 Generalized path geometries
As we have briefly mentioned in Sect.2.1, path geometries can be viewed as smooth families
of curves on a manifold M with exactly one curve through each point in each direction. More
formally, let M be a smooth manifold of dimension m, and let N := PT M be the projectivized
tangent bundle of M. This is a smooth fiber bundle over M with fiber the projective space
RPm−1. In particular, there is a canonical projection π : N → M and we have the vertical
subbundle V N ⊂ T N. Next, by definition a point in N is a line ⊂ Tx M, where x = π( ).
This leads to a smooth subbundle H ⊂ T N, called the tautological subbundle. By definition,
a tangent vector ξ ∈ T N lies in the subspace H if and only if T π · ξ ∈ ⊂ Tx M. By construction,
H ⊂ T N is a smooth subbundle of rank m, which contains the vertical subbundle
V that has rank m − 1.
Now one defines a path geometry on M as a smooth line subbundle E ⊂ H ⊂ T N,
such that H = E ⊕ V . As a line bundle, E is integrable and hence determines a foliation of
N = PT M by 1-dimensional submanifolds. Since E ∩V = {0}, a local integral manifold for
E always projects to a local 1-dimensional submanifold of M. Hence we really obtain a family
of paths in M. Moreover, taking the integral submanifold through ∈ N, the projection
evidently passes through x = π( ) with tangent space ⊂ Tx M. Hence we see that in this
family there is exactly one path through each point in each direction. It should be mentioned,
that a path geometry can be also interpreted as describing the geometry of systems of second
order ODE’s, see e.g. [1,2].
In the spirit of filtered manifolds, one may go one step further, drop the requirement
that one deals with a projectivized tangent bundle and just keep the configuration of subbundles
with certain (non-)integrability properties: Consider an arbitrary smooth manifold
N of dimension 2m − 1 and two subbundles E, V ⊂ T N of rank 1 and m − 1, such that
E ∩V = {0}. Putting H := E ⊕V , the Lie bracket of vector fields induces a skew–symmetric
bundle map H × H → T N/H. Now the pair (E, V ) is said to define a generalized path
geometry on N if and only if this bundle map vanishes on V × V and induces an isomorphism
E ⊗ V → T N/H. It is easy to see that this is always satisfied if E and V come from
a path geometry, see [1]. Further it turns out that for m = 3, the subbundle V in a generalized
path geometry is always involutive, and then the given geometry is locally isomorphic to a
path geometry on a local leaf space for the corresponding foliation.
Any generalized path geometry on a manifold N of dimension 2m − 1 induces a canonical
normal parabolic geometry of type ( ˆG, ˆP), where ˆG = ˜G = SL(m + 1, R) and ˆP is
the subgroup of all elements which stabilize both the line spanned by the first vector in the
standard basis and the plane spanned by the first two vectors in the standard basis of Rm+1.
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Geom Dedicata
On the level of Lie algebras, we obtain a decomposition ˆg = ˆg−2 ⊕ ˆg−1 ⊕ ˆg0 ⊕ ˆg1 ⊕ ˆg2
such that ˆp = ˆg0 ⊕ ˆg1 ⊕ ˆg2 as well as decompositions ˆg±1 = ˆgE
±1 ⊕ ˆgV
±1 according to the
following block decomposition with blocks of size 1, 1, and m − 1:
⎛
⎝
ˆg0 ˆgE
1 ˆg2
ˆgE
−1 ˆg0 ˆgV
1
ˆg−2 ˆgV
−1 ˆg0
⎞
⎠
The subspaces ˆgE
−1 and ˆgV
−1 give rise to ˆP–invariant subspaces in ˆg/ˆp and the relation between
the parabolic geometry and the generalized path geometry is given by the fact these two subspaces
induce the subbundles E and V of T N, which define the generalized path geometry.
Requiring the parabolic geometry to be regular and to satisfy the normalization condition
discussed in 3.2, it is uniquely determined up to isomorphism. One obtains an equivalence
of categories between generalized path geometries and regular normal parabolic geometries
in this way.
4.2 The path geometry of chains
Let (M, H) be a contact manifold of dimension 2n + 1 endowed with a contact projective
structure, and let (G → M, ω) be the associated canonical Cartan geometry of type (G, P)
as in Theorem2.2. The chains of the contact projective structure can be described as follows:
Consider the one-dimensional subspace g−2 ⊂ g and the corresponding rank one subbundle
ω−1(g−2) ⊂ T G. This is involutive and since the vertical subbundle corresponds to p ⊂ g,
local integral submanifolds project to local one-dimensional immersed submanifolds in M.
Alternatively, the chains can be viewed as the projections of the flow lines of the constant
vector fields ω−1(X) with X ∈ g−2. This concept generalizes to all parabolic contact structures.
In that setting, it was shown in Sect. 4 of [7] that chains are available through each
point in M tangent to each line ∈ Tx M which is not contained in Hx ⊂ Tx M and, as an
unparametrized curve, a chain is uniquely determined by its tangent in one point.
This nicely fits into the picture of generalized path geometries. The subset P0T M ⊂ PT M
of lines not contained in the contact distribution evidently is open, and it is a fiber bundle
over M with fiber the complement of a hyperplane in RP2n. It is also clear that the chains
give rise to a generalized path geometry on P0T M. A description of this geometry in terms
of (G → M, ω) can be found in Sect. 2.4 of [8]. In our situation, there is, however, a simple
way to describe the parabolic geometry corresponding to the path geometry of chains, at
least in the case of vanishing contact torsion. Namely, we know that the chains are actually
among the paths of the canonical subordinate projective structure associated to the contact
projective structure.
Applying the generalized Fefferman construction from 3.1 to (G → M, ω), we obtain
a Cartan geometry ( ˜G → M, ˜ω), which induces the subordinate projective structure on M.
To get the associated (generalized) path geometry, one applies the correspondence space
construction from [1]: By construction, the subgroup ˆP ⊂ ˆG = ˜G is contained in ˜P. Hence
one can form N := ˜G/ ˆP, which can be identified with the total space of the fiber bundle
˜G × ˜P ( ˜P/ ˆP) over M. One immediately verifies that ˜G × ˜P ( ˜P/ ˆP) ∼= PT M. By construction
( ˜G → N, ˜ω) is a Cartan geometry of type ( ˆG, ˆP). In the case of vanishing contact torsion,
( ˜G → M, ˜ω) is torsion free and normal. One easily verifies that torsion freeness implies that
the parabolic geometry ( ˜G → N, ˜ω) is regular and by Proposition 2.4 of [1] it is normal,
too. Hence it is the canonical parabolic geometry associated to the underlying path geometry,
whose paths are the geodesics of the connections in the projective class, see Sect. 4.7 of [1].
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Geom Dedicata
Of course, the path geometry of chains can be recovered from this as the restriction to the
open subset P0T M ⊂ PT M = N. Having made these observations, the first part of the
following result is obvious, while the second essentially follows from the general theory of
correspondence spaces.
Proposition 4.2 Let (M, H) be a contact manifold endowed with a contact projective struc-
ture.
(1) There is a linear connection on T M that has chains among its geodesics.
(2) If the given contact projective structure has vanishing contact torsion, then the associated
path geometry is torsion free if and only if it is locally flat, which is equivalent to
local flatness of the initial contact projective structure.
Proof (2) In Theorem3.2 we have observed that local flatness of (G → M, ω) is equivalent
to local flatness of ( ˜G → M, ˜ω). Since ( ˜G → M, ˜ω) and ( ˜G → N, ˜ω) share the same curvature
function, it is equivalent to local flatness of the latter geometry, too. Since P0T M ⊂ N
is a dense open subset, we get the equivalence to local flatness of the path geometry of
chains. Finally, it has been proved in Theorem 4.7 of [1] that for path geometries induced by
projective structures torsion freeness implies local flatness.
Remark 4.2 We have pointed out part (1) of this proposition only because it is in sharp contrast
with the case of other parabolic contact structures. In [8] it is shown that for integrable
Lagrangean contact structures as well as CR structures of hypersurface type, the chains can
never be obtained as geodesics of a linear connection. Also part (2) is significantly different
for those structures. While local flatness of the initial structure is equivalent to torsion freeness
of the path geometry of chains, these path geometries are always non-flat for integrable
Lagrangean contact and CR structures.
4.3 Chain preserving contactomorphisms
As we have mentioned in the remark above, for the parabolic contact structures studied in [8]
the path geometry of chains is always non-flat. It is proved there, that the parabolic contact
structure can essentially be recovered from the curvature of the path geometry of chains. This
leads to a conceptual proof of the fact that contactomorphisms which map chains to chains
(as unparametrized curves) are homomorphisms (or anti-homomorphisms in an appropriate
sense) of the underlying parabolic contact structure.
For contact projective structures, the situation is different of course, since for a locally flat
contact projective structure, also the path geometry of chains is locally flat. Still we can show
that, assuming vanishing contact torsion, contactomorphisms which map chains to chains are
morphisms of contact projective structures.
Theorem 4.3 For i = 1, 2 let (Mi , Hi ) be a contact manifolds endowed with contact projective
structures with vanishing contact torsion. Let f : M1 → M2 be a contact diffeomorphism
which maps chains to chains. Then f is an isomorphism of contact projective structures.
Proof Put Ni := PT Mi and let (pi : ˜Gi → Ni , ˜ωi ) be the (regular normal) parabolic
geometries associated to the path geometries determined by the subordinate projective structures.
Consider P0T Mi ⊂ Ni and the restricted parabolic geometries (p−1
i (P0T Mi ) →
P0T Mi , ˜ωi ), which describe the path geometries of chains. By assumption, the contactomorphism
f induces a morphism of these path geometries, so it lifts to a morphism Ψ
between the Cartan geometries.
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Geom Dedicata
Our aim is to extend Ψ to a morphism between the Cartan geometries ( ˜Gi → Ni , ˜ωi ).
Choose a local smooth section σ of the principal bundle ˜p1 : ˜G1 → M1, which has values
in p−1
1 (P0T M1), and let U ⊂ M1 be its domain of definition. Then there is a unique
˜P–equivariant map Ψσ : ˜p−1
1 (U) → ˜G2 such that Ψσ (σ(x)) = Ψ (σ(x)) for any x ∈ U. We
claim that Ψσ is an extension of Ψ to ˜p−1
1 (U). To prove this take, a point x ∈ U and consider
Ax := {u ∈ p−1
1 (x) : Ψ (u) = Ψσ (u)} ⊆ ˜p−1
1 (x).
By definition σ(x) ∈ Ax , so this set is non-empty. Further, since both Ψ and Ψσ are equivariant
for the principal right action of ˆP ⊂ ˜P, the set Ax is ˆP–invariant, and it is closed by
definition.
For A ∈ ˜p, the fundamental vector fields ζi
A ∈ X( ˜Gi ) are given by ˜ω−1
i (A). Since Ψ ∗ ˜ω2 =
˜ω1, we conclude that T Ψ ◦ ζ1
A = ζ2
A ◦ Ψ , so Ψ also intertwines the flows of these vector
fields, whenever they are defined. Otherwise put, for any u ∈ p−1
1 (P0T M1) there is a neighbourhood
V of e ∈ ˜P such that Ψ (ug) = Ψ (u)g for all g ∈ V . Since Ψσ is ˜P–equivariant by
definition, this implies that for any u ∈ Ax a neighborhood of u is contained in Ax , so Ax is
open. Since we have noted above that Ax is ˆP-equivariant, we can prove that Ax = p−1
1 (x)
and hence our claim by showing that the image of Ax surjects onto P0Tx M1 ⊂ PT M1. But
the projection to P0Tx M1 is a surjective submersion and hence an open mapping. Since both
Ax and its complement are open, also the image of Ax in P0Tx M1 is open and closed. Since
P0Tx M1 is the complement of a hyperplane in projective space and hence connected, the
proof of the claim is complete.
By construction, Ψσ : ˜p−1(U) → ˜G2 covers f |U : U → M2, so we can view it as
a morphism between the Cartan geometries ( ˜p−1
1 (U), ˜ω1) and ( ˜p−1
2 ( f (U)), ˜ω2). But this
exactly means that f |U is a morphism between the subordinate projective structures, so in
particular it locally preserves the contact geodesics. Hence locally and thus globally f is a
morphism of contact projective structures.
Acknowledgments First author supported by project P19500–N13 of the Fonds zur Förderung der wissenschaftlichen
Forschung (FWF).Second author supported by grant201/06/P379 of theCzech ScienceFoundation
(GA ˇCR). Discussions with Boris Doubrov have been very helpful. We also thank the referee for several helpful
comments.
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