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Ann. I. H. Poincaré – AN 31 (2014) 985–1014
www.elsevier.com/locate/anihpc
Partially hyperbolic geodesic flows
Fernando Carneiro
a,∗, Enrique Pujals
b,1aUERJ, R. São Francisco Xavier, 524, sala d6021 – Maracanã, Rio de Janeiro, RJ, cep 20550-013, Brazil bIMPA, Estrada Dona Castorina 110, Rio de Janeiro, RJ, cep 22460-320, Brazil
Received 12 November 2011; received in revised form 10 March 2013; accepted 31 July 2013 Available online 5 September 2013
Abstract
We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such an example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.
©2013 Published by Elsevier Masson SAS.
1. Introduction
The theory of hyperbolic dynamics has been one of the extremely successful stories in dynamical systems. Origi- nated by studying dynamical properties of geodesic flows on manifolds with negative curvature[1]and geometrical properties of homoclinic points[33], hyperbolicity is the cornerstone of uniform and robust chaotic dynamics; it char- acterizes the structural stable systems; it provides the structure underlying the presence of homoclinic points; a large category of rich dynamics are hyperbolic (geodesic flows in negative curvature, billiards with negative curvature, linear automorphisms, some mechanical systems, etc.); the hyperbolic theory has been fruitful in developing a geometrical approach to dynamical systems; and, under the assumption of hyperbolicity one obtains a satisfactory (complete) de- scription of the dynamics of the system from a topological and statistical point of view. Moreover, hyperbolicity has provided paradigms or models of behavior that can be expected to be obtained in specific problems.
Nevertheless, hyperbolicity was soon realized to be a property less universal than it was initially thought: it was shown that there are open sets in the space of dynamics which are nonhyperbolic. To overcome these difficulties, the theory moved in different directions; one being to develop weaker or relaxed forms of hyperbolicity, hoping to include a larger class of dynamics.
There is an easy way to relax hyperbolicity, called partial hyperbolicity. Letf:M→Mbe a diffeomorphism from a smooth manifoldMto itself. We say thatf is partially hyperbolic if the tangent bundle ofMsplit intoDf invariant subbundlesT M=Es⊕Ec⊕Eu, such that the behavior of vectors inEs, Euare contracted and expanded respectively
* Corresponding author.
E-mail addresses:carneiro@ime.uerj.br(F. Carneiro),enrique@impa.br(E. Pujals).
1 Second author was supported by CNPq and CAPES.
0294-1449/$ – see front matter ©2013 Published by Elsevier Masson SAS.
http://dx.doi.org/10.1016/j.anihpc.2013.07.009
byDf, but vectors inEcmay be neutral for the action of the tangent map, i.e.,|dxfnv|contracts exponentially fast ifv∈Es,|dxfnv|expands exponentially fast ifv∈Euand|dxfnv|neither contracts nor expands as fast as for the other two invariant subbundles ifv∈Ec. The Anosov condition is equivalent toEc(x)= {0}for allx∈M. This notion arose in a natural way in the context of time one-maps of Anosov flows, frame flows or group extensions. See [7,32,24,5,9]for examples of these systems and[21,29]for an overview.
However, and differently from hyperbolic ones, partially hyperbolic non-Anosov systems were unknown in the context of geodesic flows induced by Riemannian metrics. As far as we know, the way to produce partially hyperbolic systems in discrete dynamics are the following: time-one maps of Anosov flows, skew-products over hyperbolic dynamics, products and derived of Anosov deformations (DA). The two last approaches can be adapted to flows.
Our work shows that one is able to deform a specific metric that provides an Anosov geodesic flow to get a partially hyperbolic geodesic flow.Theorem Ais inspired by the Mañé’s DA construction of a partially hyperbolic diffeomorphism[24].
We prove the following theorems:
Theorem A.There are Riemannian metrics such that their geodesic flows are partially hyperbolic but not Anosov.
Moreover, some of them are transitive.
More precisely, we prove:
Theorem B. Let (M, g) be a compact locally symmetric Riemannian manifold of nonconstant negative sectional curvature. There is a C2-open set of C∞ Riemannian metrics onM such that their geodesic flows are partially hyperbolic but not Anosov. Those metrics which are on the C2-boundary of the Anosov Riemannian metrics are transitive.
Remark 1.1.These metrics areC1-close andC2-far fromg, and some areC2-far from Anosov.
Remark 1.2.Cartan classified the symmetric spaces of negative curvature (see[19,20]). They are:
i. the hyperbolic spaceRHn of constant curvature−c2, which is the canonical space form of negative constant curvature;
ii. the hyperbolic space CHn of curvature−4c2K−c2, which is the canonical Kähler hyperbolic space of constant negative holomorphic curvature−4c2[17];
iii. the hyperbolic spaceHHnof curvature−4c2K−c2, which is the canonical quaternionic Kähler symmetric space of negative curvature[4,36];
iv. the hyperbolic spaceCaH2of curvature−4c2K−c2, which is the canonical hyperbolic symmetric space of the octonions of constant negative curvature.
The theorem works for the quotient of the Kähler hyperbolic space of constant negative holomorphic curvature by a cocompact lattice, and also for the quotient of the quaternionic Kähler symmetric space of negative curvature by a cocompact lattice. Both of these locally symmetric spaces are even-dimensional. In these cases, for a fixedv∈T M and if we supposec=1 in the classification above, there are subspaces AandB of v⊥ such that if w∈Athen K(v, w)= −1, ifw∈BthenK(v, w)= −14andv⊥=A⊕B. This property implies that the geodesic flow of these manifolds is Anosov with many invariant subbundles (see Section6).
Remark 1.3.Of course, if we multiply the metric by a constant, the Anosov or the partially hyperbolic splitting remain the same, but the curvature does not. So, we consider the maximal sectional curvature of the locally symmetric space to be−1, which is true after multiplication of the metric by a constant.
LetX:N→T N be a vector field onN without singularities, i.e., for anyp∈N,X(p)=0, letφt :N →N be its flow, letp∈N be a point such thatφT(p)=pfor a positive real numberT, and letλi,i=1, . . . ,dim(N )be the eigenvalues ofdpφT :TpN→TpN. Letλ1=1 be the eigenvalue associated toX(p). We saypis hyperbolic if there is no eigenvalue in the unit circle, besidesλ1, i.e.,λi =1 fori=2, . . . ,dim(N ). We saypis quasi-elliptic if there are
eigenvalues in the unit circle, besidesλ1. We saypis nondegenerate if there is no eigenvalue equal to one or to a root of the unity besidesλ1. The next two corollaries are given by the persistence of quasi-elliptic nondegenerate periodic points.
Corollary C.1.There is aC2-open setU of metrics in the set of metrics ofMsuch that forg∈U, the geodesic flow of gis partially hyperbolic but not Anosov, for(M, g)as in the previous theorem. There is also an open setU of metrics such that forg∈U , the geodesic flow ofgis partially hyperbolic non-Anosov and with conjugate points.
Remark 1.4. A classical Mañé theorem [26]says that if, for a geodesic flow of a Riemannian manifold there is an invariant Lagrangian subbundle, then this Riemannian manifold does not have conjugate points. The existence of a partially hyperbolic non-Anosov geodesic flow implies that this theorem does not extend to invariant isotropic subbundles. Corollary C.1 states that some partially hyperbolic non-Anosov do have conjugate points.
Corollary C.2.There is aC2-open setV of Hamiltonians in the set of Hamiltonians of(T M, ωTM), near geodesic Hamiltonians, such that forh∈U, the Hamiltonian flow ofhis partially hyperbolic but not Anosov.
It is easy to construct partially hyperbolic Hamiltonians by suspensions; but they are not close to geodesic flows.
We also show that product metrics of Anosov geodesic flows are not examples with the partially hyperbolic prop- erty:
Theorem D.If(M1, g1)and(M2, g2)are Riemannian manifolds such that the geodesic flow of at least one of them is Anosov, then the geodesic flow of(M1×M2, g1+g2)is not partially hyperbolic.
For compact locally symmetric spaces of nonpositive curvature the following holds:
Theorem E.If the geodesic flow of a compact locally symmetric space of nonpositive curvature is partially hyperbolic, then its geodesic flow is Anosov.
The proof ofTheorem DandTheorem Eimply the following:
Theorem F.If(M, g)is a compact Riemannian manifold with nonpositive sectional curvature and partially hyper- bolic geodesic flow then(M, g)has rank one.
Moreover,
Theorem G. If(Mn, g)is a Riemannian manifold with partially hyperbolic geodesic flow then n is even, and if n≡2 mod 4, thendimEs=1orn−1.
Roughly speaking, the strategy of the proof ofTheorem Afollows these steps:
1. We chose a metric whose geodesic flow is Anosov and whose hyperbolic invariant splitting is of the form T (U M)=Ess⊕Es⊕ X ⊕Eu⊕Euu(Section4).
2. We take a closed geodesicγwithout self-intersections (Section5.2).
3. We change the metric in a tubular neighborhood ofγinM, such that alongγthe strong subbundles (EssandEuu) remain invariant and the weak subbundles disappear, becoming a central subbundle with no hyperbolic behavior (Section5.3).
4. Outside the tubular neighborhood ofγ, the dynamics remains hyperbolic.
5. We show that for the geodesics that intersect the tubular neighborhood the cones associated to the extremal subbundles (EssandEuu) are preserved (Sections5.3.3,5.3.4,5.3.5,5.3.6).
6. We prove that for vectors in the unstable cones there is expansion, and for vectors inside the stable cones there is contraction, under the action of the derivative of the new geodesic flow (Section5.3.7).
Remark 1.5.In the case of the geodesic flow of a compact locally symmetric Riemannian manifold(M, g)of non- constant negative curvature, the tangent bundle of the unitary tangent bundleU Msplits in many invariant subbundles:
T (U M)=Ess⊕Es⊕ X ⊕Eu⊕Euu. Ifθ∈U M,ζ ∈Ess(θ )then|dθζ|contracts exponentially fast ast→ ∞, faster than ifζ ∈Es. Ifζ ∈Euuthen|dθζ|expands exponentially fast ast→ ∞, faster than ifζ ∈Eu(see Section4).
We would like to recall that in the symplectic context, the existence of dominated splitting with two subbundles of equal dimension implies hyperbolicity. This was first observed by Newhouse for surfaces maps[27], later by Mañé in any dimension [25] for symplectic maps, by Ruggiero in the context of geodesic flows[30] and Contreras for symplectic and contact flows[12]. We want to point out that these results do not contradict ours: the splitting for the examples ofTheorem AandTheorem Bcontain more than two invariant subbundles.
There are partially hyperbolicΣ-geodesic flows, defined over a distributionΣT M which arise in the study of the dynamics of free particles in a system with constraints (see[11]). However, if the distribution is involutive then the leaves of the distribution have negative curvature, and we are again in the Anosov geodesic flows case. If the distribution is not involutive, theΣ-geodesic flow is not a geodesic flow.
The article is organized as follows:
In the second section of the article, we introduce basic results about geodesic flows, partial hyperbolicity and the equivalent property of the proper invariance of cone fields[28,21].
In the third section we proveTheorem D. We show that product metrics are not examples of partially hyperbolic non-Anosov geodesic flows.
In the fourth section we introduce properties and the classification of locally symmetric spaces of negative curvature which are the natural candidates to deform into partially hyperbolic non-Anosov geodesic flows.
In the fifth section we proveTheorem B, and soTheorem A, except for the transitivity of some of the examples.
We show that the deformed metric has a partial hyperbolic non-Anosov geodesic flow. We give a proof of the proper invariance of the strong cones, i.e., the derivative of the geodesic flow brings the strong unstable cone field properly inside itself, and the derivative of the inverse of the geodesic flow brings the stable cone field properly inside itself.
The proof is based on the calculation of the variation of the opening of the cones of an appropriate cone field, and then we prove the exponential expansion or contraction for vectors in the strong unstable and stable cones.
In the sixth section we proveTheorem E. We show that compact locally symmetric spaces of nonpositive sectional curvature are not partially hyperbolic. The geodesic flow of a locally symmetric spaces of nonconstant negative cur- vature is Anosov with at least four invariant subbundles in their dominated splitting, so they are the candidates for the deformation, since they have the property mentioned in the first item of the strategy above[14,15,22].
In the last section we proveTheorems F and Gand the transitivity of some of the examples, which is stated in Theorems A and B. We show some obstructions to the existence of a partially hyperbolic geodesic flow. In general, there are obstructions for the rank of the manifold if the Riemannian manifold has nonpositive sectional curvature, and for the dimension of the Riemannian manifold and the dimension of the hyperbolic invariant subbundles.
2. Preliminaries
In this section, we give some preliminary definitions. In the first subsection, the definitions are about geodesic flows. The basic reference for this subsection is the book by Paternain [28]. In the second subsection, we give the main definitions about partial hyperbolicity and the basic reference is the survey by Hasselblatt and Pesin[21].
2.1. Geodesic flows
A Riemannian manifold(M, g)is aC∞-manifold with a Euclidean inner productgxin eachTxMwhich varies smoothly with respect to x ∈ M. So a Riemannian metric is a smooth section g:M→ Symm+2(T M), where Symm+2(T M)is the set of positive definite bilinear and symmetric forms inT M. Along the article we will consider the topology of the space of metrics of a manifoldMto be theC2-topology on the space of these sections.
The geodesic flow of the metricgis the flow φt:T M→T M,
φt(x, v)=
γ(x,v)(t), γ(x,v)(t) ,
such that γ(x,v) is the geodesic for the metric g with initial conditions γ(x,v)(0)=x and γ(x,v)(0)=v, x ∈M, v∈TxM. Since the speed of the geodesics is constant, we can consider the flow restricted to U M := {(x, v)∈ T M: gx(v, v)=1}.
Definition 2.1.LetπM :T M→Mbe the canonical projection of the tangent bundle. The vertical subbundle,πV : V (T M)→T M, is the bundle whose fiber atθ∈TxMis given byV (θ )=ker(dθπM).
Definition 2.2. K:T (T M)→T M, which is called the connection map associated to the metricg, is defined as follows: givenξ ∈TθT Mletz:(−, )→T Mbe an adapted curve toξ; letα:(−, )→M:t→πM◦z(t ), andZ the vector field alongαsuch thatz(t )=(α(t), Z(t)); thenKθ(ξ ):=(∇αZ)(0).πH:H (T M)→T M, the horizontal subbundle, is given byH (θ ):=ker(Kθ).
Some properties ofHandV are:
1. H (θ )∩V (θ )=0,
2. dθπandKθ give identifications ofH (θ )andV (θ )withTxM, 3. TθT M=H (θ )⊕V (θ ).
The geodesic vector fieldG:T M→T (T M)is given byG(v)=(v,0)in the decompositionH⊕V ≈π∗T M⊕ π∗T M.
The decomposition in horizontal and vertical subbundles allows us to define the Sasaki metric onT M: gθ(ξ, η):=gx
dθπ(ξ ), dθπ(η) +gx
Kθ(ξ ), Kθ(η)
=gx(ξh, ηh)+gx(ξv, ηv)
forξ andη∈TθT M, withξ=(ξh, ξv)andη=(ηh, ηv)in the decompositionTθT M =H (θ )⊕V (θ ), withξh and ηh∈TxM∼=H (θ ),ξvandηv∈TxM∼=V (θ ).
It also allows us to define a symplectic 2-form and an almost complex structure JonT M and a contact form onU M:
Ωθ(ξ, η):=gx
dθπ(ξ ), Kθ(η)
−gx
Kθ(ξ ), dθπ(η)
=gx(ξh, ηv)−gx(ηh, ξv), J (ξ h, ξv):=(−ξv, ξh),
αθ(ξ ):=gθ
ξ, G(θ )
=gx
dθπ(ξ ), v
=gx(ξh, v).
Definition 2.3. The geodesic flow leaves U M invariant, and we can define a contact form on U M such that its Reeb vector field is the geodesic vector field:πS:S(U M)→U Mis the contact structure bundle onU M, with fiber S(θ ):=ker(αθ). It is an invariant subbundle for the geodesic flow, andR·G⊕S=T (U M). The vector bundle S(U M)also has a decomposition on horizontal and vertical subbundles.
In the next definition, we relate the derivative of the geodesic flow with the Jacobi fields of the metric that generates the flow.
Definition 2.4.Letγθbe a geodesic on the Riemannian manifold(M, g)with initial conditionsγθ(0)=x,γθ(0)=v, whereθ=(x, v),x∈M,v∈TxM. A Jacobi fieldζ (t)along a geodesicγθ is a vector field obtained by a variation of the geodesicγθthrough geodesics:
ζ (t):= ∂
∂s
s=0
π◦φt z(s)
,
wherez(0)=θ,z(0)=ξ andz(s)=(α(s), Z(s))(wherez, α, Zwere introduced inDefinition 2.2). It satisfies the following equation:
ζ +R γθ, ζ
γθ=0.
Its initial conditions are:
ζ (0)= ∂
∂s
s=0
π◦z(s)=dθπ ξ=ξh, ζ (0)= D
dt
∂
∂s
t=0,s=0
π◦φt z(s)
= D
∂s
∂
∂t
s=0,t=0
π◦φt z(s)
= D
∂s
s=0
Z(s)=Kθξ=ξv.
Observe that the choice ofξ determines a Jacobi fieldζξ along γθ and so the derivative of a geodesic flow is:
dθφt(ξ )=(ζξ(t), ζξ(t)).
We can restrict the action of the derivative of the geodesic flow to the contact structure: dθφt(ξ ):S(U M)→ S(U M). In this case the Jacobi fields associated with the contact structure are the orthogonal to the geodesics onM:
ζ :R→γθ∗T M such thatζ (0)=ξh,ζ (0)=ξv, whereξ∈TθT M,ξh⊥θ,ξv⊥θ.
Remark 2.5.We define the curvature tensorR:Γ (T M)×Γ (T M)×Γ (T M)→Γ (T M)as it is done in do Carmo’s book[10]:
R(X, Y )Z:= ∇Y∇XZ− ∇X∇YZ+ ∇[X,Y]Z.
2.2. Partial hyperbolicity
Definition 2.6.A partially hyperbolic flowφt:N→Nin the manifoldN generated by the vector fieldX:N→T N is a flow such that its quotient bundleT N/Xhas an invariant splittingT N/X =Es⊕Ec⊕Eusuch that these subbundles are non-trivial and have the following properties:
dxφt Es(x)
=Es φt(x)
, dxφt
Ec(x)
=Ec φt(x)
, dxφt
Eu(x)
=Eu φt(x)
, dxφt|EsCexp(t λ), dxφ−t|EuCexp(t λ),
Cexp(t μ)dxφt|EcCexp(−t μ),
for some constantsλ < μ <0< C and for allx∈N. In the context of the present paper,N is the unitarian tangent bundleU M.
Definition 2.7.A splittingE⊕F of the quotient bundleT N/Xis called a dominated splitting if:
dxφt E(x)
=E φt(x)
, dxφt
F (x)
=F φt(x)
,
dxφt|E(x) · dφt(x)φ−t|F (φt(x))< Cexp(−t λ) for some constantsC,λ >0 andx∈N.
We only need to prove the existence of dominated splitting because of a well-known result in symplectic dynamics – Theorem A of[12], Proposition 2.1 of[30]:
Lemma 2.8.Let(N, ω)be a symplectic manifold,ωits symplectic2-form andφt :N →N a flow onN such that LXω=0, i.e., it preserves the symplectic structure ofN. If there is a dominated splittingT N/X =E⊕Ec⊕F such thatdim(E)=dim(F ), then for allx∈N there is a positive real numberCand a negative real numberλsuch that
dxφt|ECexp(t λ), dxφ−t|FCexp(t λ).
2.2.1. Partial hyperbolicity and cone fields
There is a useful criterion for verifying partial hyperbolicity, called the cone criterion:
Givenx∈N, a subspaceE(x)⊂TxNand a numberδ, we define the cone atxcentered aroundE(x)with angleδ as
C
x, E(x), δ
=
v∈TxN:
v, E(x)
< δ ,
where (v, E(x))is the angle that the vectorv∈TxN makes with its own projection to the subspaceE(x)⊂TxN. Sometimes, the constantδinvolves in the definition of the coneC(x, E(x), δ)is called theopening of the cone.
A flow is partially hyperbolic if there areδ >0, some timeT >0, and two continuous cone familiesC(x, E1(x), δ) andC(x, E2(x), δ)such that:
dxφ−t
C
x, E1(x), δ C
x, E1
φ−t(x) , δ
, dxφt
C
x, E2(x), δ C
x, E2
φt(x) , δ
, dxφtξ1< Kexp(t λ),
dxφ−tξ2< Kexp(t λ),
forξ1∈C(x, E1(x), δ),ξ2∈C(x, E2(φt(x)), δ), some constantsK >0,λ <0 and allt >0.
2.2.2. Partial hyperbolicity and angle cone variation
LetPrE:T N→Ebe the orthogonal projection toE, whereπE:E→N is a vector subbundle ofT N. We define ΘE(v):=g(PrEv,PrEv)
g(v, v) . (1)
To prove that the proper invariance of cones holds, it is enough to check the following inequality:
d
dtΘE(φt(x))
dxφt(v)
>0 (2)
forv∈∂C(x, E(x), δ):= {w∈TxN: (w, E(x))=δ}.
Remark 2.9.The quantityΘE(v)equals twice the square of the cosine of the angle betweenvand the subspaceE, so if it increases along the flow, then the cone field is properly invariant. We call the derivative above theangle cone variation. This calculation is inspired by the calculations in[35], although we do not use quadratic forms here.
The proper invariance of the cones by the derivative of the geodesic flow implies the existence of a dominated splitting. For the exponential expansion or contraction in the unstable and stable directions, respectively, we only need to check exponential expansion or contraction inside the unstable and stable cones, respectively.
Lemma 2.10. For a fixed δ >0, and a fixed subbundle E→N, E(x)⊂TxN, if inequality (2) holds for v∈
∂C(x, E(x), δ), then the cone field is proper invariant for the geodesic flow.
Proof. Letc∈(1,2)be such that 2 cos2(δ)=c. Then C
x, E(x), δ
=
ΘE(v)c ,
∂C
x, E(x), δ
=
ΘE(v)=c .
Notice that the quantity on the left side of(2)is the same forvand forkvfor everyk >0. Then we can calculate for vsuch thatg(v, v)=1. Define
∂1C
x, E(x), δ
=
g(PrEv,PrEv)=c, g(v, v)=1 .
Then the set of vectors in the above defined boundary of the cones is compact, which implies that its derivative is bounded away from zero:
d
dtΘE(φt(x))
dxφt(v)
a >0.
Its immediate consequence is that the cone field is properly invariant by the flow. 2
3. The geodesic flow of a product metric is not partially hyperbolic
Now, we are going to show that some simple candidates for partially hyperbolic geodesic flows are not partially hyperbolic. In particular, we are going to prove that product metrics are neither Anosov nor partially hyperbolic.
A natural candidate for symplectic partially hyperbolic dynamics is the following: for any hyperbolic symplectic actionΦ:R→Sp(E, ω),π:E→Ba symplectic bundle withωas its symplectic 2-form, one can produce another symplectic action Φ∗:R→Sp(E, ω)⊕Sp(B×R2, ω0):t →Φ(t )⊕Id, where ω0=dx∧dy is the canonical symplectic form ofR2= {(x, y): x, y∈R}. The symplectic flow associated with this symplecticR-action is partially hyperbolic with a central direction of dimension 2. For geodesic flows the above described construction does not work.
Theorem 3.1.Let(M, g)be a Riemannian manifold whose geodesic flow is Anosov. Then, the product Riemannian manifold(M×Tn, g+g0)where(Tn, g0)isTnwith its canonical flat metric, is not partially hyperbolic.
Proof. Notice that for anyx∈M,{x} ×Tn is a totally geodesic submanifold of(M×Tn, g+g0). So, its second fundamental form is identically zero. Since the metric inTnis flat this implies that, for anyx∈Mand anyy∈Tn:
R
γ(x,y,0,v), (0, w)
γ(x,y,0,v)=0,
whereγ(x,y,0,v)is the geodesic of(M×Tn, g+g0)starting at(x, y)∈M×Tnwithγ(x,y,0,v)(0)=(0, v)∈TxM×Rn, the covariant derivative of the geodesic starting at(x, y)with tangent vector(0, v).
For a product metric in(M1×M2, g1+g2), let us sayRis the curvature tensor of the product Riemannian manifold with the product metric andR1the curvature tensor of the Riemannian manifoldM1. Then the following properties hold:
i. R(X, Y, Z, W )=R1(X, Y, Z, W ), forX, Y, Z, Wtangent toM1, because of Gauss’ equation and the fact that the second fundamental form is zero[10];
ii. R(X, Y, Z, N )=0, forX, Y, Ztangent toM1andN tangent toM2, because of Codazzi’s equation and the fact that the second fundamental form is zero[10];
iii. R(X, N, X,N ) =0, forX, Y tangent toM1andN,Ntangent toM2, becauseK(X, N )=0[10].
Then, for a submanifold{x} ×Tnwith the flat metric:
R
γ(x,y,0,v),·
γ(x,y,0,v)≡0.
So, the derivative of the geodesic flow along geodesics in {x} ×Tn does not have any exponential contraction or expansion. So, there is no partially hyperbolic splitting for its geodesic flow. 2
Theorem 3.2. Let (M1, g1) and (M2, g2) be two Riemannian manifolds whose geodesic flows are Anosov. The geodesic flow of the Riemannian manifold(M1×M2, g1+g2)is not Anosov.
Proof. The proof that this geodesic flow is not Anosov is easy. It is a classical result that (x0, γ(y,v)(t)) and (γ(x,u)(t), y0) are geodesics of the product metric, x0∈M1, y0∈M2, u∈TxM1, v ∈TyM2, γ(x,u)(0)=x and γ(x,u)(0)=u,γ(y,v)(0)=y andγ(y,v)(0)=v. So, we choosex0andx1∈M1close enough. Then(x0, γ(y,v)(t))and (x1, γ(y,v)(t))are two geodesics of the product metric with initial conditions(x0, y,0, v)and(x1, y,0, v). Letdistbe the distance function for the Sasaki metric ofU (M1×M2)anddist1be the distance function for the Sasaki metric of U M1. The geodesic flow is not expansive, becausedist (φt(x0, y,0, v), φt(x1, y,0, v))=dist1((x0,0), (x1,0))and therefore, for any >0, ifx0andx1are close enough, thendist1((x0,0), (x1,0)) < , and so the geodesic flow is not Anosov. 2
Theorem 3.3.The geodesic flow of the product metric of a product manifold of two Riemannian manifolds with Anosov geodesic flows is not partially hyperbolic.
Proof. Take local coordinates for the geodesic flow of the product metric. Letx∈M1,y∈M2,u∈TxM1,v∈TyM2, and letγ(x,y,u,v)(t )be the geodesic with initial conditionsγ(x,y,u,v)(0)=(x, y)andγ(x,y,u,v)(0)=(u, v). Since the product metric is a sum of the two metrics, we have thatπi:M1×M2→Mi,i=1,2, the natural projection from the product manifold toMi, is an isometric submersion. Soγ(x,y,u,v)(t )=(γ(x,u)(t ), γ(y,v)(t )).
Let us construct an orthonormal basis of parallel vector fields for γ(x,y,u,v)(t ). Suppose g1x(u, u)=1 and g2y(v, v)=1. So, to have(x, y, u, v)in the unitary tangent bundle ofM1×M2we take(x, y, αu, βv), and
g(x,y)
(αu, βv), (αu, βv)
=α2gx1(u, u)+β2gy2(v, v)=α2+β2=1.
Then
γ(x,y,αu,βv)(t )=
γ(x,αu)(t ), γ(y,βv)(t )
, γ(x,y,αu,βv)(t )=
αγ(x,u)(t ), βγ(y,v)(t ) .
LetEi,i=2, . . . ,dim(M1), be an orthogonal frame of parallel vector fields along the geodesicγ(x,u). LetFj,j=2, . . . ,dim(M2), be an orthogonal frame of parallel vector fields along the geodesicγ(y,v).
Notice that along the geodesicγ(x,y,αu,βv), since its components areγ(x,αu)andγ(y,βv), the following holds:
g1γ
(x,αu)(t )
γ(x,αu)(t ), γ(x,αu)(t )
=α2, gγ2
(y,βv)(t )
γ(y,βv)(t ), γ(y,βv)(t )
=β2, so the proportion(α, β)is preserved along the geodesic.
So{(αγ(x,u)(t ), βγ(y,v)(t )), (βγ(x,u)(t ),−αγ(y,v)(t )), (Ei(t ),0), (0, Fj(t ))}i,j is an orthonormal frame of parallel vector fields along the geodesicγ(x,y,αu,βv)(t ).
The fact that the second fundamental form of the submanifolds{p}×M2andM1×{q}is zero, together with Gauss and Codazzi equations, imply that:
R
(u1,0), (u2,0), (u3,0), (u4,0)
=R1(u1, u2, u3, u4), R
(0, v1), (0, v2), (0, v3), (0, v4)
=R2(v1, v2, v3, v4), R
(u1,0), (u2,0), (u3,0), (0, v1)
=0, R
(0, v1), (0, v2), (0, v3), (u1,0)
=0.
Also the fact that the curvature is zero for planes generated by one vector tangent toM1and another tangent toM2
implies:
R
(u1,0), (0, v1), (u2,0), (0, v2)
=0.
All these equations imply that along the geodesicγ(x,y,αu,βv)(t ):
R
γ(x,y,αu,βv), (Ei,0), γ(x,y,αu,βv), (Ek,0)
=α2R1
γ(x,u), Ei, γ(x,u), Ek
, R
γ(x,y,αu,βv), (0, Fj), γ(x,y,αu,βv), (0, Fl)
=β2R2
γ(y,v), Fj, γ(y,v), Fl , R
γ(x,y,αu,βv), (Ei,0), γ(x,y,αu,βv), (0, Fj)
=0.
Now, we are going to write a system of Jacobi fields. First we take an orthonormal basis of parallel fieldsUi:R→ Tγ (t )U (M1×M2)along a geodesicγ:R→U (M1×M2),i=1, . . . ,dim(M1)+dim(M2), such that{Ui(t )}is the basis
αγ(x,u)(t ), βγ(y,v)(t ) ,
βγ(x,u)(t ),−αγ(y,v)(t ) ,
Ei(t ),0 ,
0, Fj(t )
k,j,
such that j =2, . . . ,dim(M2),k=2, . . . ,dim(M1). We write the Jacobi field ζ :R→ with respect to this base:
ζ (t )= ni=2fi(t )Ui(t ). Thenζ (t )= ni=2fi (t )Ui(t )and 0=
n j=2
fj +
n i=2
fiR
γ , Ui, γ , Uj Uj. So, the Jacobi equation can be written as:
f f
=
0 I
−K 0 f f
whereKij=R(γ , Ui, γ , Uj).
In the case of the product metric we have:
f f
=
⎡
⎢⎢
⎣
0 0 I 0
0 0 0 I
−α2K1 0 0 0 0 −β2K2 0 0
⎤
⎥⎥
⎦ f
f
.
With a change in the order of the basis of parallel vector fields we have:
F =
⎡
⎢⎢
⎣
0 I 0 0
−α2K1 0 0 0
0 0 0 I
0 0 −β2K2 0
⎤
⎥⎥
⎦F.
So the systems decouples and the solutions are given immediately by the solutions forM1andM2.
Now suppose the geodesic flow of the product metric is partially hyperbolic with splitting Es ⊕Ec ⊕Eu, dimEs =p, dimEu=q. So the geodesic flow of each metricg1 andg2is partially hyperbolic and each geodesic flow inherits a partially hyperbolic splitting:
E1s⊕E1c⊕Eu1,
along geodesics inM1× {y}(β=0), such thatE1s⊕E1u⊂TxM1⊕ {0} ⊂TxM1⊕TyM2, and E2s⊕E2c⊕Eu2,
along geodesics in{x} ×M2(α=0), such thatE2s⊕E2u⊂ {0} ⊕TyM2⊂TxM1⊕TyM2.
For geodesics of the product metric which haveα=0=β, we get a splitting into five invariant subbundlesE1s⊕ E2s⊕Ec⊕E1u⊕E2u, without the domination, sinceαandβ multiply the Lyapunov exponents of each subbundle.
Since we already have a splitting,Es andEuare necessarily one of a combination of subbundles ofE1s andE2s,E1u andEu2, respectively:
Es∈
E⊕F: E⊂E1s, F⊂E2s, dimE+dimF =p , Eu∈
E⊕F: E⊂Eu1, F ⊂E2u, dimE+dimF =q .
So there is no way to go from the caseα=0 toβ=0 without breaking the continuity of the splitting, because one cannot go from the case dimE=0 forβ=0, to dimF=0 forα=0 in a continuous way. 2
4. Anosov geodesic flow with many invariant subbundles
In this section, we introduce the metric which we are going to deform to produce the example of a partially hyperbolic and non-Anosov geodesic flow.
The candidate for the deformation is a compact locally symmetric space which is a quotient of the symmetric space of nonconstant negative curvatureM:=G/Kby a cocompact latticeΓ, whereGis a Lie group that acts transitively onMandKis a compact subgroup ofG[6].
Cartan classified the symmetric spaces of negative curvature (see[19,20]). They are:
i. the hyperbolic spaceRHn of constant curvature−c2, which is the canonical space form of negative constant curvature;
ii. the hyperbolic space CHn of curvature−4c2K−c2, which is the canonical Kähler hyperbolic space of constant negative holomorphic curvature−4c2[17];
iii. the hyperbolic spaceHHnof curvature−4c2K−c2, which is the canonical quaternionic Kähler symmetric space of negative curvature[4,36];
iv. the hyperbolic spaceCaH2of curvature−4c2K−c2, which is the canonical hyperbolic symmetric space of the octonions of constant negative curvature.
Their geodesic flows are all Anosov, but the geodesic flow of the first one does not have more than the two invariant subbundles, the stable and the unstable, which cannot be decomposed in other subbundles. The others have more invariant subbundles, as in the first item of the strategy written in Section1. So, the metrics which are the candidates to produce a partially hyperbolic geodesic flow which is not Anosov are the metrics in items [ii.], [iii.] and [iv.].
Through the article we are going to considerc=12.
For these type of metrics we need the following properties to hold:
i. For allv∈TxM, the subspace{w∈TxM: K(v, w)= −1}is parallel alongγvin the sense that the derivative of the projection to this subspace ofTxMalong geodesics is zero.
ii. For closed geodesicsγ: [0, T] →M,(γ (0), γ (0))=(γ (T ), γ (T )), the parallel translation fromγ (0)toγ (T ) along γ of these subspaces {w∈TxM: K(v, w)= −1}and {w∈TxM: K(v, w)= −14}, where v=γ (0), preserves orientation.
The examples that satisfy the properties above are:
i. compact Kähler manifolds of negative holomorphic curvature−1 (see[17]),
ii. compact locally symmetric quaternionic Kähler manifolds of negative curvature (see[4]).
In the next subsection we describe a series of properties of the hyperbolic subbundles of the Anosov geodesic flows for the two cases listed above. More precisely, in Section4.1we describe the strong and weak stable and unstable subbundles, in Section4.1.1we study the cone variation (as defined in Section2.2.2) for the hyperbolic subbundles and in Section4.1.2we study the orientability of those subbundles.
4.1. Subspaces ofS(U M)andU M
Since the candidate has nonconstant negative curvature, then its sectional curvature, up to multiplication of the metric by a constant, has planes of sectional curvature −1 and planes of sectional curvature−14. Actually, every vectorv∈T Mis in a plane with curvature−1 and in another with curvature−14.
We define A(x, v):=
w∈TxM: K(v, w)= −1
, (3)
B(x, v):=
w∈TxM: K(v, w)= −1 4
. (4)
If we restrict the derivative of the geodesic flow to the subbundleS(U M)=kerα→U M, whereαis the contact form onU M, then S(x, v)=H (x, v) ⊕V (x, v), and H (x, v) andV (x, v) are identified with {v}⊥=A(x, v)⊕ B(x, v)⊂TxM,(x, v)∈U M. The subbundlesAandBare invariant by parallel translation along geodesics.
Lemma 4.1. The geodesic flow of the symmetric spaces of nonconstant negative curvature induces a hyperbolic splitting of the contact structure defined onU M:S(U M)/X =Ess⊕Es⊕Eu⊕Euu.
Proof. We can define the invariant subbundlesPCu(v), PCs(v)⊂T(x,v)U M,C=A, Bsuch that PCu(v)=
(w, αCw)∈S(x, v): w∈C(x, v) , PCs(v)=
(w,−αCw)∈S(x, v): w∈C(x, v) , whereαA=1 andαB=12.
That invariant subbundles are exactly the subbundles of the decomposition in the first item of the strategy stated in the introduction:
Euu(x, v)=PAu(x, v), Ess(x, v)=PAs(x, v), Es(x, v)=PBs(x, v), Eu(x, v)=PBu(x, v).
Above subbundles are invariants and the splitting is dominated: Jacobi fields inEuuandEsscontract for the past and the future, respectively, at ratee−t and Jacobi fields inEu andEs contract for the past and the future, respectively, at ratee−t /2. 2
4.1.1. Angle cone variation for the Anosov flow with many subbundles
Let us calculate the proper invariance of the cones in the case of the geodesic flow of the compact locally symmetric Riemannian manifold of nonconstant negative sectional curvature.
We use the following family of trajectories for the system:
q(t, u)=π◦φt z(u)
,
|u|< . We consider the geodesicv(t ):=φt(z(0))=γz(0)(t).
The Jacobi system along the geodesicγz(0)is given by(ξ(t), η(t)), where ξ(t )=dq
du(t,0), η(t)=Dv
du(t,0)= D du
dq dt(t,0).
So the following equations hold:
Dξ
dt =η, Dη
dt = −R(v, ξ )v. (5)
The quantity(1), which in this case is
ΘAu(ξ, η):=g(PrA(ξ+η),PrA(ξ+η)) g(ξ, ξ )+g(η, η) ,
indicates twice the square of the cosine of the angle between the vector (ξ, η)∈T(x,v)U M and its projection to PAu(x, v). The cone in this case is
C
v, PAu(x, v), c
=
(ξ, η)∈T(x,v)U M:ΘAu(ξ, η)=g(PrPAu(v)(ξ, η),PrPAu(v)(ξ, η)) g((ξ, η), (ξ, η)) c
,
wheregis the Sasaki metric andc=2 cos2δ, whereδis the angle between the vectors in the boundary of the cone and the subspacePAu(v). So, as explained in Section2.2.2, to prove that the cone fields are properly invariant is equivalent to prove that the cosine of this angle increases under the action of the derivative of the geodesic flow, for vector in the boundary of the cone fields,
∂C
v, PAu(x, v), c
=
(ξ, η)∈T(x,v)U M: ΘAu(ξ, η)=c∈(1,2) .
Remember that for any Riemannian manifold (M, g) and any u(t ), v(t )∈Tγ (t )M vector fields along a geodesic γ:R→MonM
d
dtg(u, v)=g Du
dt , v
+g
u,Dv dt
.
If(M, g)is locally symmetric, and ifξ(t )∈Tγ (t )Mis a vector field along a geodesicγ (t)onM, then D
dtPrAξ =PrA
D dtξ,
i.e. the subspaceA(γ (t))is parallel along the geodesicγ:R→Mof(M, g). Let us call, to simplify the equations, ξA:=PrAξ, ξB:=PrBξ,
ξA=PrA
D
dtξ=(ξA), ξB=PrB
D
dtξ=(ξB), and remember thatξ =ξA+ξB. Then, for
g(ξA+ηA, ξA+ηA)
g(ξ, ξ )+g(η, η) =ΘAu(ξ, η)=c∈(1,2),
