Jacobi fields of the Tanaka-Webster connection on Sasakian manifolds

Abstract

We build a variational theory of geodesics of the Tanaka-Webster connection on a strictly pseudoconvex CR manifold.

Full text

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arXiv:math/0605393v1 [math.DG] 15 May 2006
Jacobi fields of the Tanaka-Webster
connection on Sasakian manifolds
Elisabetta Barletta1
Sorin Dragomir
Abstract. We build a variational theory of geodesics of the Tana-
ka-Webster connection ∇ on a strictly pseudoconvex CR manifold
M. Given a contact form θ on M such that (M,θ) has nonpositive
pseudohermitian sectional curvature (kθ(σ) ≤ 0) we show that
(M,θ) has no horizontally conjugate points. Moreover, if (M,θ) is
a Sasakian manifold such that kθ(σ) ≥ k0> 0 then we show that
the distance between any two consecutive conjugate points on a
lengthy geodesic of ∇ is at most π/(2√k0). We obtain the first and
second variation formulae for the Riemannian length of a curve in
M and show that in general geodesics of ∇ admitting horizontally
conjugate points do not realize the Riemannian distance.
1. Introduction
Sasakian manifolds possess a rich geometric structure (cf. [5], p. 73-
80) and are perhaps the closest odd dimensional analog of K¨ ahlerian
manifolds. In particular the concept of holomorphic sectional curva-
ture admits a Sasakian counterpart, the so called ϕ-sectional curvature
H(X) (cf. [5], p. 94) and it is a natural problem (as well as in K¨ ahlerian
geometry, cf. e.g. [18], p. 171, and p. 368-373) to investigate how re-
strictions on H(X) influence upon the topology of the manifold. An
array of findings in this direction are described in [5], p. 77-80. For in-
stance, by a result of M. Harada, [12], for any compact regular Sasakian
manifold M satisfying the inequality h > k2the fundamental group
π1(M) is cyclic. Here h = inf{H(X) : X ∈ Tx(M), ?X? = 1, x ∈ M}
and it is also assumed that the least upper bound of the sectional cur-
vature of M is 1/k2. Moreover, if additionally M has minimal diameter
π then M is isometric to the standard sphere S2n+1, cf. [13], p. 200.
1Authors’ address:
di Matematica,
barletta@unibas.it, dragomir@unibas.it
Universit` a degli Studi della Basilicata, Dipartimento
Campus Macchia Romana, 85100 Potenza,Italy,e-mail:
1

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In the present paper we embrace a different point of view, that of
pseudohermitian geometry (cf. [28]). To describe it we need to intro-
duce a few basic objects (cf. [5], p. 19-28). Let M be a (2n + 1)-
dimensional C∞manifold and (ϕ,ξ,η,g) a contact metric structure i.e.
ϕ is an endomorphism of the tangent bundle, ξ is a tangent vector field,
η is a differential 1-form, and g is a Riemannian metric on M such that
ϕ2= −I + η ⊗ ξ,
g(ϕX,ϕY ) = g(X,Y ) − η(X)η(Y ),
and Ω = dη (the contact condition) where Ω(X,Y ) = g(X,ϕY ). Any
contact Riemannian manifold (M,(ϕ,ξ,η,g)) admits a natural almost
CR structure
ϕ(ξ) = 0,η(ξ) = 1,
X,Y ∈ T(M),
T1,0(M) = {X − iJX : X ∈ Ker(η)}
√−1) i.e. it satisfies (2) below. By a result of S. Ianu¸ s, [14],
if (ϕ,ξ,η) is normal (i.e. [ϕ,ϕ] + 2(dη) ⊗ ξ = 0) then T1,0(M) is in-
tegrable, i.e. it obeys to (3) in Section 2. Cf. [5], p. 57-61, for the
geometric interpretation of normality, as related to the classical embed-
dability theorem for real analytic CR structures (cf. [1]). Integrability
of T1,0(M) is required in the construction of the Tanaka-Webster con-
nection of (M,η), cf. [26], [28] and definitions in Section 2 (although
many results in pseudohermitian geometry are known to carry over to
arbitrary contact Riemannian manifolds, cf. [27] and more recently [2],
[6]). A manifold carrying a contact metric structure (ϕ,ξ,η,g) whose
underlying contact structure (ϕ,ξ,η) is normal is a Sasakian mani-
fold (and g is a Sasakian metric). The main tool in the Riemannian
approach to the study of Sasakian geometry is the availability of a vari-
ational theory of geodesics of the Levi-Civita connection of (M,g) (cf.
e.g. [13], 194-197). In this paper we start the elaboration of a similar
theory regarding the geodesics of the Tanaka-Webster connection ∇ of
(M,η) and give a few applications (cf. Theorems 6-7 and 13 below).
Our motivation is twofold. First, we aim to study the topology of
Sasakian manifolds under restrictions on the curvature of ∇ and con-
jecture that Carnot-Carath´ eodory complete Sasakian manifolds whose
pseudohermitian Ricci tensor ρ satisfies ρ(X,X) ≥ (2n−1)k0?X?2for
some k0> 0 and any X ∈ Ker(η) must be compact. Second, the rela-
tionship between the sub-Riemannian geodesics of the sub-Riemannain
manifold (M,Ker(η),g) and the geodesics of ∇ (emphasized by our
Corollary 1) together with R.S. Strichartz’s arguments (cf. [24], p. 245
and 261-262) clearly indicates that a variational theory of geodesics of
∇ is the key requirement in bringing results such as those in [25] or [23]
(i =

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into the realm of subelliptic theory. In [3] one obtains a pseudohermi-
tian version of the Bochner formula (cf. e.g. [4], p. 131) implying a
lower bound on the first nonzero eigenvalue λ1of the sublaplacian ∆b
of a compact Sasakian manifold
(1)−λ1≥ 2nk/(2n − 1)
(a CR analog to the Lichnerowicz theorem, [20]). It is likely that a
theory of geodesics of ∇ may be employed to show that equality in (1)
implies that M is CR isomorphic to a sphere S2n+1(the CR analog to
Obata’s result, [23]).
Acknowledgements. The Authors are grateful to the anonymous Ref-
eree who pointed out a few errors in the original version of the manuscript.
The Authors acknowledge support from INdAM (Italy) within the inter-
disciplinary project Nonlinear subelliptic equations of variational origin in
contact geometry.
2. Sub-Riemannian geometry on CR manifolds
Let M be an orientable (2n + 1)-dimensional C∞manifold. A CR
structure on M is a complex distribution T1,0(M) ⊂ T(M) ⊗ C, of
complex rank n, such that
(2)T1,0(M) ∩ T0,1(M) = (0)
and
(3) Z,W ∈ T1,0(M) =⇒ [Z,W] ∈ T1,0(M)
(the formal integrability property).
bars denote complex conjugates). The integer n is the CR dimension.
The pair (M,T1,0(M)) is a CR manifold (of hypersurface type). Let
H(M) = Re{T1,0(M) ⊕ T0,1(M)} be the Levi distribution. It carries
the complex structure J : H(M) → H(M) given by J(Z + Z) =
i(Z − Z) (i =√−1). Let H(M)⊥⊂ T∗(M) the conormal bundle, i.e.
H(M)⊥
tian structure on M is a globally defined nowhere zero cross-section θ in
H(M)⊥. Pseudohermitian structures exist as the orientability assump-
tion implies that H(M)⊥≈ M × R (a diffeomorphism) i.e. H(M)⊥
is a trivial line bundle. For a review of the main notions of CR and
pseudohermitian geometry one may see [8].
Let (M,T1,0(M)) be a CR manifold, of CR dimension n. Let θ be a
pseudohermitian structure on M. The Levi form is
Here T0,1(M) = T1,0(M) (over-
x= {ω ∈ T∗
x(M) : Ker(ω) ⊇ H(M)x}, x ∈ M. A pseudohermi-
Lθ(Z,W) = −i(dθ)(Z,W),Z,W ∈ T1,0(M).

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M is nondegnerate if Lθis nondegenerate for some θ. Two pseudoher-
mitian structures θ andˆθ are related by
(4)
ˆθ = f θ
for some C∞function f : M → R \ {0}. Since Lˆθ= fLθ nonde-
generacy of M is a CR invariant notion, i.e. it is invariant under a
transformation (4) of the pseudohermitian structure. The whole set-
ting bears an obvious analogy to conformal geometry (a fact already
exploited by many authors, cf. e.g. [10], [26]-[28]). If M is nondegen-
erate then any pseudohermitian structure θ on M is actually a contact
form, i.e. θ ∧ (dθ)nis a volume form on M. By a fundamental result
of N. Tanaka and S. Webster (cf. op. cit.) on any nondegenerate CR
manifold on which a contact form θ has been fixed there is a canonical
linear connection ∇ (the Tanaka-Webster connection of (M,θ)) com-
patible to the Levi distribution and its complex structure, as well as
to the Levi form. Precisely, let T be the globally defined nowhere zero
tangent vector field on M, transverse to H(M), uniquely determined
by θ(T) = 1 and T ⌋dθ = 0 (the characteristic direction of dθ). Let
Gθ(X,Y ) = (dθ)(X,JY ),X,Y ∈ H(M),
(the real Levi form) and consider the semi-Riemannian metric gθon M
given by
gθ(X,Y ) = Gθ(X,Y ),gθ(X,T) = 0,gθ(T,T) = 1,
for any X,Y ∈ H(M) (the Webster metric of (M,θ)). Let us extend J
to an endomorphism of the tangent bundle by setting JT = 0. Then
there is a unique linear connection ∇ on M such that i) H(M) is
parallel with respect to ∇, ii) ∇gθ= 0, ∇J = 0, and iii) the torsion
T∇of ∇ is pure, i.e.
(5)T∇(Z,W) = T∇(Z,W) = 0,T∇(Z,W) = 2iLθ(Z,W)T,
for any Z,W ∈ T1,0(M), and
(6)τ ◦ J + J ◦ τ = 0,
where τ(X) = T∇(T,X) for any X ∈ T(M) (the pseudohermitian
torsion of ∇). The Tanaka-Webster connection is a pseudohermitian
analog to both the Levi-Civita connection in Riemannian geometry and
the Chern connection in Hermitian geometry.
A CR manifold M is strictly pseudoconvex if Lθis positive definite for
some θ. If this is the case then the Webster metric gθis a Riemannian
metric on M and if we set ϕ = J, ξ = −T, η = −θ and g = gθ
then (ϕ,ξ,η,g) is a contact metric structure on M. Also (ϕ,ξ,η,g) is

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normal if and only if τ = 0. If this is the case gθis a Sasakian metric
and (M,θ) is a Sasakian manifold.
We proceed by recalling a few concepts from sub-Riemannian ge-
ometry (cf. e.g. R.S. Strichartz, [24]) on a strictly pseudoconvex CR
manifold. Let (M,T1,0(M)) be a strictly pseudoconvex CR manifold,
of CR dimension n. Let θ be a contact form on M such that the Levi
form Gθ is positive definite. The Levi distribution H(M) is bracket
generating i.e. the vector fields which are sections of H(M) together
with all brackets span Tx(M) at each point x ∈ M, merely as a conse-
quence of the nondegeneracy of the given CR structure. Indeed, let ∇
be the Tanaka-Webster connection of (M,θ) and let {Tα: 1 ≤ α ≤ n}
be a local frame of T1,0(M), defined on the open set U ⊆ M. By the
purity property (5)
Γγ
(7)
αβTγ− Γγ
βαTγ− [Tα,Tβ] = 2igαβT,
where ΓA
BCare the coefficients of ∇ with respect to {Tα}
∇TBTC= ΓA
and gαβ= Lθ(Tα,Tβ).Our conventions as to the range of indices
are A,B,C,··· ∈ {0,1,··· ,n,1,··· ,n} and α,β,γ,··· ∈ {1,··· ,n}
(where T0= T). Note that {Tα,Tα,T} is a local frame of T(M)⊗C on
U. If Tα= Xα−iJXαare the real and imaginary parts of Tαthen (7)
shows that {Xα,JXα} together with their brackets span the whole of
Tx(M), for any x ∈ U. Actually more has been proved. Given x ∈ M
and v ∈ H(M)x\{0} there is an open neighborhood U ⊆ M of x and a
local frame {Tα} of T1,0(M) on U such that T1(x) = v − iJxv, hence v
is a 2-step bracket generator so that H(M) satisfies the strong bracket
generating hypothesis (cf. the terminology in [24], p. 224).
Let x ∈ M and g(x) : T∗
Gθ,x(v,g(x)ξ) = ξ(v),v ∈ H(M)x, ξ ∈ T∗
Note that the kernel of g is precisely the conormal bundle H(M)⊥.
In other words Gθ is a sub-Riemannian metric on H(M) and g its
alternative description (cf. also (2.1) in [24], p. 225). Ifˆθ = euθ is
another contact form such that Gˆθis positive definite (u ∈ C∞(M))
then ˆ g = e−ug. Clearly if the Levi form Lθis only nondegenerate then
(M,H(M),Gθ) is a sub-Lorentzian manifold, cf. the terminology in
[24], p. 224.
Let γ : I → M be a piecewise C1curve (where I ⊆ R is an interval).
Then γ is a lengthy curve if ˙ γ(t) ∈ H(M)γ(t) for every t ∈ I such
that ˙ γ(t) is defined. For instance, any geodesic of ∇ (i.e. any C1
curve γ(t) such that ∇˙ γ˙ γ = 0) of initial data (x,v), v ∈ H(M)x, is
BCTA
x(M) → H(M)xdetermined by
x(M).

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lengthy (as a consequence of ∇gθ= 0 and ∇T = 0). A piecewise C1
curve ξ : I → T∗(M) is a cotangent lift of γ if ξ(t) ∈ T∗
g(γ(t))ξ(t) = ˙ γ(t) for every t (where defined). Clearly cotangent lifts
of a given lengthy curve γ exist (cf. also Proposition 1 below). Also,
cotangent lifts of γ are uniquely determined modulo sections of the
conormal bundle H(M)⊥along γ. That is, if η : I → T∗(M) is another
cotangent lift of γ then η(t) − ξ(t) ∈ H(M)⊥
of a lengthy curve γ : I → M is given by
?
The definition doesn’t depend upon the choice of cotangent lift ξ of
γ. The Carnot-Carath´ eodory distance ρ(x,y) among x,y ∈ M is the
infimum of the lengths of all lengthy curves joining x and y. That ρ is
indeed a distance function on M follows from a theorem of W.L. Chow,
[7], according to which any two points x,y ∈ M may be joined by a
lengthy curve (provided that M is connected).
Let gθ be the Webster metric of (M,θ). Then gθ is a contraction
of the sub-Riemannian metric Gθ(Gθis an expansion of gθ), cf. [24],
p. 230. Let d be the distance function corresponding to the Webster
metric. The length L(γ) of a lengthy curve γ is precisely its length
with respect to gθhence
γ(t)(M) and
γ(t)for every t. The length
L(γ) =
I
{ξ(t)[g(γ(t))ξ(t)]}1/2dt.
(8)d(x,y) ≤ ρ(x,y),x,y ∈ M.
While ρ and d are known to be inequivalent distance functions, they
do determine the same topology. For further details on Carnot-Cara-
th´ eodory metrics see J. Mitchell, [22].
Let (U,x1,··· ,x2n+1) be a system of local coordinates on M and let
us set Gij= gθ(∂i,∂j) (where ∂iis short for ∂/∂xi) and [Gij] = [Gij]−1.
Using
Gθ(X,g dxi) = (dxi)(X),X ∈ H(M),
for X = ∂k− θkT (where θi= θ(∂i)) leads to
(9)gij(Gjk− θjθk) = δi
where g dxi= gij∂j and T = Ti∂i.
θ(g dxi) = 0 so that (9) yields
k− θkTi
On the other hand gijθj =
(10)gij= Gij− TiTj.
As an application we introduce a canonical cotangent lift of a given
lengthy curve on M.

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Proposition 1. Let γ : I → M be a lengthy curve and let ξ : I →
T∗(M) be given by ξ(t)Tγ(t)= 1 and ξ(t)X = gθ(˙ γ,X), for any X ∈
H(M)γ(t). Then ξ is a cotangent lift of γ.
Proof. Let xi(t) be the components of γ with respect to the chosen
local coordinate system. By the very definition of ξ
dxi
(11)ξj= Gij
dt+ θj.
Hence
g ξ = ξjgij∂i= gij(Gjkdxk
dt
+ θj)∂i= gijGjkdxk
dt∂i=
= (Gij− TiTj)Gjkdxk
dt∂i= (δi
k− Tiθk)dxk
dt∂i=
= ˙ γ(t) − θ(˙ γ(t))T = ˙ γ(t).
We recall (cf. [24], p. 233) that a sub-Riemannian geodesic is a C2
curve γ(t) in M satisfying the Hamilton-Jacobi equations associated to
the Hamiltonian function H(x,ξ) =1
2gij(x)ξiξjthat is
(12)
dxi
dt
= gij(γ(t))ξj(t),
(13)
dξk
dt
= −1
2
∂gij
∂xk(γ(t))ξi(t)ξj(t),
for some cotangent lift ξ(t) ∈ T∗(M) of γ(t). Our purpose is to show
that
Theorem 1. Let M be a strictly pseudoconvex CR manifold and θ
a contact form on M such that Gθ is positive definite. A C2curve
γ(t) ∈ M, |t| < ǫ, is a sub-Riemannian geodesic of (M,H(M),Gθ) if
and only if γ(t) is a solution to
b′(t) = A(˙ γ, ˙ γ),(14)∇˙ γ˙ γ = −2b(t)J ˙ γ,|t| < ǫ,
with ˙ γ(0) ∈ H(M)γ(0), for some C2function b : (−ǫ,ǫ) → R. Here
A(X,Y ) = gθ(τX,Y ) is the pseudohermitian torsion of (M,θ).
According to the terminology in [24], p. 237, the canonical cotangent
lift ξ(t) of a given lengthy curve γ(t) is the one determined by the
orthogonality requirement
Vj(ξ)Γj(ξ,v) = 0,
for any v ∈ H(M)⊥
Vk(ξ) =dξk
(15)
γ(t)and any |t| < ǫ, where
dt+1
2
∂gij
∂xkξiξj,

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Γi(ξ,v) = Γijkξjvk,Γijk=1
2(gℓj∂gik
∂xℓ+ gℓk∂gij
∂xℓ− gℓi∂gjk
∂xℓ).
Let γ(t) be a lengthy curve and ξ0(t) the cotangent lift of γ(t) furnished
by Proposition 1. Then any other cotangent lift ξ(t) is given by
(16)ξ(t) = ξ0(t) + a(t)θγ(t),|t| < ǫ,
for some a : (−ǫ,ǫ) → R. We shall need the following result (a replica
of Lemma 4.4. in [24], p. 237)
Lemma 1. The unique cotangent lift ξ(t) of γ(t) satisfying the orthog-
onality condition (15) is given by (16) where
a(t) = −1
2|˙ γ(t)|−2gθ(∇˙ γ˙ γ , J ˙ γ) − 1,|t| < ǫ.
Proof. By (11) and (16)
Vk(ξ) = Vk(ξ0) + a′(t)θk+ a(t)∂θk
∂xℓ
dxℓ
dt+
+1
2
∂gij
∂xk[a(t)(ξ0
iθj+ ξ0
jθi) + a(t)2θiθj]
(where ξ0= ξ0
idxi) and using
∂gij
∂xkθiθj= 0
we obtain
(17)Vi(ξ) = Vi(ξ0) + a′(t)θi+ 2a(t)(dθ)(˙ γ,∂i).
Note that Γi(ξ,v) = Γi(ξ0,v) and Γijkθjvk= 0, for any v ∈ H(M)⊥
Let us contract (17) with Γi(ξ,v) and use (15) and Γi(ξ0,v)θi= 0. This
ought to determine a(t). Indeed
γ(t).
(18)Vi(ξ0)Γi(ξ0,v) + 2a(t)(dθ)(˙ γ,Γ(ξ0,v)) = 0,
where Γ(ξ,v) = Γi(ξ,v)∂i. On the other hand, a calculation based on
(10)-(11) shows that
Vk(ξ0) = Gkℓ(d2xℓ
dt2+
????
ℓ
ij
????
dxi
dt
dxj
dt) + 2(dθ)(˙ γ,∂k),
????
ℓ
ij
????= Gℓk|ij,k|,
Vi(ξ0) = Gij(D˙ γ˙ γ)j+ 2(dθ)(˙ γ,∂i),
|ij,k| =1
2(∂Gik
∂xj+∂Gjk
∂xi−∂Gij
∂xk),
hence
(19)

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where D is the Levi-Civita connection of (M,gθ). Then (18)-(19) yield
gθ(D˙ γ˙ γ , Γ(ξ0,v)) + 2(a(t) + 1)(dθ)(˙ γ , Γ(ξ0,v)) = 0,
for any v ∈ H(M)⊥
gθ(Γ(ξ0,θ), D˙ γ˙ γ + 2(a(t) + 1)J ˙ γ) = 0
and
Γi(ξ0,θ) = −Gij(dθ)(˙ γ , ∂j),
(because of T ⌋dθ = 0) yields
(20) 2(a(t) + 1)|˙ γ(t)|2+ gθ(D˙ γ˙ γ , J ˙ γ) = 0.
Lemma 1 is proved. At this point we may prove Theorem 1. Let
γ(t) ∈ M be a sub-Riemannian geodesic of (M,H(M),Gθ).
there is a cotangent lift ξ(t) ∈ T∗(M) of γ(t) (given by (16) for some
a : (−ǫ,ǫ) → R) such that V (ξ) = 0 (where V (ξ) = Vi(ξ)∂i). In
particular the orthogonality condition (15) is identically satisfied, hence
a(t) is determined according to Lemma 1. Using (17) and (19) the sub-
Riemannian geodesics equations are
γ(t). Yet H(M)⊥is the span of θ hence
Then
Gij(D˙ γ˙ γ)j+ a′(t)θi+ 2(a(t) + 1)(dθ)(˙ γ , ∂i) = 0
or
(21)D˙ γ˙ γ + a′(t)T + 2(a(t) + 1)J ˙ γ = 0.
We recall (cf. e.g. [10]) that D = ∇−(dθ +A)⊗T on H(M)⊗H(M)
hence (by the uniqueness of the direct sum decomposition T(M) =
H(M) ⊕ RT) the equations (21) become
∇˙ γ˙ γ + 2(a(t) + 1)J ˙ γ = 0,
(and we set b = a + 1). Theorem 1 is proved.
a′(t) = A(˙ γ, ˙ γ),
Corollary 1. Let M be a strictly pseudoconvex CR manifold and θ
a contact form on M with vanishing pseudohermitian torsion (τ =
0). Then any lengthy geodesic of the Tanaka-Webster connection ∇
of (M,θ) is a sub-Riemannian geodesic of (M,H(M),Gθ). Viceversa,
if every lengthy geodesic γ(t) of ∇ is a sub-Riemannian geodesic then
τ = 0.
Indeed, if ∇˙ γ˙ γ = 0 then the equations (14) (with b = 0) are identically
satisfied.
Proposition 2. Let γ(t) ∈ M be a sub-Riemannian geodesic and
s = φ(t) a C2diffeomorphism. If γ(t) = γ(φ(t)) then γ(s) is a sub-
Riemannian geodesic if and only if φ is affine, i.e. φ(t) = αt + β, for

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some α,β ∈ R. In particular, every sub-Riemannian geodesic may be
reparametrized by arc length φ(t) =?t
Proof. Set k = |˙ γ(0)|2> 0. By taking the inner product of the first
equation in (14) by ˙ γ(t) it follows that d|˙ γ(t)|2/dt = 0, hence |˙ γ(t)|2=
k, |t| < ǫ.
quantities associated to γ(s). In particular k = φ′(0)−2k. Locally
d2xi
dt2+ Γi
dtdt
On the other hand, using (20) and
d2xi
dt2+ Γi
dtdt
we obtain
k(a + 1) = k(a + 1)φ′(t)3.
Then (22) may be written
kφ′′(t)dγ
0|˙ γ(u)|du.
Throughout the proof an overbar indicates the similar
(22)
jk
dxj
dxk
= −2(a + 1)Ji
j
dxj
dt.
jk
dxj
dxk
= φ′′(t)d2xi
ds2+ φ′(t)2(d2xi
ds2+ Γi
jk
dxj
ds
dxk
ds)
ds+ 2(a + 1)φ′(t)2[kφ′(t)2− k]Jdγ
ds= 0
hence φ′′(t) = 0. Proposition 2 is proved.
Let S1→ C(M)
e.g. [8], p. 104). Let Σ be the tangent to the S1-action. Next, let us
consider the 1-form σ on C(M) given by
1
n + 2{dr + π∗(iωα
where r is a local fibre coordinate on C(M) (so that locally Σ = ∂/∂r)
and R = gαβRαβis the pseudohermitian scalar curvature of (M,θ).
Then σ is a connection 1-form in S1→ C(M) → M. Given a tangent
vector v ∈ Tx(M) and a point z ∈ π−1(x) we denote by v↑its horizontal
lift with respect to σ, i.e. the unique tangent vector v↑∈ Ker(σz) such
that (dzπ)v↑= v. The Fefferman metric of (M,θ) is the Lorentz metric
on C(M) given by
Fθ= π∗˜Gθ+ 2(π∗θ) ⊙ σ,
where˜Gθ = Gθ on H(M) ⊗ H(M) and˜Gθ(X,T) = 0, for any X ∈
T(M). Also ⊙ is the symmetric tensor product. We close this sec-
tion by demonstrating the following geometric interpretation of sub-
Riemannian geodesics (of a strictly pseudoconvex CR manifold).
π
−→ M be the canonical circle bundle over M (cf.
σ =
α−i
2gαβdgαβ−
R
4(n + 1)θ)},
Theorem 2. Let M be a strictly pseudoconvex CR manifold, θ a con-
tact form on M such that Gθis positive definite, and Fθthe Fefferman
metric of (M,θ). For any geodesic z : (−ǫ,ǫ) → C(M) of Fθ if the

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projection γ(t) = π(z(t)) is lengthy then γ : (−ǫ,ǫ) → M is a sub-
Riemannian geodesic of (M,H(M),Gθ). Viceversa, let γ(t) ∈ M be a
sub-Riemannian geodesic. Then any solution z(t) ∈ C(M) to the ODE
(23)˙ z(t) = ˙ γ(t)↑+ ((n + 2)/2)b(t)Σz(t),
where b(t) = a(t) + 1 is given by (20), is a geodesic of Fθ.
Here ˙ γ(t)↑∈ Ker(σz(t)) and (dz(t)π)˙ γ(t)↑= ˙ γ(t). To prove Theorem 2
we shall need the following
Lemma 2. For any X,Y ∈ H(M)
∇C(M)
X↑
Y↑= (∇XY )↑− (dθ)(X,Y )T↑− (A(X,Y ) + (dσ)(X↑,Y↑))ˆΣ,
∇C(M)
X↑
T↑= (τX + φX)↑,
∇C(M)
T↑
X↑= (∇TX + φX)↑+ 2(dσ)(X↑,T↑)ˆΣ,
∇C(M)
X↑
ˆΣ = ∇C(M)
ˆΣ
X↑= (JX)↑,
∇C(M)
T↑
T↑= V↑, ∇C(M)
ˆΣ
ˆΣ = 0,
∇C(M)
ˆΣ
T↑= ∇C(M)
T↑
ˆΣ = 0,
where φ : H(M) → H(M) is given by Gθ(φX,Y ) = (dσ)(X↑,Y↑),
and V ∈ H(M) is given by Gθ(V,Y ) = 2(dσ)(T↑,Y↑). AlsoˆΣ =
((n + 2)/2)Σ.
This relates the Levi-Civita connection ∇C(M)of Fθ to the Tanaka-
Webster connection of (M,θ). Cf. [9] for a proof of Lemma 2.
Proof of Theorem 2. Let z(t) ∈ C(M) be a geodesic of ∇C(M)and
γ(t) = π(z(t)). Assume that ˙ γ(t) ∈ H(M)γ(t). Note that ˙ z(t)− ˙ γ(t)↑∈
Ker(dz(t)π) hence ˙ z(t) is given by (23), for some b : (−ǫ,ǫ) → R. Then
(by Lemma 2)
0 = ∇C(M)
˙ z
˙ z = ∇C(M)
˙ γ↑
˙ γ↑+ b′(t)ˆΣ + 2b(t)(J ˙ γ)↑=
= (∇˙ γ˙ γ)↑+ [b′(t) − A(˙ γ , ˙ γ)]ˆΣ + 2b(t)(J ˙ γ)↑
hence (by T(C(M)) = Ker(σ) ⊕ RΣ) γ(t) satisfies the equations (14),
i.e. γ(t) is a sub-Riemannian geodesic. The converse is obvious.

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3. Jacobi fields on CR manifolds
Let M be a strictly pseudoconvex CR manifold endowed with a con-
tact form θ such that Gθ is positive definite. Let ∇ be the Tanaka-
Webster connection of (M,θ).Let γ(t) ∈ M be a geodesic of ∇,
parametrized by arc length. A Jacobi field along γ is vector field X on
M satisfying to the second order ODE
(24)∇2
˙ γX + ∇˙ γT∇(X, ˙ γ) + R(X, ˙ γ)˙ γ = 0.
Let Jγbe the real linear space of all Jacobi fields of (M,∇). Then Jγis
(4n + 2)-dimensional (cf. Prop. 1.1 in [18], Vol. II, p. 63). We denote
by ˆ γ the vector field along γ defined by ˆ γγ(t)= t˙ γ(t) for every value of
the parameter t. Note that ˙ γ, ˆ γ ∈ Jγ. We establish
Theorem 3. Every Jacobi field X along a lengthy geodesic γ of ∇ can
be uniquely decomposed in the following form
(25)X = a˙ γ + bˆ γ + Y
where a,b ∈ R and Y is a Jacobi field along γ such that
(26)gθ(Y, ˙ γ)γ(t)= −
?t
0
θ(X)γ(s)A(˙ γ, ˙ γ)γ(s)ds.
In particular, if i) Xγ(t) ∈ H(M)γ(t) for every t, or ii) (M,θ) is a
Sasakian manifold (i.e. τ = 0), then Y is perpendicular to γ.
We need the following
Lemma 3. For any Jacobi field X ∈ Jγ
d
dt{gθ(X, ˙ γ)} + θ(X)γ(t)A(˙ γ, ˙ γ)γ(t)= const.
Proof. Let us take the inner product of the Jacobi equation (24) by
˙ γ and use the skew symmetry of gθ(R(X,Y )Z,W) in the arguments
(Z,W) (a consequence of ∇gθ= 0) so that to get
d2
dt2{gθ(X, ˙ γ)} +d
On the other hand, let us set XH= X −θ(X)T (so that XH∈ H(M)).
Then
dt{gθ(T∇(X, ˙ γ), ˙ γ)} = 0.
gθ(T∇(X, ˙ γ), ˙ γ) = −2Ω(XH, ˙ γ)gθ(T, ˙ γ) + θ(X)gθ(τ(˙ γ), ˙ γ)
or (as γ is lengthy)
gθ(T∇(X, ˙ γ), ˙ γ) = θ(X)A(˙ γ, ˙ γ).

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Lemma 3 is proved. Throughout the section we adopt the notation
X′= ∇˙ γX and X′′= ∇2
Proof of Theorem 3. We set by definition
˙ γX.
a = gθ(X, ˙ γ)γ(0),b = gθ(X′, ˙ γ)γ(0)+ θ(X)γ(0)A(˙ γ, ˙ γ)γ(0),
and Y = X − a˙ γ − bˆ γ. Clearly Y ∈ Jγ. Then, by Lemma 3
d
dt{gθ(Y, ˙ γ)} + θ(Y )A(˙ γ, ˙ γ) = α,
for some α ∈ R. Next we integrate from 0 to t
gθ(Y, ˙ γ)γ(t)− gθ(Y, ˙ γ)γ(0)+
?t
0
θ(Y )γ(s)A(˙ γ, ˙ γ)γ(s)ds = αt
and substitute Y from (25) (and use ˙ γ, ˆ γ ∈ H(M)). Differentiating
the resulting relation with respect to t at t = 0 gives α = 0. Hence
?t
The existence statement in Theorem 3 is proved. We need the following
terminology. Given X ∈ Jγa Jacobi field Y ∈ Jγsatisfying (26) is said
to be slant at γ(t) relative to X. Also Y is slant if it slant at any
point of γ. To check the uniqueness statement let X = a′˙ γ + b′ˆ γ + Z
be another decomposition of X, where a′, b′∈ R and Z ∈ Jγis slant
(relative to X). Then
gθ(Y, ˙ γ) +
0
θ(X)γ(s)A(˙ γ, ˙ γ)γ(s)ds = 0.
(a + bt)˙ γ(t) + Yγ(t)= (a′+ b′t)˙ γ(t) + Zγ(t)
and taking the inner product with ˙ γ(t) yields a + bt = a′+ b′t, i.e.
a = a′, b = b′and Yγ(t)= Zγ(t). Q.e.d.
Corollary 2. Suppose a Jacobi field X ∈ Jγ is slant at γ(r) and at
γ(s) relative to itself, for some r ?= s. Then X is slant. In particular,
if i) Xγ(t)∈ H(M)γ(t)for every t, or ii) (M,θ) is a Sasakian manifold,
and X is perpendicular to γ at two points, it is perpendicular to γ at
every point of γ.
Proof. By Theorem 3 we may decompose X = a˙ γ + bˆ γ + Y , where
Y ∈ Jγis slant (relative to X). Taking the inner product of Xγ(r)=
(a + br)˙ γ(r) + Yγ(r)with ˙ γ(r) gives a + br = 0. Similarly a + bs = 0
hence (as r ?= s) a = b = 0, so that X = Y . Q.e.d.
4. CR manifolds without conjugate points
Two points x and y on a lengthy geodesic γ(t) are horizontally con-
jugate if there is a Jacobi field X ∈ Jγsuch that Xγ(t)∈ H(M)γ(t)for

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every t and Xx= Xy= 0. As T∇is pure, the Jacobi equation (24) may
also be written
(27) X′′− 2Ω(X′, ˙ γ)T + θ(X′)τ(˙ γ) + θ(X)(∇˙ γτ)˙ γ + R(X, ˙ γ)˙ γ = 0.
Given X ∈ Jγone has (by (27))
d
dt{gθ(X′,X)} = gθ(X′′,X) + gθ(X′,X′) =
= |X′|2+ 2θ(X)Ω(X′, ˙ γ) − θ(X′)A(˙ γ,X)−
−θ(X)gθ(∇˙ γτ ˙ γ,X) − gθ(R(X, ˙ γ)˙ γ,X).
On the other hand (again by ∇gθ= 0)
θ(X′)A(˙ γ,X) + θ(X)gθ(∇˙ γτ ˙ γ,X) =
= θ(X′)A(˙ γ,X) + θ(X)d
dt{A(˙ γ,X)} − θ(X)A(˙ γ,X′) =
=d
dt{θ(X)A(˙ γ,X)} − θ(X)A(˙ γ,X′)
hence
d
dt{gθ(X′,X) + θ(X)A(˙ γ,X)} =
= |X′|2− gθ(R(X, ˙ γ)˙ γ,X) + θ(X)[A(˙ γ,X′) + 2Ω(X′, ˙ γ)].
S. Webster (cf. [28]) has introduced a notion of pseudohermitian sec-
tional curvature by setting
kθ(σ) =1
4Gθ(X,X)−2gθ,x(Rx(X,JxX)JxX,X),
for any holomorphic 2-plane σ (i.e. a 2-plane σ ⊂ H(M)xsuch that
Jx(σ) = σ), where {X,JxX} is a basis of σ.
makes the sphere ι : S2n+1⊂ Cn+1(endowed with the contact form
θ0 = ι∗[i
a pseudohermitian analog to the notion of holomorphic sectional cur-
vature in Hermitian geometry. On the other hand, for any 2-plane
σ ⊂ Tx(M) one may set
kθ(σ) =1
4gθ,x(Rx(X,Y )Y,X)
where {X,Y } is a gθ,x-orthonormal basis of σ. Cf. [18], Vol. I, p. 200,
the definition of kθ(σ) doesn’t depend upon the choice of orthonormal
basis in σ because the curvature R(X,Y,Z,W) = gθ(R(Z,W)Y,X) of
the Tanaka-Webster connection is skew symmetric in both pairs (X,Y )
and (Z,W). We refer to kθas the pseudohermitian sectional curvature
of (M,θ). A posteriori the restriction (29) of kθto holomorphic 2-planes
(28)
(29)
The coefficient 1/4
2(∂ − ∂)|z|2]) have constant curvature +1. Clearly, this is

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is referred to as the holomorphic pseudohermitian sectional curvature
of (M,θ). As an application of (28) we may establish
Theorem 4. If (M,θ) has nonpositive pseudohermitian sectional cur-
vature then (M,θ) has no horizontally conjugate points.
We need
Lemma 4. For every Jacobi field X ∈ Jγ
d
dt{θ(X)} − 2Ω(X, ˙ γ)γ(t)= c = const.
To prove Lemma 4 one merely takes the inner product of (27) by T.
Proof of Theorem 4. The proof is by contradiction. If there is a
lengthy geodesic γ(t) ∈ M (parametrized by arc length) and a Jacobi
field X ∈ Jγsuch that Xγ(a)= Xγ(b)= 0 for two values a and b of the
parameter then we may integrate in (28) so that to obtain
(30)
?b
a
{|X′|2−gθ(R(X, ˙ γ)˙ γ,X)+θ(X)[A(˙ γ,X′)+2Ω(X′, ˙ γ)]}dt = 0.
On the other hand
θ(X)Ω(X′, ˙ γ) = θ(X)d
dt{Ω(X, ˙ γ)} =
=d
dt{θ(X)Ω(X, ˙ γ)} − Ω(X, ˙ γ)θ(X′).
Then (by Lemma 4)
2
?b
a
θ(X)Ω(X′, ˙ γ)dt = −2
?b
a
Ω(X, ˙ γ)d
dt{θ(X)}dt =
= c
?b
a
d
dt{θ(X)}dt −
?b
a
θ(X′)2dt = −
?b
a
θ(X′)2dt
hence (30) becomes
?b
a
{|X′|2− gθ(R(X, ˙ γ)˙ γ,X) + θ(X)A(˙ γ,X′) − θ(X′)2}dt = 0.
Finally, if X ∈ H(M) then X′∈ H(M) and then (under the assump-
tions of Theorem 4) X′= 0, a contradiction.