Directory UMM :Journals:Journal_of_mathematics:BJGA:

Ioan Bucataru, Matias F. Dahl

Dedicated to the 70-th anniversary of Professor Constantin Udriste

Abstract. From a spray space S on a manifold M we construct a new geometric spaceP of larger dimension with the following properties: (i) geodesics inP are in one-to-one correspondence with parallel Jacobi fields ofM; (ii)P is complete if and only ifS is complete; (iii) if two geodesics in P meet at one point, the geodesics coincide on their common domain, andP has no conjugate points; (iv) there exists a submersion that maps geodesics inP into geodesics onM.

The spaceP is constructed by first taking two complete lifts of spray S. This will give a spray Scc _{on the second iterated tangent bundle T T M.} Then spaceP is obtained by restricting tangent vectors of geodesics for

Scc _{onto a suitable (2 dimM + 2)-dimensional submanifold of T T T M.} Due to the last restriction, the space P is not a spray space. However, the construction shows that conjugate points can be removed if we add dimensions and relax assumptions on the geometric structure.

M.S.C. 2000: 53C22, 53C17.

Key words: spray space; geodesic spray; geodesic variation; complete lift; conjugate points; sub-spray.

1

Introduction

SupposeS is a spray on a manifold M. In this paper we show how to construct a new geometric spaceP that is based onS, but such thatP has no conjugate points. This is done in three steps:

(i) We start with a spraySon a manifoldM. For example,Scould be the geodesic spray for a Riemannian metric, a Finsler metric, or a non-linear connection [5, 7, 23, 24].

(ii) Next we take two complete lifts of S (see below). The first complete lift Sc gives a spray on T M whose geodesics are Jacobi fields on M. Similarly, the second complete lift gives a sprayScc_{onT T M whose geodesics can be described} as Jacobi fields for geodesics for Sc_{. That is, geodesics of S}c _{describe linear} deviation of nearby geodesics inM, and geodesics ofScc_{describe second order} deviation of nearby geodesics inM.

∗

Balkan Journal of Geometry and Its Applications, Vol.15, No.1, 2010, pp. 17-40. c

(iii) In the last step, we restrict tangent vectors of geodesics ofScc_{onto a submanifold} ∆ ⊂ T T T M that is invariant under the geodesic flow of Scc_{. By choosing} ∆ in a suitable way, we obtain a space P where geodesics are in one-to-one correspondence with parallel Jacobi fields inM.

In step (ii) the original sprayS is lifted twice using thecomplete lift. Essentially, the complete lift can be seen as a geometrization of the Jacobi equation. For example, if we start with a (pseudo-)Riemannian metricg onM, the complete lift ofggives a pseudo-Riemannian metricgc _{on T M whose geodesics are Jacobi fields onM. This} means that Jacobi fields onM can be treated as solutions to a geodesic equation on

T M, whence there is no need for a separate Jacobi equation. In this work we will use the complete lift of a spray. For affine sprays, this complete lift was introduced by A. Lewis [20]. In the Riemannian context, the complete lift is also known as theRiemann extension, and for a discussion about the complete lift in other contexts, see [6]. In step (ii), we need to study sprays on manifoldsM, T M, andT T M and also complete lifts of sprays onM and T M. To avoid studying all these cases separately we first study sprays and complete lifts on iterated tangent bundles of arbitrary order. This is the topic of Sections 2-6. One advantage of studying the Jacobi equation using the complete lift is that it simplifies certain aspects of Jacobi fields. For example, using the complete lift the Jacobi equation for a spray is essentially a direct consequence of the chain rule and the canonical involutions onT T M andT T T M (see remark after Proposition 6.3).

In step (iii) the phase space of sprayScc _{onT T M is restricted to a submanifold} ∆⊂T T T M. By choosing ∆ suitably, we define a geometryP where geodesics are in one-to-one correspondence with parallel Jacobi fields. The geometry of sprays that have been restricted in this way is described in Section 7. For previous work on sprays with restricted phase space, see [2, 18, 19]. The spaceP is constructed and discussed in Section 8. Here we show thatP has no conjugate points. We also show that the canonical submersionπ:T T M →M maps geodesics inP into geodesics inM. Hence the geometry ofP can be used to study dynamical properties ofM.

Let us emphasize that due to the restriction in step (iii), spaceP is not a spray space. It seems that to remove conjugate points, some relaxation of the underlying geometric structure is needed. For example, in Riemannian geometry the assumption that a manifold has no conjugate points can have strong implications.

(i) Suppose M is an n-torus with a Riemannian metric. Then the no-conjugate assumption implies thatM is flat [8, 14].

(ii) Suppose M is a Riemannian manifold such that M is complete, simply con-nected, dimM ≥3, andM is flat outside a compact set. Then the no-conjugate assumption implies thatM is isometric toRn [10].

See also [9, 11, 22]. If one relaxes the assumption on the geometric structure, then the no-conjugate assumption becomes weaker; on the 2-torus, there are non-flat affine con-nections without conjugate points [15], and on then-torus there are non-flat Finsler metrics without conjugate points [11].

tomography a typical assumption is that the manifold has no conjugate points. See [9, 25]. Another example is geometric optics, where conjugate points are problematic since they lead to caustics, where the amplitude becomes infinite.

2

Preliminaries

We assume thatM is a smooth manifold without boundary and with finite dimension

n≥1. By smooth we mean that M is a topological Hausdorff space with countable base that is locally homeomorphic toRn, and transition maps are C∞_{-smooth. All} objects are assumed to beC∞_{-smooth on their domains.}

By (T M, π0, M) we mean the tangent bundle of M. For r ≥ 1, let TrM =

T· · ·T M be the r:th iterated tangent bundle, and for r = 0 let T0_{M = M. For}

example, when r = 2 we obtain the second tangent bundle T T M [4, 13], and in generalTr+1_{M =T T}r_{M forr≥0.}

For a tangent bundle Tr+1_{M where r ≥ 0, we denote the canonical projection}

operator byπr: Tr+1M →TrM. Occasionally we also write πT T M→M,πT M→M, . . . instead of π0◦π1, π0, . . .. Unless otherwise specified, we always use canonical local

coordinates (induced by local coordinates onM) for iterated tangent bundles. Ifxi are local coordinates forTr_{M for somer ≥0, we denote induced local coordinates} forTr+1_M,Tr+2_{M, andT}r+3_{M by}

(x, y), (x, y, X, Y), (x, y, X, Y, u, v, U, V).

As above, we usually leave out indices for local coordinates and write (x, y) instead of (xi_{, y}i_).

Forr ≥1, we treat Tr_{M as a vector bundle over the manifold T}r−1_{M with the}

vector space structure induced by projectionπr−1:TrM →Tr−1M unless otherwise

specified. Thus, if{xi_{:i= 1, . . . ,2}r−1_{n}are local coordinates forT}r−1_{M, and (x, y)}

are local coordinates for Tr_{M, then vector addition and scalar multiplication are} given by

(x, y) + (x,y˜) = (x, y+ ˜y),

(2.1)

λ·(x, y) = (x, λy).

(2.2)

Ifx∈Tr_{M andr≥0 we define}

Txr+1M = {ξ∈T

r+1_{M :πr(ξ) =x}.}

Forr≥0, avector fieldon an open setU ⊂Tr_{M is a smooth mapX:U →T}r+1_M

such thatπr◦X = idU. The set of all vector fields onU is denoted byX(U). Suppose that γ is a smooth map γ: (−ε, ε)k → Tr_{M where k ≥ 1 and r ≥ 0.} Suppose also thatγ(t1, . . . , tk_{) = (x}i_(t1_{, . . . , t}k_{)) in local coordinates forT}r_{M. Then} thederivativeof γ with respect to variabletj _{is the curve∂}

tjγ: (−ε, ε) k

→Tr+1_M

defined by∂tjγ=

¡

xi_{, ∂x}i_/∂tj¢_{. Whenk= 1 we also writeγ}′ _=∂

tγ and say thatγ′ is thetangent ofγ.

between manifolds, we denote the tangent mapT M →T N byDφ, and ifc:I→M

is a curve, then

(φ◦c)′(t) =Dφ◦c′(t), t∈I.

(2.3)

2.1

Transformation rules in

T

r

M

Suppose that x = (xi_{) and ˜x = (˜x}i_{) are overlapping coordinates for T}r_{M where}

r ≥0. It follows that if ξ ∈ Tr+1_{M has local representations (x, y) and (˜x,y˜), we}

have transformation rules

˜

xi_{= ˜x}i_{(x), y˜}i₌ ∂x˜i

∂xa(x)y a_.

Now (x, y) and (˜x,y˜) are overlapping coordinates for Tr+1_{M. It follows that if ξ∈}

Tr+2_{M has local representations (x, y, X, Y) and (˜x,y,˜ X,˜ Y˜), we have transformation}

rules

˜

xi _{= x˜}i_(x),

˜

yi ₌ ∂˜x i

∂xa(x)y a_,

˜

Xi = ∂˜x i

∂xa(x)X a

,

˜

Yi ₌ ∂˜xi

∂xa(x)Y

a₊ ∂2x˜i

∂xa_∂xb(x)y a_Xb_.

3

Lifts on iterated tangent bundles

3.1

Canonical involution on

T

r

M

Whenr≥2 there are two canonical projections Tr_{M →T}r−1_{M given by}

πr−1:TrM →Tr−1M, Dπr−2:TrM →Tr−1M.

(3.1)

This means thatTr_{M contains two copies ofT}r−1_{M, and there are two ways to treat}

Tr_{M as a vector bundle overT}r−1_{M. Unless otherwise specified, we always assume}

thatTr_{M is vector bundle (T}r_{M, π}

r−1, Tr−1M), whence the vector structure ofTrM

is locally given by equations (2.1)-(2.2). However, there is also another vector bundle structure induced by projectionDπr−2:TrM →Tr−1M. If xi are local coordinates

forTr−2_{M and (x, y, X, Y) are local coordinates forT}r_{M, this structure is given by}

(x, y, X, Y) + (x,y, X,˜ Y˜) = (x, y+ ˜y, X, Y + ˜Y),

(3.2)

λ·(x, y, X, Y) = (x, λy, X, λY).

(3.3)

following diagram commutes.

Tr_M oo κr _//

Tr_M

Dπr−2

}

}

{{{{ {{{{

{{{{

Tr−!! 1_M

πr−1

DD DD

DD DD

DD_D

OnT T M, this involution map is well known [4, 13, 16, 21, 23].

Definition 3.1 (Canonical involution on Tr_{M). Forr≥2, thecanonical} invo-lutionκr:TrM →TrM is the unique diffeomorphism that satisfies

∂t∂sc(t, s) = κr◦∂s∂tc(t, s) (3.4)

for all mapsc: (−ε, ε)2_→Tr−2_{M. Forr= 1, we defineκ}

1= idT M.

Let r ≥ 2, let xi _{be local coordinates for T}r−2_{M, and let (x, y, X, Y) be local}

coordinates forTr_{M. Then}

κr(x, y, X, Y) = (x, X, y, Y).

For example, in local coordinates forT T M andT T T M we have

κ2(x, y, X, Y) = (x, X, y, Y),

κ3(x, y, X, Y, u, v, U, V) = (x, y, u, v, X, Y, U, V).

Forr≥1, we have identities

κ2r = idTr_M, (3.5)

πr◦Dκr = κr◦πr, (3.6)

Dπr−1 = πr◦κr+1,

(3.7)

πr−1◦Dπr−1 = πr−1◦πr, (3.8)

Dπr−1◦πr+1 = πr◦DDπr−1,

(3.9)

DDπr−1◦κr+2 = κr+1◦DDπr−1,

(3.10)

πr−1◦πr◦κr+1 = πr−1◦πr. (3.11)

Let us point out that the two projections in (3.1) are not the only projections from Tr+1_{M →T}r_{M. For example, whenr = 3, there are (at least) 6 projections}

T3_{M →T}2_M;π

2, κ2◦π2,Dπ1,κ2◦Dπ1,DDπ0, andκ2◦DDπ0.

Letγ0 be a curveγ0:I→Tr−1M for somer≥1, and let

X_{(γ0) = {η:I→T}r_{M :πr−1◦η=γ0}.}

Elements in X_{(γ0) are called vector fields along γ0, and} X_{(γ0) has a natural vector} space structure induced by the vector bundle structure ofTr_{M in equations} (2.1)-(2.2).

Ifη∈X_{(γ0) andC∈R, then}

κr+1◦(Cη)′ = C(κr+1◦η′),

(3.12)

and ifη1, η2∈X(γ0), then

κr+1◦(η1+η2)′ = κr+1◦η1′ +κr+1◦η2′.

(3.13)

3.2

Slashed tangent bundles

T

r

M

\ {

0

}

Theslashed tangent bundle is the open set inT M defined as

T M \ {0} = {y∈T M :y6= 0}.

For an iterated tangent bundleTr_{M wherer≥2 we define theslashed tangent bundle} as the open set

Tr_{M \ {0} = {ξ∈T}r_{M : (Dπ}

Tr−1M→M)(ξ)∈T M\ {0}}.

For example,

T T M\ {0} = {(x, y, X, Y)∈T T M :X 6= 0},

T T T M\ {0} = {(x, y, X, Y, u, v, U, V)∈T T T M :u6= 0},

where, say,T T M\ {0}=T2_{M\ {0}. Whenr= 0, let us also defineT}r_{M\ {0}=M,} and for any setA⊂Tr_{M wherer≥0, let}

A\ {0} = A∩TrM\ {0}.

Forr≥1 we have

κr+1(Tr+1M\ {0}) = T(TrM\ {0}),

(3.14)

(Dπr−1)(Tr+1M\ {0}) = TrM \ {0},

(3.15)

(Dκr)(Tr+1_{M\ {0}) = T}r+1_{M \ {0}.}

(3.16)

Before proving these equations, we define theLiouville vector fieldEr∈X(TrM). Forr≥1, it is given by

Er(ξ) = ∂s((1 +s)ξ)|s=0, ξ∈TrM.

If r ≥ 1, and (x, y) and (x, y, X, Y) are local coordinates for Tr_{M and T}r+1_M,

respectively, then

Er(x, y) = (x, y,0, y).

Equation (3.14) follows using equation (3.7) and by writing

πTrM→M = πTr−1M→M ◦πr−1, r≥1.

(3.17)

Ifr≥1, we have

ξ = Dπr−1◦κr+1◦Er(ξ), ξ∈TrM,

(3.18)

3.3

Lifts for functions

Supposef ∈C∞_{(M) is a smooth function. Then we can liftf using the vertical lift} or thecomplete liftand obtain functionsfv_{, f}c_∈C∞_{(T M) defined by}

fv_{(ξ) =f◦π}

0(ξ), fc(ξ) =df(ξ), ξ∈T M.

(3.19)

Heredf is the exterior derivative off. In local coordinates (x, y) forT M, it follows that

fv_{(x, y) =f(x), f}c_{(x, y) =} ∂f

∂xi(x)y i_.

Using these lifts one can define vertical and complete lift for tensor fields onM of arbitrary order. For a full development of these issues, see [26].

Next we generalize the vertical and complete lifts to functions defined on iterated tangent bundlesTr_{M of arbitrary orderr≥0.}

Definition 3.2. Forr≥0, thevertical liftof a functionf ∈C∞(Tr_{M \ {0}) is the} functionfv_∈C∞_(Tr+1_{M \ {0}) defined by}

fv(ξ) = f◦πr◦κr+1(ξ), ξ∈Tr+1M \ {0}.

Ifr= 0, Definition 3.2 reduces to liftfv _{in equation (3.19), and ifr≥1, equation} (3.7) implies thatfv_=f◦Dπ

r−1, and equation (3.15) implies thatfv is smooth. For

r≥1, letxi_{be local coordinates forT}r−1_{M, and let (x, y, X, Y) be local coordinates}

forTr+1_{M. Then}

fv(x, y, X, Y) = f(x, X), f ∈C∞(TrM\ {0}).

Definition 3.3. Forr≥0, thecomplete liftof a functionf ∈C∞_(Tr_{M\ {0}) is the} functionfc_∈C∞_(Tr+1_{M \ {0}) defined by}

fc(ξ) = (df)◦κr+1(ξ), ξ∈Tr+1M\ {0}.

If r = 0, then Definition 3.3 reduces to lift fc _{in equation (3.19), and if r ≥ 1,} then equation (3.14) implies thatfc _{is smooth. Forr≥1, letx}i _{be local coordinates} forTr−1_{M, and let (x, y, X, Y) be local coordinates forT}r+1_{M. Then}

fc_{(x, y, X, Y) =} ∂f

∂xa(x, X)y a₊ ∂f

∂ya(x, X)Y

a_{, f ∈C}∞_(Tr_{M\ {0}).}

Taking two complete lifts off ∈C∞_(Tr_{M \ {0}) yields}

fcc = fc(x, Y, u, V) (3.20)

+

µ

∂f ∂xa

¶c

(x, y, u, v)Xa+

µ

∂f ∂ya

¶c

(x, y, u, v)Ua,

where argument (x, y, X, Y, u, v, U, V)∈Tr+2_{M\ {0}has been suppressed.}

Iff ∈C∞_(Tr_{M\ {0}) for somer≥1, then}

fvv _{= f}vv_◦(Dκ r+1),

(3.21)

fvc _{= f}cv_◦(Dκ r+1),

(3.22)

fcc = fcc◦(Dκr+1).

(3.23)

4

Sprays

AsprayonM is a vector fieldSonT M\{0}that satisfies two conditions. Essentially, these conditions state that (i) an integral curve ofS is of the form c′_{: I → T M \}

{0} for a curve c: I → M, and (ii) integral curves of S are closed under affine reparametrizations t 7→ Ct+t0. Then curve c: I → M is a geodesic of S. The

motivation for studying sprays is that they provides a unified framework for studying geodesics for Riemannian metrics, Finsler metrics, and non-linear connections. See [5, 7, 23, 24]. Next we generalize the definition of a spray to iterated tangent bundles

Tr_{M for anyr≥0.}

4.1

Sprays on

T

r

M

Definition 4.1 (Spray space). SupposeS is a vector field S ∈ X_(Tr+1_{M \ {0})} wherer≥0. ThenS is a sprayonTr_{M if}

(i) (Dπr)(S) = idTr+1_M\{0},

(ii) [Er+1, S] =S for Liouville vector fieldEr+1∈X(Tr+1M).

LetS be a vector fieldS ∈X_(Tr+1_{M \ {0}) wherer ≥0. Then condition (i) in} Definition 4.1 states that if (x, y, X, Y) are local coordinates forTr+2_{M, then locally}

S(x, y) = ¡xi, yi, yi,−2Gi(x, y)¢ (4.1)

= yi ∂

∂xi

¯ ¯ ¯ ¯

(x,y)

−2Gi_{(x, y)} ∂

∂yi

¯ ¯ ¯ ¯

(x,y)

,

whereGi_{are locally defined functionsG}i_:Tr+1_{M\{0} →R. Condition (ii) in}

Defini-tion 4.1 states that funcDefini-tionsGi_{arepositively2-homogeneous; if (x, y)∈T}r+1_M\{0},

then

Gi_{(x, λy) = λ}2_Gi_{(x, y), λ >0.}

This is a consequence of Euler’s theorem for homogeneous functions [3].

Conversely, if S is a vector fieldS ∈ X_(Tr+1_{M\ {0}) that locally satisfies these} two conditions, thenS is a spray onTr_{M. FunctionsG}i _{in equation (4.1) are called} spray coefficientsforS.

Whenr= 0, Definition 4.1 is equivalent to the usual definition of a spray [7, 24]. However, when r ≥ 1, Definition 4.1 makes a slightly stronger assumption on the smoothness ofS. Namely, ifr≥1 andSis a spray onTr_{M (in the sense of Definition} 4.1) thenS is smooth onTr+1_{M \ {0}, but ifS is a spray on manifoldT}r_{M (in the} usual sense) thenSis smooth onT(Tr_{M)\ {0}. SinceT(T}r_{M)\ {0} ⊃T}r+1_{M\ {0},}

it follows that ifS is a spray onTr_{M (in the sense of Definition 4.1), then S is also} a spray on manifoldTr_{M (in the usual sense). The stronger assumption onS will} be needed in Section 5 to prove that the complete lift of a spray onTr_{M is a spray} onTr+1_{M. In this work we only consider sprays onT}r_{M that arise from complete} lifts of a spray on M. Therefore we do not distinguish between the weaker and stronger definitions of a spray. These comments motivate the slightly non-standard terminology in Definition 4.1.

Proposition 4.2. IfS is a spray onTr+1_{M wherer≥0, then}

S∗ = (DDπr)◦S◦κr+2◦Er+1

is a spray onTr_M.

Proof. Equations (3.7), (3.9), and (3.17) imply that maps

κr+2◦Er+1:Tr+1M\ {0} → Tr+2M\ {0},

DDπr: T(Tr+2M\ {0}) → T(Tr+1M \ {0}),

are smooth, so S∗_:Tr+1_{M \ {0} → T(T}r+1_{M \ {0}) is a smooth map. Let (x, y)}

be local coordinates forTr+1_{M, and let (x, y, X, Y) be local coordinates forT}r+2_M.

ThenS can be written as

S(x, y, X, Y) = ¡x, y, X, Y, X, Y,−2Gi_{(x, y, X, Y),−2H}i_{(x, y, X, Y)}¢

for locally defined functionsGi_{, H}i_:Tr+2_{M\{0} →Rthat are positively 2-homogeneous}

with respect to (X, Y). It follows that

S∗(x, y) = ¡x, y, y,−2Gi_{(x,0, y, y)}¢_,

whence S∗ _{is a vector field S}∗ _{∈ X(T}r+1_{M \ {0}), and (Dπr)(S}∗_{) = idT}

r+1_M\{0}. Since functions (x, y)7→Gi_{(x,0, y, y) are positively 2-homogeneous,S}∗ _{is a spray. ¤}

4.2

Geodesics on

T

r

M

Suppose γ is a curve γ: I → Tr_{M where r ≥ 0. Then we say that γ is regular} if γ′_{(t) ∈ T}r+1_{M \ {0} for all t ∈ I. When r = 0, this coincides with the usual}

definition of a regular curve, and whenr≥1, curveγ is regular if and only if curve

πTr_M→M ◦γ:I→M is regular.

Definition 4.3 (Geodesic). SupposeS is a spray on Tr_{M where r ≥0. Then a} regular curveγ: I→Tr_{M is a geodesicofS if and only if}

γ′′ = S◦γ′.

SupposeS is a spray on Tr_{M and locally S is given by equation (4.1). Then a} regular curveγ: I→Tr_{M,γ= (x}i_{), is a geodesic ofS if and only if}

¨

xi = −2Gi◦γ′.

(4.2)

In Definition 4.3 we have defined geodesics on open intervals. Ifγ is a curve on a closed interval we say thatγis a geodesic ifγcan be extended into a geodesic defined on an open interval.

5

Complete lifts for a spray

characteristic feature of spray Sc _{is that its geodesics are essentially in one-to-one} correspondence with Jacobi fields of S. This correspondence will be the topic of Section 6. In this section, we define the complete lift for a spray on an iterated tangent bundleTr_{M of arbitrary orderr≥0. This makes it possible to take iterated complete} lifts; if we start with a sprayS onM we can take iterated liftsSc_{, S}cc_{, S}ccc_{, . . .and} liftS onto an arbitrary iterated tangent bundle.

The definition below for the complete lift of a spray can essentially be found in [20, Remark 5.3]. For a further discussion about related lifts, see [6].

Definition 5.1 (Complete lift of spray). SupposeS is a spray onTr_{M for some}

r≥0. Then thecomplete liftofS is the spraySc _onTr+1_{M defined by}

Sc _{= Dκ}

r+2◦κr+3◦DS◦κr+2,

whereDS is the tangent map ofS,

DS:T(Tr+1M \ {0}) → T2(Tr+1M\ {0}).

Let us first note that equations (3.6), (3.7), (3.11), and (3.17) imply that

Dκr+2◦κr+3:T2(Tr+1M\ {0}) → T(Tr+2M \ {0})

is a smooth map. ThusSc _{is a smooth map T}r+2_{M \ {0} →T(T}r+2_{M \ {0}), and}

by equations (3.6), (3.7), and (3.14),Sc _{is a vector fieldS}c _∈X(Tr+2_{M\ {0}). IfS}

is the spray in equation (4.1), then locally

Sc ₌ ³_{x, y, X, Y, X, Y,−2(G}i₎v_,−2¡_Gi¢c´ (5.1)

= Xi ∂

∂xi +Y i ∂

∂yi −2(G i₎v ∂

∂Xi −2(G i₎c ∂

∂Yi,

andSc _{is a spray onT}r+1_M.

Suppose thatSis a spray onTr_{M for somer≥0, and suppose thatγis a regular} curveγ:I→Tr+1_{M, γ= (x, y). Thenγis a geodesic ofS}c _{if and only if}

¨

xi = −2Gi◦(πr◦γ)′, (5.2)

¨

yi _{= −2(G}i₎c_◦γ′_. (5.3)

It follows thatπr◦γ= (xi) is a geodesic ofS. In fact, ifS∗is the spray in Proposition 4.2, then

S = (Sc)∗.

(5.4)

Thus a spray can always be recovered from its complete lift. What is more, ifS is a spray onTr+1_{M forr≥0, thenS}∗c_{=Sif and only ifS=A}c _{for a sprayAonT}r_M.

Thegeodesic flowof a spray S is defined as the flow ofS as a vector field. Proposition 5.2 (Geodesic flow for the complete lift of a spray). SupposeS

is a spray onTr_{M wherer≥0andS}c _{is the complete lift ofS. Suppose furthermore} that

are the geodesic flows of spraysS andSc_{, respectively, with maximal domains}

D(S)⊂Tr+1_{M \ {0} ×}

R, D(Sc_)⊂Tr+2_{M\ {0} ×}

R.

Then

((Dπr)×idR)D(Sc) = D(S),

(5.5)

and

φc

t(ξ) = κr+2◦Dφt◦κr+2(ξ), (ξ, t)∈ D(Sc),

(5.6)

whereDφt is the tangent map of the mapξ7→φt(ξ)wheret is fixed.

Proof. To prove inclusion “⊂” in equation (5.5), let (ξ, t)∈ D(Sc_{), and letγ:I →}

Tr+2_{M \ {0} be an integral curve ofS}c _{such thatγ(0) =ξandt∈I. Then}

DDπr◦Sc = S◦Dπr,

so Dπr◦γ: I → Tr+1M \ {0} is an integral curve of S, and ((Dπr)(ξ), t) ∈ D(S). The other inclusion follows similarly sinceγ′ _{is an integral curve of S}c _{whenγ is an} integral curve ofS.

Supposev is a curve v: (−ε, ε)→Tr+1_{M \ {0}, and suppose that curve ξ:I×}

(−ε, ε)→Tr+2_{M \ {0},}

ξ(t, s) = κr+2◦∂s(φt◦v(s)) (5.7)

is defined for some intervalI andε >0. For (t, s)∈I×(−ε, ε) we then have

Sc_{◦ξ(t, s) = Dκ}

r+2◦κr+3◦DS◦∂s(φt◦v(s)) = Dκr+2◦κr+3◦∂s(S◦φt◦v(s)) = Dκr+2◦κr+3◦∂s∂t(φt◦v(s)) = Dκr+2◦∂t∂s(φt◦v(s)) = Dκr+2◦∂t(κr+2◦ξ(t, s))

= ∂tξ(t, s).

To prove equation (5.6), let (ξ0, t0)∈ D(Sc). Letj(t) =φct(ξ0) be the integral curve

j: I∗ _{→ T}r+2_{M \ {0} of S}c _{with maximal domain I}∗ _{⊂ R. Then t}

0 ∈ I∗. For a

compact subsetK⊂I∗_{with 0∈K we show that}

j(t) =κr+2◦Dφt◦κr+2(ξ0), t∈K

(5.8)

whence equation (5.6) follows. Sinceξ0 ∈ Tr+2M \ {0}, it follows that κr+2(ξ0) =

∂sv(s)|s=0 for a curve v: (−ε, ε) →Tr+1M \ {0}. Suppose τ ∈ K. Then (ξ0, τ)∈

D(Sc_{), and by equation (5.5), (v(0), τ)∈ D(S). Since D(S) is open [1], there is an} open intervalI∋τ and an ε >0 such that curveξ(t, s) in equation (5.7) is defined onI×(−ε, ε). Then, asKis compact, we can shrinkεand assume thatK⊂I. Now equation (5.8) follows sinceξ(t,0) =κr+2◦Dφt◦κr+2(ξ0),Sc◦ξ(t,0) =∂tξ(t,0) for

6

Jacobi fields for a spray

Definition 6.1 (Jacobi field). Suppose S is a spray on Tr_{M where r ≥ 0, and} suppose that γ: I → Tr_{M is a geodesic of S. Then a curve J:I → T}r+1_{M is a}

Jacobi field alongγ if (i) J is a geodesic ofSc_,

(ii) πr◦J =γ.

In Proposition 6.3 we will show that Definition 6.1 is equivalent with the usual characterization of a Jacobi field in terms of geodesic variations. In view of Proposition 5.2, this should not be surprising. For example, in Riemannian geometry it is well known that Jacobi fields are closely related to the tangent map of the exponential map.

Definition 6.2 (Geodesic variation). SupposeS is a spray onTr_{M wherer≥0,} and suppose thatγ:I→Tr_{M is a geodesic of S. Then ageodesic variationofγis a} smooth mapV:I×(−ε, ε)→Tr_{M such that}

(i) V(t,0) =γ(t) for allt∈I,

(ii) t7→V(t, s) is a geodesic for alls∈(−ε, ε).

Suppose that I is a closed interval. Then we say that a curve J: I → Tr_{M is} a Jacobi field if we can extend J into a Jacobi field defined on an open interval. Similarly, a mapV:I×(−ε, ε)→Tr_{M is a geodesic variation if there is a geodesic} variationV∗_:I∗_×(−ε∗_{, ε}∗_)→Tr_{M such thatV =V}∗ _{on the common domain ofV} andV∗_{and I⊂I}∗_.

The next proposition motivates the above non-standard definition for a Jacobi field using the complete lift of a spray.

Proposition 6.3 (Jacobi fields and geodesic variations). Let S be a spray on

Tr_{M wherer≥0, letJ: I→T}r+1_{M be a curve, whereI is open or closed, and let}

γ:I→Tr_{M be the curveγ=π} r◦J. (i) IfJ can be written as

J(t) = ∂sV(t, s)|_s=0, t∈I (6.1)

for a geodesic variation V:I×(−ε, ε)→Tr_{M, then J is a Jacobi field along}

γ.

(ii) If J is a Jacobi field along γ and I is compact, then there exists a geodesic variationV:I×(−ε, ε)→Tr_{M such that equation (6.1)holds.}

Proof. For (i), let us first assume that Iis open. Fort∈I we then have

Sc◦∂tJ(t) = Dκr+2◦κr+3◦DS◦κr+2◦∂t∂sV(t, s)|s=0

= Dκr+2◦κr+3◦∂s(S◦∂tV(t, s))|s=0

= Dκr+2◦κr+3◦∂s∂t∂tV(t, s)|s=0

= Dκr+2◦∂t∂s∂tV(t, s)|s=0

= ∂t∂t∂sV(t, s)|s=0

IfI is closed, we can extendV and J so that I is open and the result follows from the case whenI is open.

For (ii), we have J′_{(0) ∈ T}r+2_{M \ {0}, so we can find a curve w: (−ε, ε) →}

Tr+1_{M \ {0} such that κ}

r+2(J′(0)) = ∂sw(s)|s=0. Then w(0) = γ′(0). Since I is

compact and D(S) is open, we can extend I into an open interval I∗ _{and find an}

ε >0 such thatV(t, s) =πr◦φt◦w(s) is a mapV: I∗×(−ε, ε)→TrM. We have

V(t,0) =γ(t) fort∈I, and for eachs∈(−ε, ε), the mapt7→V(t, s) is a geodesic of

S. Proposition 5.2 and equations (3.7) and (2.3) imply that fort∈I,

J(t) = πr+1◦φct◦J ′₍₀₎

= πr+1◦κr+2◦Dφt◦∂sw(s)|s=0

= ∂s(πr◦φt◦w(s))|s=0.

We have shown thatV is a geodesic variation for Jacobi fieldJ. ¤

Remark 6.4. Supposec: I→M is a geodesic for a Riemannian metric, whereI is compact. Then one can characterize Jacobi fields alongcusing geodesic variations as in Proposition 6.3 [12]. Using the complete lift, we can therefore write the traditional Jacobi equation in Riemannian geometry asJ′′ _{= S}c_◦J′_{. It is interesting to note} that the derivation of the latter equation only uses the definition ofSc_{, the geodesic} equation for S, the commutation rule (3.4) for κr, and the chain rule in equation (2.3). In particular, there is no need for local coordinates, covariant derivatives, nor curvature. For comparison, see the derivations of the Jacobi equations in Riemannian geometry [23], in Finsler geometry [3], and in spray spaces [24]. All of these derivations are considerably more involved than the proof of Proposition 6.3 (i). For semi-sprays, see also [6] and [7].

6.1

Geodesics of

S

cc

LetS be a spray on Tr_{M for some r ≥ 0. We know that a regular curve γ: I →}

Tr+1_{M, γ= (x, y), is a geodesic ofS}c _{if and only ifγlocally solves equations} (5.2)-(5.3). Let us next derive corresponding geodesic equations for sprayScc_.

LetSbe given by equation (4.1) in local coordinates (x, y) forTr+1_{M. Then the}

complete lift ofSc _{is the sprayS}cc _onTr+2_{M given by}

Scc _{= (x, y, X, Y, u, v, U, V, u, v, U, V,}

−2(Gi₎vv_,−2(Gi₎cv_,−2(Gi₎vc_,−2(Gi₎cc¢_.

SupposeJ is a regular curveJ:I →Tr+2_{M, J = (x, y, X, Y). By equations (3.20)}

and (3.22),J is a geodesic ofScc_{if and only if}

¨

xi _{= −2G}i_◦c′_, ¨

yi = −2(Gi)c◦J1′,

¨

Xi _{= −2(G}i₎c_◦J′

2,

¨

Yi = −2(Gi)cc◦J′

= −2(Gi₎c_(xi_{, Y}i_,x˙i_,Y˙i₎

−2

µµ

∂Gi

∂xa

¶c

(J1′)Xa+

µ

∂Gi

∂ya

¶c (J1′) ˙Xa

¶

where curvesc:I→Tr_M,J

1:I→Tr+1M, andJ2:I→Tr+1M are given by

c=πTr+2_M→TrM◦J, J1=πr+1◦J, J2= (Dπr)(J),

and in local coordinatesc= (xi_{), J}

1= (xi, yi), andJ2= (xi, Xi).

We have shown that ifJis a geodesic of sprayScc_{, thenJcontains two independent} Jacobi fields J1 and J2 along c. The interpretation of this is seen by writing J =

(x, y, X, Y) using a geodesic variation. Then J1 = (x, y) is the base geodesic of

Sc_{, and J}

2 = (x, X) describes the variation of geodesic c:I → M. A geometric

interpretation of componentsYi _{seems to be more complicated. For example, (x, Y)} does not define a vector field alongc. However, for fixed local coordinates,Yi_describe the variation of the vector components of Jacobi fieldJ1 = (x, y). If J2 = 0, that

is, the variation does not vary the base geodesic c, then equations for Yi _simplify and (xi_{, Y}i_{) is a Jacobi field. In this case, curve (x}i_{, Y}i_{) is also independent of local} coordinates (see transformation rules in Section 2.1).

6.2

Iterated complete lifts

LetS0 _{be a spray onM. For r≥1, let S}r _{be the rth iterated complete lift ofS}0_,

that is, forr≥1, let

Sr _{= (S}r−1₎c_.

ThenS0_{, S}1_{, S}2_{, . . .are sprays onM, T M, T T M, . . ., and in general,S}r _{is a spray on}

Tr_M.

Equation (5.4) shows that eachSr _{contains all geometry of the original sprayS}0_.

A more precise description is given by equation (5.1). It shows that spraysS1_{, S}2_{, . . .}

also contain new geometry obtained from derivatives of spray coefficientsGi _ofS0_.

Namely, the rth complete lift Sr _{depends on derivatives of G}i _{to order r. This} phenomena can also be seen from the geodesic flows of higher order lifts. Ifφis the flow ofS0_{, then up to a permutation of coordinates, the flow ofS}1_{isDφ, the flow of}

S2_{isDDφ, and, in general, the flow ofS}r_{is therth iterated tangent mapD· · ·Dφ.} This means that the flow of S1 _{describes the linear deviation of nearby geodesic of}

S. That is, the flow ofS1 _{describes the evolution of Jacobi fields. Similarly, flows of}

higher order lifts describe higher order derivatives of geodesic deviations.

Proposition 6.5 (New Jacobi fields from old ones). SupposeS0_{, S}1_{, S}2_{, . . .are}

defined as above, and suppose thatj:I→Tr_{M is a geodesic for someS}r_. (i) Ifr≥0,t0∈R, andC >0, thenj(Ct+t0)is a geodesic ofSr.

(ii) If r ≥ 1 and k: I → Tr_{M is another geodesic of S}r _{such that π}

r−1◦j(t) =

πr−1◦k(t), then

αj+βk, α, β∈R is a geodesic ofSr_.

(v) If r≥2, then(Dπr−2)(j) :I→Tr−1M is a geodesic ofSr−1.

(vi) Ifr≥0, thenj′_:I→Tr+1_{M is a geodesic ofS}r+1_.

(vii) If r≥0, thentj′_{(t) :I→T}r+1_{M is a geodesic ofS}r+1_.

(viii) Ifr≥1, thenEr◦j:I→Tr+1M is a geodesic ofSr+1.

Proof. Properties (i), (ii), and (iv) follow using equations (4.2), (5.2), and (5.3). Pro-perties (vi), (vii), and (viii) follow by locally studying geodesic variations

V(t, s) = j(t+s), V(t, s) = j((1 +s)t), V(t, s) = (1 +s)j(t),

and using Proposition 6.3 (i). Property (iii) follows using geodesic equations forScc in Section 6.1 and equation (3.23). Property (v) follows using equation (3.7). ¤

6.3

Conjugate points

SupposeS is a spray onTr_{M for somer≥0. Ifa, bare distinct points inT}r_{M that} can be connected by a geodesicγ: [0, L]→Tr_{M, thenaandbareconjugate pointsif} there is a Jacobi fieldJ: [0, L]→Tr+1_{M alongγ that vanishes ataand b, butJ is}

not identically zero (with respect to vector space structure in equations (2.1)-(2.2)). The next proposition shows that S has conjugate points if and only if Sc _has conjugate points. Thus the complete lift alone does not remove conjugate points.

Proposition 6.6 (Conjugate points and complete lift). SupposeS is a spray onTr_{M for somer≥0.}

(i) If a, b ∈ Tr_{M are conjugate points for S, then zero vectors in T}r+1

a M and

Tbr+1M are conjugate points for S c_.

(ii) If a, b ∈ Tr_{M are conjugate points for S, then there are non-zero conjugate} points inTr+1

a M andT r+1

b M forS c_.

(iii) If a, b ∈ Tr+1_{M are conjugate points for S}c_{, then πr(a), πr(b) are conjugate} points for S.

Proof. For property (i), supposeJ: [0, L]→Tr+1_{M is a Jacobi field ofS that shows}

that a andb are conjugate points. Then the claim follows by studying Jacobi field

Er+1 ◦J. For property (ii), suppose that J: [0, L] → Tr+1M is as in (i), and let

γ: [0, L]→Tr_{M be the geodesic γ=π}

r◦J forS. We will show that γ′(0), γ′(L)∈

Tr+1_{M \ {0} are conjugate points for S}c_{. This follows by considering Jacobi field}

j: [0, L]→Tr+2_M,

j(t) = ∂s(γ′(t) +sJ(t))|s=0.

For property (iii), suppose J: [0, L] → Tr+2_{M is a Jacobi field of S}c _{that shows} thataandb are conjugate points. ThenJ is a geodesic ofScc_{, and locallyJ satisfy} equations in Section 6.1. If Jacobi fieldj = (Dπr)(J) does not vanish identically, the claim follows. Otherwise (Dπr)(J) vanishes identically, and the result follows by the

7

Sprays restricted to a semi-distribution

From a spraySonMone can construct a new geometric space by restricting the spray to a geodesically invariant distribution ∆⊂T M. This is done by requiring that all geodesics are tangent to the distribution. For example, geodesics in Euclidean space R3 can in this way be constrained toxy-planes. See [2, 18, 19].

In this section we study a slightly more general geometry, where one can not only restrict possible directions, but also basepoints for geodesics. For example, geodesics inR3 can in this way be constrained to one line or one plane. For a spray on Tr_M, this is done by requiring that geodesics are tangent to a suitable geodesically invariant submanifold ∆⊂Tr+1_{M. Such a submanifold will be called asemi-distributionand}

the restricted geometry will be called a sub-spray. There does not seem to be any work on this type of geometry. The terms semi-distribution and sub-spray neither seem to have been used before.

Definition 7.1. A set ∆⊂Tr+1_{M wherer≥0 is asemi-distribution onT}r_{M if}

(i) πr(∆) is a submanifold inTr_M.

(ii) B=πr◦κr+1(∆) is a submanifold inTrM.

(iii) There is a k≥1 such that every b∈B has an open neighborhoodU ⊂B, and there arekmapsV1, . . . , Vk:U →Tr+1M such that

(a) πr◦Vi=ιfori= 1, . . . , k, whereιis inclusion U ֒→TrM, (b) Vi are pointwise linearly independent,

(c) for allu∈U we have

κr+1(∆)∩π−r1(u) = span{V1(u), . . . , Vk(u)}.

(In (b) and (c), the linear structure of Tr+1_{M is with respect to equations}

(2.1)-(2.2).)

We say thatk is therank of ∆ and write rank ∆ =k.

In condition (ii),B=π0(∆) whenr= 0, andB= (Dπr−1)(∆) whenr≥1. Thus,

ifr= 0 andπ0(∆) =M, a semi-distribution is a distribution in the usual sense.

Condition (iii) states that there is akdimensional vector space associated to each

b∈B, and 1≤k≤2r_{dimM. Whenr= 0, the structure of these vector spaces in ∆} is given by equations (2.1)-(2.2), and whenr≥1, the structure is given by equations (3.2)-(3.3). The next example motivates the use of vector space structure in equations (3.2)-(3.3) whenr ≥1. Namely, these equations describe the natural vector space structure for tangents to Jacobi fields.

fields alongγ, such that locallyγ= (x), J1= (x, y),andJ2= (x, z). Forα, β∈Rwe

then have

αJ1+βJ2 = (x, αy+βz),

(αJ1+βJ2)′ = (x, αy+βz,x, α˙ y˙+βz˙)

= α·J1′ +β·J2′,

where on the last line, + and · are as in equations (3.2)-(3.3). Thus, if we define the vector space structure for Jacobi fields by equations (2.1)-(2.2), then the natural vector structure for tangents (and initial values) is given by equations (3.2)-(3.3). On the other hand, the multiplication operator in equation (2.2) appears naturally when reparametrizing a curve. IfJ: I→Tr_{M is a curve for r≥0, andj(t) =J(Ct+t}

0),

thenj′_{(t) =C·J}′_(Ct+t

0), where multiplication ·is as in equation (2.2).

Proposition 7.3. Suppose∆ is a semi-distribution onTr_{M andB=π}

r◦κr+1(∆).

Then∆ is a sub-manifold in Tr+1_{M and}

dim ∆ = dimB+ rank ∆.

The proof of Proposition 7.3 follows by settingA=κr+1(∆) in the lemma below.

We also use this lemma to prove Proposition 8.3.

Lemma 7.4. SupposeAis a subsetA⊂Tr+1_{M for somer≥0 such that}

(i) πr(A)is a submanifold inTr_M.

(ii) There is ak≥1such that everyb∈πr(A)has an open neighborhoodU ⊂πr(A), and there are kmapsV1, . . . , Vk: U →Tr+1M such that

(a) πr◦Vi=ιfori= 1, . . . , k, whereι is inclusion U ֒→TrM, (b) V1, . . . , Vk are pointwise linearly independent inU,

(c) for all u∈U we have

A∩πr−1(u) = span{V1(u), . . . , Vk(u)}.

(In (b) and (c), the linear structure ofTr+1_{M is with respect to equations}

(2.1)-(2.2).)

ThenAis a submanifold ofTr+1_{M of dimensiondimπr(A) +k. Moreover, if we can}

assume thatU =πr(A), thenA is diffeomorphic toπr(A)×Rk_.

Proof. Let ξ ∈ A. Then πr(ξ) has an open neighborhood U ⊂ πr(A) with k maps

V1, . . . , Vk:U →Tr+1M such that (a), (b), and (c) hold. By possibly shrinkingU we can find maps Vk+1, . . . , VN:U → Tr+1M, where N = dimT_πr+1

r(ξ)M, such that

πr◦Vi=ι fori= 1, . . . , N, and

π−1

r (u) = span{V1(u), . . . , VN(u)}. Letf be the diffeomorphismf: U×RN →π−1

r (U) defined as

Letg: U×Rk _→π−1

r (U) be the restriction of f onto U×Rk. Theng is a smooth injection and immersion such thatg(U×Rk_{) =A∩π}−1

r (U), and mapf−1◦g:U×Rk→

U×RN _{is the inclusion (u, α}

1, . . . , αk)7→(u, α1, . . . , αk,0, . . . ,0). Since a closed set in a compact Hausdorff space is compact, f−1_{◦g is proper. Thusg is proper, and}

the claim follows from the following result: Iff:M →N is a smooth map between manifolds that is proper, injective, and an immersion, then f(M) is a submanifold inN of dimension dimM, andf restricts to a diffeomorphismf:M →F(M). See

results 7.4, 8.3, and 8.25 in [17]. ¤

7.1

Geodesics in a sub-spray

Definition 7.5 (Geodesically invariant set). Let S be a spray on Tr_{M where}

r≥0. Then a set ∆⊂Tr+1_{M is ageodesically invariant set for S provided that:}

Ifγ:I→Tr_{M is a geodesic ofSwithγ}′_(t

0)∈∆ for somet0∈I, thenγ′(t)∈∆

for allt∈I.

Definition 7.6 (Sub-spray). SupposeS is a spray on Tr_{M for some r ≥0, and} ∆ is a geodesically invariant semi-distribution on Tr_{M. Then we say that triple} Σ = (S, Tr_{M,∆) is asub-spray. A curveγ:I→T}r_{M is a geodesic inΣ if}

(i) γ:I→Tr_{M is a geodesic ofS,} (ii) γ′_(t

0)∈∆ for somet0∈I(whence γ′(t)∈∆ for allt∈I).

By taking ∆ =Tr+1_{M, we may treat any spray as a sub-spray. Let us also note}

that if ∆⊂Tr+1_{M\ {0}wherer≥0, then}

πr(∆)⊂Tr_{M, π}

r◦κr+1(∆)⊂TrM\ {0}.

Then

∆\ {0} = {γ′(0) : γ: (−ε, ε)→Tr_{M is a geodesic in Σ},}

πr(∆\ {0}) = {γ(0) : γ: (−ε, ε)→Tr_{M is a geodesic in Σ}.}

In other words, a vector ξ∈ Tr+1_{M is in ∆\ {0} if and only if there is a geodesic}

in Σ whose tangent passes throughξ, and a pointx∈Tr_{M is inπr(∆\ {0}) if and} only if there is a geodesic in Σ that passes throughx. We therefore say that ∆\ {0}

isphase space for Σ, andπr(∆\ {0}) isconfiguration spacefor Σ. When r≥1, the setB= (Dπr−1)(∆) satisfies

B\ {0} = {(πr−1◦γ)′(0) : γ: (−ε, ε)→TrM is a geodesic in Σ}.

and we can interpretB\{0}as phase space of geodesics in Σ that have been projected ontoTr−1_M.

Example 7.7 (Geodesics through a point). Let Σ = (S, Tr_{M,∆) be a sub-spray} for somer≥0, and letz ∈πr(∆\ {0}) be a point in configuration space. Then the set

parameterizes initial values for geodesics that pass throughz. Let us study the struc-ture and the degrees of freedom for ∆(z).

When r = 0, the structure of ∆(z) is easy to understand; the set ∆(z) is a punctured vector subspace ofTzM whose dimension is the rank of ∆.

Whenr ≥1, the structure of ∆(z) becomes more complicated. For example, in Section 8, we construct a sub-spray where configuration space and phase space are diffeomorphic, and ∆(z) contains only one vector. To understand this, let us assume that ∆ is represented in canonical local coordinates (x, y, X, Y) forTr+1_{M. That is,}

we here only consider coordinates (x, y, X, Y) that belong to ∆. Then coordinates (x, y, X, Y) have dim ∆ = dimB+ rank ∆ degrees of freedom. Coordinates (x, X) represent submanifoldB. They have dimBdegrees of freedom, and once (x, X)∈Bis fixed, coordinates (y, Y) parameterize the rank ∆ dimensional vector space associated with (x, X). Ifz= (x0, y0), then geodesics that pass throughz are parameterized by

(x0, y0, X, Y), but very little can be said about possible values for (X, Y). Coordinates

(x, X) have dimB degrees of freedom, but we do not know how these divide between

x- and X-coordinates. Similarly, coordinates (y, Y) have rank ∆ degrees of freedom, but we do not know how these divide betweeny- andY-coordinates.

The next proposition shows that geodesics in a sub-spray on Tr_{M have a} lin-ear structure when r ≥ 1, but geodesics are not necessarily invariant under affine reparametrizations.

Proposition 7.8. LetΣ = (S, Tr_{M,∆) be a sub-spray wherer≥0.} (i) Suppose that j: I → Tr_{M is a geodesic in Σ. If t}

0 ∈ R, and C > 0, then

j(Ct+t0)is a geodesic in Σif r= 0 orC= 1.

(ii) Suppose thatr≥1. Ifj, k:I→Tr_{M are geodesics inΣsuch thatπ}

r−1◦j(t) =

πr−1◦k(t), then

αj+βk, α, β∈R is a geodesic in Σ.

Proof. Property (i) follows since reparametrizations scale tangent vectors as in equa-tion (2.2), and this multiplicaequa-tion is only compatible with the vector structure of ∆ whenr= 0 orC= 1. Property (ii) follows using equations (3.12)-(3.13). ¤

7.2

Jacobi fields for a sub-spray

Proposition 6.3 shows that for sprays, Jacobi fields on compact intervals can be char-acterized using geodesic variations. For sub-sprays, we take this characterization as the definition of a Jacobi field.

Definition 7.9 (Jacobi field in a sub-spray). Letγ:I→Tr_{M be a geodesic in} a sub-spray Σ = (S, Tr_{M,∆) wherer≥0. Suppose thatJ:K→T}r+1_{M is a curve}

whereK⊂I is compact, andV is a mapV:I×(−ε, ε)→Tr_{M such that} (i) t7→V(t, s),t∈I is a geodesic in sub-spray Σ for alls∈(−ε, ε), (ii) V(t,0) =γ(t) fort∈I,

ThenJ:K→Tr+1_{M is aJacobi field along γ.}

By Proposition 6.3 (ii), a Jacobi field for a sub-spray (S, Tr_{M,∆) is a Jacobi field} for the sprayS. The converse also holds when ∆ =Tr+1_M.

8

A sub-spray for parallel Jacobi fields

This section contains the main results of this paper. We construct a sub-spray P

whose geodesics are in one-to-one correspondence with parallel Jacobi fields, and study its geodesics.

Definition 8.1 (Parallel Jacobi field). LetSbe a spray onM, and letJ:I→T M

be a curve. ThenJ is called a parallel Jacobi field forS if there areα, β∈R, and a geodesicc:I→M such that

J(t) = αc′(t) +βtc′(t), t∈I.

(8.1)

If I is closed we say that a curve J:I →T M is a parallel Jacobi field ifJ can be extended into a parallel Jacobi defined on an open interval. Proposition 6.5 shows that a parallel Jacobi field is a Jacobi field.

Lemma 8.2. SupposeJ:I→T M is a parallel Jacobi field. (i) IfC >0 andt0∈R, thenJ(Ct+t0) is a parallel Jacobi field.

(ii) J can be extended to the maximal domain of geodesicc=πT M→M ◦J, and the extension is a parallel Jacobi field.

To construct sub-sprayP, letSbe a spray on a manifoldM, letScc_{be the second} complete lift ofS, and let ∆ be the geodesically invariant semi-distribution onT T M

defined in Proposition 8.3. Then we define sub-sprayP as

P = (Scc_{, T T M,∆).}

Proposition 8.3. SupposeS is a spray on a manifold M, and let∆ be the set

∆ = {(κ2◦J′)′(0) :J: (−ε, ε)→T M is parallel Jacobi field forS}.

Then

(i) ∆⊂T T T M\ {0},

(ii) ∆ is a geodesically invariant semi-distribution onT T M of rank2, (iii) phase space ∆ and configuration spaceπ2(∆)are diffeomorphic.

Proof. Let us first note that ∆ consists of points

(x(0),x˙(0), αx˙(0), αx¨(0) +βx˙(0),

˙

whereα, β∈Randc: (−ε, ε)→M is a geodesicc(t) = (x(t)). By Lemma 7.4 (and by the result used to prove Lemma 7.4), it follows that sets

π2(∆) = {αS(y) +βE1(y) :y∈T M\ {0}, α, β∈R},

(Dπ1)(∆) = {S(y) :y∈T M\ {0}},

are submanifolds inT T M diffeomorphic to T M \ {0} ×R2 and T M\ {0}, respec-tively. Letι be the inclusion B ֒→T T M, where B = (Dπ1)(∆), and let Sb be the

diffeomorphism Sb:T M \ {0} → B such that S =ι◦Sb and Sb−1 = π1◦ι. By the

geodesic equation forSc _{and the definition ofS}c _{it follows that}

κ3(∆) = (DS)(π2(∆))

= {αV1(ξ) +βV2(ξ) :ξ∈B, α, β ∈R},

whereV1, V2:B→T T T M are smooth maps

V1=DS◦ι, V2=DS◦E1◦π1◦ι.

Nowπ2◦Vi =ι fori= 1,2. A local calculation shows thatV1 andV2 are pointwise

linearly independent. Hence ∆ is a semi-distribution onT T M, and by Lemma 7.4,

κ3(∆) is diffeomorphic toB×R2.

To prove that ∆ is geodesically invariant, letγ:I →T T M be a geodesic of Scc withγ′_{(0)∈∆. By Proposition 6.5, it follows that}

γ(t) =κ2◦J′(t), t∈(−ε, ε)

(8.2)

for a parallel Jacobi fieldJ: (−ε, ε)→T M. By Lemma 8.2 (ii) we can extendJ into a parallel Jacobi fieldJ: I →T M such that (8.2) holds for all t∈ I. If t0 ∈ I, we

haveγ′_(t

0) = (κ2◦J˜′)′(0) for parallel Jacobi field ˜J(t) =J(t+t0), and (ii) follows.

Property (iii) follows since both submanifolds are diffeomorphic toB×R2. ¤

Let us note that configuration spaceπ2(∆) is a proper subset ofT T M, and

dimπ2(∆) = 2n+ 2, dim ∆ = 2n+ 2, dim(Dπ1)(∆) = 2n.

The next proposition shows that geodesics inP are in one-to-one correspondence with parallel Jacobi fields forM.

Proposition 8.4 (Geodesics in P). Supposeγ: I →T T M is a curve. Then the following are equivalent:

(i) γ is a geodesic inP.

(ii) There is a parallel Jacobi fieldJ:I→T M such that

γ(t) = κ2◦J′(t), t∈I.

(iii) There is a geodesic c:I→M andα, β∈Rsuch that

Moreover, in (ii) and (iii)J andc, α, β are uniquely determined byγ.

The next proposition shows that the geometry ofP can be used to study dynamical properties ofM.

Proposition 8.5. The projectionπT T M→M: T T M →M is a submersion that maps geodesics inP into geodesics on M.

A sub-spray (S, Tr_{M,∆) wherer≥0 iscompleteif any geodesicγ:I→T}r_{M can} be extended into a geodesicγ:R→Tr_M.

Proposition 8.6. Sub-sprayP is complete if and only ifM is complete.

Proof. Suppose P is complete. By Proposition 8.4, any geodesic c: I → M can be lifted into a geodesic c′′_{: I → T T M for P. The converse direction follows by}

Proposition 8.4 and Lemma 8.2 (ii). ¤

In general, a geodesic c:I → M for a sprayS on M is uniquely determined by

c′_{(0). The next proposition shows that in sub-spray P, a geodesic γ:I → T T M is} uniquely determined byγ(0). This is not surprising in view of Proposition 8.3 (iii).

Proposition 8.7. If γ1:I1 → T T M and γ2: I2 → T T M are geodesics in P with

γ1(0) =γ2(0), thenγ1=γ2 on their common domain.

Proof. By Proposition 8.4 we have thatγi =κ2◦Ji′ for parallel Jacobi fieldsJi:I→

T M, i= 1,2. HenceJ′

1(0) =J2′(0), and the claim follows. ¤

Proposition 8.7 imposes a strong restriction on the behavior of geodesics in P. For example, if two points inT T M can be connected with a geodesic in P, then the geodesic is unique (up to loops). Also, any piece-wise geodesic curve that is continuous must be smooth. ThereforeP has no broken geodesics nor geodesic triangles.

For a sub-spray we define conjugate points as for sprays (see Section 6.3).

Proposition 8.8. Sub-sprayP has no conjugate points.

Proof. If a Jacobi field vanishes once, Proposition 8.7 implies that the corresponding

geodesic variation is trivial. ¤

Acknowledgements. I.B. has been supported by CNCSIS grant IDEI 398 from the Romanian Ministry of Education. M.D. has been supported by Academy of Finland Centre of Excellence programme 213476, the Institute of Mathematics at the Helsinki University of Technology, and Tekes project MASIT03 — Inverse Problems and Reliability of Models.

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Authors’ addresses: Ioan Bucataru

Faculty of Mathematics, Al.I.Cuza University, B-dul Carol 11, Iasi, 700506, Romania. E-mail: bucataru@uaic.ro

http://www.math.uaic.ro/˜bucataru

Matias F. Dahl

Department of Mathematics, University College London, Gower Street, WC1E 6BT London, U.K.