Conformal vector fields on Finsler manifolds arXiv:1111.1629v1 [math.DG] 7 Nov 2011
József Szilasi
Anna Tóth
Abstract Applying concepts and tools from classical tangent bundle geometry and using the apparatus of the calculus along the tangent bundle projection (‘pull-back formalism’), first we enrich the known lists of the characterizations of affine vector fields on a spray manifold and conformal vector fields on a Finsler manifold. Second, we deduce consequences on vector fields on the underlying manifold of a Finsler structure having one or two of the mentioned geometric properties.
Mathematics Subject Classification (2010): 53C60, 53A30 Keywords: spray manifold, Finsler manifold, projective vector field, affine vector field, conformal vector field.
Introduction The theory of ‘geometrical’ – projective, affine, conformal, isometric – vector fields on a Finsler manifold has a vast literature, mainly from the period dominated technically by the classical tensor calculus, visually, ‘the debauch of indices’. Chapter VIII of K. Yano’s book ‘The theory of Lie derivatives and its applications’ presents a survey of the main achievements from the beginning of the 20th century to 1957. A good overview of the developments of the next decades can be found in R. B. Misra’s paper [15], written in 1981, revised and updated in 1993. It is important to note that in a 2-part paper, see [13],[14], M. Matsumoto clarified and improved some results of Yano in the framework of his theory of Finsler connections. From the (relatively) modern, but partly tensor calculus based literature the works of H. Akbar-Zadeh [2],[3], J. Grifone [9],[10] and R. L. Lovas [12] are worth mentioning. Grifone applies systematically the ‘τT M : T T M → T M formalism’, combining with the Frölicher–Nijenhuis calculus of vector-valued forms; Lovas formulates and proves his results in 1
◦
◦
◦
terms of the ‘pull-back formalism π : T M ×M T M → T M’. Our paper is a continuation of both Grifone’s and Lovas’s works. Although we are going to develop the greater part of the theory in terms of the pull-back bundle, the concepts and techniques of the tangent bundle geometry, including the vertical calculus on T M, also play an eminent role in our considerations. To make the paper more readable, in section 1 we summarize in a coherent way the various concepts and tools which will be indispensable in the following. We apply two types of a Lie derivative operator: beside the classical Lie derivative operator Lξ on T M (ξ ∈ X(T M)) we need a further operator, denoted by Leξ , which acts on the tensor algebra of the C ∞ (T M)-module of ◦ the sections of the vector bundle π : T M ×M T M → T M (or of the bundle π). To assure the validity of the crucial identity [Leξ , Leη ] = Le[ξ,η] in case of the ‘new’ operator, we are forced to differentiate with respect to projectable vector fields on T M. In section 2 some basic properties of the operator Leξ are established. The affine and projective properties of a Finsler manifold depend only on its canonical spray, so it is natural to examine affine and projective vector fields in the (virtual) generality of spray manifolds. A vector field X on a manifold M is said to be an affine vector field or a Lie symmetry for a spray S : T M → T T M if S is invariant under the flow of the complete lift X c of X, that is, if LX c S = [X c , S] = 0. In Lovas’s paper [12] various equivalents of this property are established. In section 3 we enrich his list with some new items, which will be technically useful in the next section. By a conformal vector field on a Finsler manifold (M, F ) we mean a vector field X on M satisfying LeX c g = ϕ g, where g is the metrical tensor of the Finsler manifold (the vertical Hessian of the energy function E = 21 F 2 ) and ϕ is a function, defined and continuous ◦
on T M, smooth on the deleted bundle T M. It turns out at once that ϕ has to be fibrewise constant, i.e., of the form ϕ = f ◦ τ , where f is a smooth function on M and τ is the tangent bundle projection. Homothetic and isometric (or Killing) vector fields are the particular cases for which ϕ is a constant function, resp. identically zero. In section 4 we present further characterizations of conformal vector fields on a Finsler manifold (Proposition 11), one of them has already been proposed by Grifone in [10]. We show that if a vector field X ∈ X(M) is both affine and conformal on a Finsler manifold (M, F ), then X c is a conformal vector field for the Sasaki extension of the metric tensor of (M, F ) (Proposition 13). At this stage, the following ‘expectable’, but non-trivial conclusions may 2
be deduced fairly easily: (a) Homothetic vector fields on a Finsler manifold are affine vector fields (Proposition 14). (b) If a vector field on a Finsler manifold is both projective and conformal, then it is a homothetic vector field (Proposition 16). (c) If a vector field preserves the Dazord volume form of a Finsler manifold and it is also projective, then it is an affine vector field (Proposition 17, (i)). (d) If a vector field is both volume-preserving (in the above sense) and conformal, then it is a Killing field (Proposition 17, (ii)).
1
Basic setup
1.1 Generalities Most of our basic notations and conventions will be the same as in [4], see also [16]. However, for the reader’s convenience, we present here a short review on the most essential things. (a) By a manifold we mean a finite dimensional smooth manifold whose underlying topological space is Hausdorff, second countable and connected. In what follows, M will be an n-dimensional manifold, where n ≥ 2. Let k ∈ N ∪ {∞}. We denote by C k (M) the set of k-times continuously differentiable real-valued functions on M, with the convention that C 0 (M) is the set of the continuous functions on M. In particular, C ∞ (M) is the real algebra of smooth functions on M. (b) TheS tangent space of M at a point p ∈ M is denoted by Tp M; T M := p∈M Tp M . The tangent bundle of M is the triplet (T M, τ, M), where the tangent bundle projection τ is defined by τ (v) := p if v ∈ Tp M. Instead of (T M, τ, M) we usually write τ : T M → M or simply τ . Similarly, the tangent bundle of T M is (T T M, τT M , T M) or τT M : T T M → T M or τT M . In general, we prefer to denote a bundle by the same symbol as we use for its projection. A vector field on M is a smooth section of the tangent bundle τ : T M → M. The vector fields on M form a C ∞ (M)-module which will be denoted by X(M). The zero vector field o on M is defined by p ∈ M 7→ o(p) := 0p := the zero vector in Tp M. ◦
◦
The deleted bundle for τ is the fibre bundle τ : T M → M, where ◦
◦
◦
T M := T M \ o(M), τ := τ ↾ T M . 3
(c) If ϕ : M → N is a smooth mapping between smooth manifolds, then we denote its derivative by ϕ∗ , which is a fibrewise linear smooth mapping of T M into T N. Two vector fields X ∈ X(M) and Y ∈ X(N) are ϕ-related if Y . A vector field ξ on T M is said to be ϕ∗ ◦ X = Y ◦ ϕ; then we write X ∼ ϕ projectable if there exists a vector field X on M such that ξ ∼ X. τ (d) TheL classical graded derivations of the graded algebra Ω(M) := nk=0 Ωk (M) of the differential forms on M are the Lie derivative LX (X ∈ X(M)), the substitution operator iX (X ∈ X(M)), the exterior derivative d,
related by H. Cartan’s ‘magic’ formula (1.1)
LX = iX ◦ d + d ◦ iX .
1.2 Canonical constructions and objects (a) By the vertical lift of a smooth function f on M we mean the function f v := f ◦ τ ∈ C ∞ (T M); the complete lift of f is the function f c ∈ C ∞ (T M) given by f c (v) := v(f ), v ∈ T M. o. The vertical vector fields form (b) A vector field ξ on T M is vertical if ξ ∼ τ a C ∞ (T M)-module Xv (T M), which is also a subalgebra of the Lie algebra X(T M). The Liouville vector field on T M is the unique vertical vector field C ∈ Xv (T M) such that (1.2)
Cf c = f c for all f ∈ C ∞ (M).
The vertical lift of a vector field X on M is the unique vertical vector field X v ∈ Xv (T M) satisfying (1.3)
X v f c = (Xf )v for all f ∈ C ∞ (M);
the complete lift X c ∈ X(T M) of X is characterized by (1.4)
X c f c = (Xf )c , f ∈ C ∞ (M)
(see [19], Ch. I.3). Then we have (1.5)
X c f v = (Xf )v , f ∈ C ∞ (M). 4
Both X v and X c are projectable: X v ∼ o, X c ∼ X. Lie brackets involving τ τ vertical and complete lifts satisfy the rules (1.6a-c) (1.7a-b) (c) Let
If
[X v , Y v ] = 0, [X c , Y v ] = [X, Y ]v , [C, X v ] = −X v , [C, X c ] = 0.
[X c , Y c ] = [X, Y ]c ,
T M ×M T M : = (u, v) ∈ T M × T M τ (u) = τ (v) , ◦ ◦ ◦ T M ×M T M : = (u, v) ∈ T M × T M τ (u) = τ (v) . ◦
π := pr1 ↾ T M ×M T M,
◦
π := pr1 ↾ T M ×M T M, ◦
◦
then both π and π are vector bundles over T M and T M, resp., with fibres ◦
{u} × Tτ (u) M ∼ = Tτ (u) M;
u ∈ T M, resp. u ∈ T M. ◦
◦
We denote by Sec(π) and Sec(π) the C ∞ (T M)-, resp. C ∞ (T M)-module of the sections of these bundles. A typical section in Sec(π) is of the form e : v ∈ T M 7−→ (v, X(v)) ∈ T M ×M T M, X
where X : T M → T M is a smooth mapping such that τ ◦ X = τ . X is called e We have a canonical section in Sec(π), denoted by the principal part of X. δ, whose principal part is the identity mapping of T M. Every vector field b in Sec(π), called a basic section, whose principal X on M yields a section X part is X ◦ τ . Locally, the C ∞ (T M)-module Sec(π) is generated by the basic sections. We denote by Tlk (π) the C ∞ (T M)-module of the type (k, l) tensors over ◦ the module Sec(π); the meaning of Tlk (π) is analogous. (d) We have a canonical C ∞ (T M)-linear injection i : Sec(π) → X(T M) given on the basic sections by (1.8)
b := X v , X ∈ X(M), i(X)
and a canonical C ∞ (T M)-linear surjection j : X(T M) → Sec(π) such that (1.9)
b j(X v ) := 0, j(X c ) := X.
Then Im(i) = Ker(j) = Xv (T M). The mapping J := i ◦ j is said to be the vertical endomorphism of X(T M). It follows immediately that Im(J) = Ker(J) = Xv (T M), J2 = 0. Due to their C ∞ (T M)-linearity, i, j and J have a natural pointwise interpretation. 5
1.3 Some vertical calculus (a) We define the vertical differential ∇v F of a function F ∈ C ∞ (T M) as a 1-form in T10 (π) given by (1.10)
e := ∇ve F := (iX)F, e e ∈ Sec(π). ∇v F (X) X X
The vertical differential ∇v Ye of a section Ye ∈ Sec(π) is the type (1, 1) tensor in T11 (π) defined by ( e := ∇v Ye := j[iX, e η], ∇v Ye (X) e X (1.11) η ∈ X(T M), j(η) = Ye .
(It is easy to check that ∇vXe Ye does not depend on the choice of η satisfying j(η) = Ye .) By the standard technique, to make sure that Leibniz’s rule holds, the operators ∇vXe may be extended to tensor derivations of the full tensor algebra of Sec(π). (b) Next we consider the graded algebra Ω(T M) of the differential forms on T M, and we define an operator dJ : Ω(T M) −→ Ω(T M) by the rules (1.12)
dJ F := dF ◦ J,
dJ dF := −d dJ F ;
F ∈ C ∞ (T M).
Then dJ is a graded derivation of degree 1 of Ω(T M), called the vertical differentiation on T M. We have (and we shall need) the following important relation: (1.13)
dJ ◦ LC − LC ◦ dJ = dJ .
For details, we refer to the book [6]. We mention that ∇v and dJ , at the level of functions, are related by dJ F = ∇v F ◦ j, F ∈ C ∞ (T M). (c) Let K be a type (1, 1) tensor on T M, interpreted as an endomorphism of the C ∞ (T M)-module X(T M). It will be convenient to denote the Lie derivative −Lη K (η ∈ X(T M)) by [K, η]. Then, for any vector field ξ on T M, [K, η]ξ = [Kξ, η] − K[ξ, η]. We have, in particular, (1.14a-c)
[J, C] = J;
[J, X v ] = 0, [J, X c ] = 0 (X ∈ X(M)). 6
In what follows, for simplicity, we shall denote also by i, j and J ◦
◦
the restrictions of these mappings to Sec(π) and X(T M). 1.4 Ehresmann connections ◦ ◦ (a) By an Ehresmann connection in T M we mean a C ∞ (T M)-linear mapping ◦
◦
H : Sec(π) −→ X(T M) such that ◦ . j ◦ H = 1Sec(π)
◦
We emphasize (cf. 1.2(d)) that the C ∞ (T M)-linearity of H makes it possible to interpret an Ehresmann connection as a strong bundle map ◦
◦
H : T M ×M T M −→ T T M as follows: ◦ ◦ e ∈ Sec(π) For each (u, v) ∈ T M ×M T M there exists a section X such e e that X(u) = (u, v). Let Hu (v) := H(X)(u). Then Hu is well-defined and ◦ e e e ∈ Sec(π). H(X)(u) = Hu (X(u)) for all X
Obviously, the mappings
◦
◦
Hu : {u} × T˚τ (u) M −→ Tu T M, u ∈ T M ◦
◦
are linear. Now we obtain the desired mapping H : T M ×M T M → T T M by setting H ↾ {u} × T˚τ (u) M := Hu . ◦
◦
◦
(b) Let H : Sec(π) → X(T M) be an Ehresmann connection in T M. Then ◦
◦
Xh (T M) := Im(H) is a submodule of X(T M), and we have the direct ◦
◦
◦
◦
decomposition X(T M) = Xv (T M)⊕Xh (T M). Vector fields on T M belonging ◦
to Xh (T M) are called horizontal. Notice that they do not form, in general, ◦
a subalgebra of the Lie algebra X(T M). The mappings h : = H ◦ j, v := 1 ◦
◦
X(T M )
− h, ◦
V : = i−1 ◦ v : X(T M) −→ Sec(π) 7
are called the horizontal projection, the vertical projection and the vertical mapping associated to H, respectively. h and v are indeed projection ◦
operators in X(T M), while the mapping V has the properties ◦ , Ker(V) = Im(H). V ◦ i = 1Sec(π)
The horizontal lift of a vector field X on M (with respect to H) is b = h(X c ). X h := H(X)
◦
b and X c are regarded here as a section in Sec(π) and a vector field on (X ◦
T M, resp.; for simplicity, we make no notational distinction.) (c) An Ehresmann connection H is said to be homogeneous if [C, X h ] = 0 for all X ∈ X(M). ◦
◦
Then H, as a strong bundle map of T M ×M T M to T T M, may be extended continuously to a mapping T M ×M T M → T T M such that H(0p , v) = (o∗ )p (v) for all p ∈ M, v ∈ Tp M. Thus, in what follows, we shall always assume that a homogeneous Ehresmann connection is defined on the entire T M ×M T M (or on Sec(π)). ◦
(d) If H is an Ehresmann connection in T M, then the mapping ◦
given by
◦
◦
∇ : X(T M) × Sec(π) −→ Sec(π), (ξ, Ye ) 7−→ ∇ξ Ye
(1.15a) (1.15b)
(1.11) ∇vξ Ye : = ∇vVξ Ye = j[vξ, HYe ] ∇hξ Ye : = ∇hjξ Ye := V[hξ, iYe ]
◦
is a covariant derivative operator in the vector bundle π, called the Berwald derivative induced by H. By the tension of H we mean the ∇h -differential t := ∇h δ of the canonical ◦ e ∈ Sec(π), section. Then, for any section X (1.16)
In particular,
e := (∇h δ)(X) e := ∇he δ = V[HX, e C]. t(X) X b = [X h , C], it(X) 8
X ∈ X(M);
therefore H is homogeneous if, and only if, its tension vanishes. With the help of the induced Berwald derivative we define the torsion T of an Ehresmann connection H by ◦ e Ye ∈ Sec(π). X,
e Ye ) := ∇ e Ye − ∇ e X e − j[HX, e HYe ]; T(X, HX HY
Evaluating on basic sections, we obtain the more expressive formula
2
b Yb ) = [X h , Y v ] − [Y h , X v ] − [X, Y ]v ; iT(X,
X, Y ∈ X(M).
Lie derivative along the tangent bundle projection
Let ξ be a projectable vector field on T M (1.1(c)). We define a Lie derivative operator Leξ on the tensor algebra of the C ∞ (T M)-module Sec(π) by the rules (2.1a)
(2.1b)
Leξ ϕ : = ξϕ, if ϕ ∈ C ∞ (T M); Leξ Ye : = i−1 [ξ, iYe ], if Ye ∈ Sec(π),
and by extending it to the whole tensor algebra in such a way that Leξ satisfies the product rule of tensor derivations. Since ξ is a projectable and iYe is a vertical vector field, it follows that the vector field [ξ, iYe ] is vertical, so Leξ Ye is well-defined. If v = i◦V is the vertical projection associated to an Ehresmann connection H in T M, then i−1 [ξ, iYe ] = V[ξ, iYe ], so we get the useful formula Leξ Ye = V[ξ, iYe ].
(2.2)
Notice, however, that the Lie derivative operator Leξ does not depend on any Ehresmann connection in T M. If, in particular, ξ := X c or ξ := X h , where X is a vector field on M, then (2.2) takes the form LeX c Ye = V[X c , iYe ],
(2.3) resp. (2.4)
(1.15b) LeX h Ye = V[X h , iYe ] = ∇hXb Ye . (1.7b)
Since [X c , iδ] = [X c , C] = 0, it follows that (2.5)
LeX c δ = 0. 9
The Lie derivative of a basic section with respect to a complete lift leads essentially to the ordinary Lie derivative. Namely, for any vector fields X, Y on M we have (2.3) (1.6b) \ \ [ LeX c Yb = V[X c , Y v ] = V[X, Y ]v = V ◦ i[X, Y ] = [X, Y]=L XY .
This relation indicates that our Lie derivative operator LeX c is a natural extension of the classical Lie derivative LX on M. Lemma 1. For any projectable vector fields ξ, η on T M, [Leξ , Leη ] = Le[ξ,η] .
(2.6)
Proof. Obviously, both sides of (2.6) act in the same way on smooth functions on T M. If Ye is a section of π, then, applying (2.2) repeatedly, [Leξ , Leη ]Ye = Leξ V[η, iYe ] − Leη V[ξ, iYe ] = V([ξ, iV[η, iYe ]] − [η, iV[ξ, iYe ]]) = V([ξ, [η, iYe ]] + [η, [iYe , ξ]]) = −V[iYe , [ξ, η]] = V[[ξ, η], iYe ] = Le[ξ,η] Ye .
Lemma 2. Let X ∈ X(M), η ∈ X(T M). Then (2.7) Proof. Since (1.14c)
LeX c jη = jLX c η.
0 = [J, X c ]η = [Jη, X c ] − J[η, X c], we find
iLeX c jη = [X c , Jη] = J[X c , η] = i(jLX c η),
which implies (2.7).
We end this section with the definition of the Lie derivative Leξ D of a covariant derivative D : X(T M) × Sec(π) → Sec(π): it is given by the rule e := Leξ (Dη Z) e − Dη (Leξ Z) e − D[ξ,η] Z, e (Leξ D)(η, Z)
where η ∈ X(T M), Ze ∈ Sec(π). Notice finally that the theory of Lie derivatives ‘along the tangent bundle projection’ sketched here works without any change also on the bundle ◦ ˚M ×M T M → T ˚M. π: T
10
3
Affine vector fields on a spray manifold
3.1 By a spray for M we mean a C 1 mapping S : T M → T T M, smooth on ◦
T M, such that (3.1) (3.2) (3.3)
τT M ◦ S = 1T M ; JS = C; [C, S] = S.
Condition (3.2) is equivalent to the requirement τ∗ ◦ S = 1T M , so a spray for M is a section also of the secondary vector bundle τ∗ : T T M → T M. In view of (3.3), a spray is a homogeneous vector field (of class C 1 ) of degree 2. We say that a manifold endowed with a spray is a spray manifold. 3.2 If H is a homogeneous Ehresmann connection in T M, then S := H ◦ δ is a spray for M, called the spray associated to H. Indeed, for any vector w in T M, S(w) = H(w, w) ∈ Tw T M, therefore τT M (S(w)) = w, so (3.1) is valid. Since J ◦ S = i ◦ j ◦ H ◦ δ = i ◦ δ = C, condition (3.2) also holds. To check (3.3), observe first that the vector field [C, S] − S is vertical, and hence h[C, S] = h S. However, hS = H ◦ j ◦ H ◦ δ = H ◦ δ =: S, so we get h[C, S] = S. On the other hand, by the homogeneity of H, 0 = −it(δ) = −v[H ◦ δ, C] = v[C, S], therefore h[C, S] = [C, S] and [C, S] = S. Finally, the C 1 differentiability of S can be shown using the ‘Observation’ in 3.11 (p. 1378) of [16]. Thus sprays exist in abundance for a manifold. Conversely, if S is a spray for M, then there exists a unique torsion-free homogeneous Ehresmann connection H in T M such that the horizontal lifts with respect to H are given by b = 1 (X c + [X v , S]), X ∈ X(M). X h := H(X) 2 For a proof of this fundamental fact we refer to [16], 3.3, or to the original source [5]. The Ehresmann connection specified by (3.4) is said to be the Ehresmann connection induced by the spray S.
(3.4)
3.3 Let (M, S) be a spray manifold. We say that a vector field X on M is a projective vector field for (M, S) (or for the spray S) if there is a continuous ◦
function ϕ on T M, smooth on T M, such that (3.5)
[X c , S] = ϕ C. 11
If, in particular, ϕ is the zero function, then we say that X is an affine vector field for (M, S), or a Lie symmetry of S. Proposition 3. Suppose (M, S) is a spray manifold. Let H be the Ehresmann connection induced by S, and let ∇ be the Berwald derivative arising from H. For a vector field X on M, the following conditions are equivalent: (i) X is a Lie symmetry of S; (ii) [h, X c ] = 0; (iii) [v, X c ] = 0; (iv) LeX c ∇ = 0;
(v) [X c , Y h ] = [X, Y ]h , for any vector field Y on M;
(vi) [LeX c , LeY h ] = Le[X,Y ]h , Y ∈ X(M);
(vii) LeX c ◦ V = V ◦ LX c .
Proof. The equivalence of conditions (i), (ii) and (iv) has already been proved in [12]. (ii) ⇐⇒ (iii) This is evident, since v = 1 −h (1 := 1X(T M ) ) and [1, ξ] = 0 for all ξ ∈ X(T M). (ii) ⇐⇒ (v) For any vector field Y on M, [h, X c ]Y c = [hY c , X c ] − h[Y c , X c ] = [Y h , X c ] − h[Y, X]c = [Y h , X c ] − [Y, X]h , so the vanishing of [h, X c ] implies that [X c , Y h ] = [X, Y ]h . The converse is also true, since [h, X c ] annihilates the module of vector fields: for any vector field ξ on T M we have [h, X c ]Jξ = [h ◦ J(ξ), X c] − h[Jξ, X c ] = 0. (v) ⇐⇒ (vi)
(see Lemma 1). (iii) ⇐⇒ (vii)
This is an immediate consequence of the identity [LeX c , LeY h ] = Le[X c ,Y h ]
For any vector field ξ on T M, iLeX c (Vξ) = [X c , vξ],
iV(LX c ξ) = v[X c , ξ],
hence LeX c (Vξ) = V(LX c ξ) if, and only if,
0 = [vξ, X c] − v[ξ, X c] = [v, X c ]ξ.
12
4
Conformal vector fields on a Finsler manifold
4.1 Let (M, F ) be a Finsler manifold. We recall that the Finsler function ◦
F : T M → R here is assumed to be smooth on T M, positive (F (v) > 0, if ◦
v ∈ T M), positive-homogeneous of degree 1 (F (λ v) = λ F (v) for all v ∈ T M and positive real number λ), and it is also required that the metric tensor 1 g := ∇v ∇v F 2 2 is fibrewise non-degenerate. The function E := 21 F 2 is the energy function of ◦
(M, F ). The homogeneity of F implies that over T M we have CF = F,
CE = 2E.
The Hilbert 1-form of (M, F ) is θe := ∇v E = F ∇v F – in the pull-back formalism, θ := dJ E – in the τT M formalism. It is easy to check that ◦ e X) e = g(X, e δ) for each X e ∈ Sec(π). θ(
θe and θ are related by
θ = θe ◦ j.
(4.1)
The 2-form
ω := dθ = ddJ E ◦
on T M is said to be the fundamental 2-form of (M, F ). Its relation to the metric tensor is given by ◦
(4.2)
ω(Jξ, η) = g(jξ, jη);
ξ, η ∈ X(T M).
The non-degeneracy of g implies the non-degeneracy of ω – and vice versa. Lemma 4. With the notations introduced above, let (M, F ) be a Finsler manifold, and let X be a vector field on M. Then (4.3) (4.4)
e ◦ j = LX c θ; (LeX c θ)
(LeX c g)(jξ, jη) = (LX c ω)(Jξ, η); 13
◦
ξ, η ∈ X(T M).
Proof. We check only the less trivial second relation: (LX c ω)(Jξ, η) = X c ω(Jξ, η) − ω(LX c Jξ, η) − ω(Jξ, LX c η) (2.7), (4.2)
=
X c g(jξ, jη) − ω(LX c Jξ, η) − g(jξ, LeX c jη).
Since LX c Jξ = [X c , Jξ] = −[J, X c ]ξ + J[X c , ξ] = JLX c ξ, the second term at the right-hand side of the above relation takes the form (2.7)
(4.2)
ω(LX c Jξ, η) = ω(JLX c ξ, η) = g(jLX c ξ, jη) = g(LeX c jξ, jη).
So we obtain
(LX c ω)(Jξ, η) = X c g(jξ, jη) − g(LeX c jξ, jη) − g(jξ, LeX c jη) = (LeX c g)(jξ, jη). 4.2 We continue to assume that (M, F ) is a Finsler manifold. The 2n-form n(n−1) 2
(−1) σ := n!
ωn, ◦
where ω n = ω ∧ ... ∧ ω (n factors) is a volume form on T M, called the Dazord ◦
volume form of (M, F ). By the divergence of a vector field ξ on T M (with ◦
respect to σ) we mean the unique function div ξ ∈ C ∞ (T M) such that Lξ σ = (div ξ) σ. Lemma 5. If (M, F ) is a Finsler manifold, then the divergence of the ◦
Liouville vector field C on T M with respect to the Dazord volume form is n = dim M. (1.13)
Proof. LC ω = LC ddJ E = dLC dJ E = ddJ LC E −ddJ E = 2ddJ E −ddJ E = ω. From this it follows by induction that LC ω n = n ω n , whence our claim. 4.3 If (M, F ) is a Finsler manifold, then there exists a unique spray S for M such that ◦
(4.5)
iS ddJ E = −dE
over T M, and S ↾ o(M) = 0.
We say that S is the canonical spray of (M, F ); the Ehresmann connection induced by S according to (3.4) is said to be the canonical connection of (M, F ). It may be characterized as the unique torsion-free homogeneous 14
Ehresmann connection H for M which is compatible with the Finsler function in the sense that dF ◦ H = 0, or, equivalently, X h F = 0 for all
X ∈ X(M).
With the help of the canonical connection, we define the Sasaki extension G of the metric tensor g of (M, F ) by the rule ◦
(4.6)
G(ξ, η) := g(jξ, jη) + g(Vξ, Vη);
ξ, η ∈ X(T M),
where V is the vertical mapping associated to H. Then G is a Riemannian ◦
metric tensor on T M. For subsequent applications, we collect here some further technical results. e in Sec(π), we have Lemma 6. For any section X e ∇vXe δ = X.
(4.7)
Proof. Let H be a homogeneous Ehresmann connection for M and let S := H ◦ δ be the spray associated to H (3.2). Then, applying the so-called Grifone identity ([8], Prop. I.7), we find that e Hδ] = j[iX, e S] = X. e ∇vXe δ := j[iX, Lemma 7. The energy function of a Finsler manifold can be obtained from the metric tensor by (4.8)
g(δ, δ) = 2E;
from the fundamental 2-form by (4.9)
ω(C, S) = 2E,
where S is a spray for the base manifold. Proof. g(δ, δ) = ∇v (∇v E)(δ, δ) = ∇vδ (∇v E)(δ) = ∇vδ (∇v E(δ)) − ∇v E(∇vδ δ) (4.7)
= ∇vδ (CE) − ∇v E(δ) = C(CE) − CE = 4E − 2E = 2E;
ω(C, S) = ddJ E(C, S) = C dJ E(S) − S (dJ E(C)) − dJ E([C, S]) = C(CE) − dJ E(S) = 4E − 2E = 2E. 15
Lemma 8. The divergence of the canonical spray of a Finsler manifold vanishes. (1.1)
(4.5)
Proof. LS ω = LS ddJ E = iS dddJ E + diS ddJ E = −ddE = 0, which implies our claim. 4.4 Let (M, F ) be a Finsler manifold. We say that a vector field X on M is a projective, resp. an affine vector field of (M, F ), if it is a projective vector field, resp. a Lie symmetry for the canonical spray of (M, F ). A vector field X on M is said to be a conformal vector field, if the Lie derivative of the metric tensor of (M, F ) with respect to the complete lift of X satisfies the relation LeX c g = ϕ g
(4.10)
◦
for a continuous function ϕ : T M → R, of class C 1 on T M, called the conformal factor of X. Particular cases of conformal vector fields are homothetic vector fields for which the conformal factor is a constant function and isometric vector fields, also called Killing vector fields, for which the conformal factor is the zero function on T M. Lemma 9. If X is a conformal vector field on a Finsler manifold (M, F ) with conformal factor ϕ, then X c E = ϕ E. (2.5) (4.8) Proof. 2X c E = X c (g(δ, δ)) = (LeX c g)(δ, δ) + 2 g(LeX c δ, δ) = (LeX c g)(δ, δ)
(4.10)
(4.8)
= ϕ g(δ, δ) = 2 ϕE.
Lemma 10. If X is a conformal vector field on a Finsler manifold (M, F ), then the conformal factor of X is the vertical lift of a smooth function on M. Proof. In view of the previous lemma, X c E = ϕ E, where ◦
ϕ ∈ C 0 (T M) ∩ C 1 (T M). Acting on both sides of this relation by the Liouville vector field, we get on the one hand C(X c E) = C(ϕ E) = (Cϕ)E + 2ϕE, on the other hand C(X c E) = [C, X c ]E + X c (CE) = 2X c E = 2ϕE, so it follows that (Cϕ)E = 0, and hence C ϕ = 0. This means that ϕ is positive-homogeneous of degree 0, which implies (see, e.g., [16], 2.6, Lemma 2) that ϕ is of the form ϕ = f ◦ τ, f ∈ C ∞ (M). 16
Proposition 11. Let (M, F ) be a Finsler manifold. For a vector field X on M, the following conditions are equivalent: (i) X is a conformal vector field with conformal factor ϕ; (ii) X c E = ϕ E; (iii) LX c θ = ϕ θ; e (iv) LeX c θe = ϕ θ;
(v) LX c ω = ϕ ω + dϕ ∧ dJ E;
ϕ = f ◦ τ,
f ∈ C ∞ (M). ◦
In conditions (ii) − (iv), ϕ ∈ C 0 (T M) ∩ C 1 (T M). Proof. The arrangement of our reasoning follows the scheme ⇐=
=⇒
(i) =⇒ (ii) (v) ⇐= (iii) ⇐⇒ (iv). (i) =⇒ (ii) (ii) =⇒ (iii)
This is just a restatement of Lemma 9. Let Y be a vector field on M. We have on the one hand (1.6b)
(LX c θ)(Y v ) = X c (θ(Y v )) − θ([X c , Y v ]) = X c (θ(Y v )) − θ([X, Y ]v ) = 0 = (ϕ θ)(Y v ), since the vertical vector fields are annullated by the 1-form θ = dJ E. On the other hand, (1.6c)
(LX c θ)(Y c ) = X c (dJ E(Y c )) − dJ E([X c , Y c ]) = X c (Y v E) − [X, Y ]v E (1.6b)
(ii)
(∗)
= X c (Y v E) − [X c , Y v ]E = Y v (X c E) = Y v (ϕ E) = ϕ(Y v E) = (ϕ dJ E)(Y c ) = (ϕ θ)(Y c ).
At step (∗) we used the fact that our condition X c E = ϕE implies, as it turns out from the proof of Lemma 10, that ϕ is a vertical lift. Thus LX c θ = ϕθ, as we claimed. (iii) =⇒ (v) (iii) LX c ω = LX c d θ = dLX c θ = d(ϕ θ) = d ϕ ∧ θ + ϕdθ = ϕ ω + dϕ ∧ dJ E. To check that the function ϕ here is a vertical lift, we evaluate both sides of (iii) at a spray S. Then θ(S) = dJ E(S) = d E(C) = 2E, while (LX c θ)(S) = X c (dJ E(S)) − dJ E([X c , S]) = 2X c E − J[X c , S]E = 2X c E, 17
since [X c , S] is vertical (see, e.g., [16], p. 1350). Thus we obtain that X c E = ϕ E, which implies, as we have just remarked, that ϕ = f ◦ τ , f ∈ C ∞ (M). ◦
(v) =⇒ (i)
For any vector fields ξ, η on T M,
(v) (4.4) (LeX c g)(jξ, jη) = (LX c ω)(Jξ, η) = (ϕ ω + d ϕ ∧ dJ E)(Jξ, η) = ϕω(Jξ, η) + dJ ϕ(ξ)dJ E(η) − dϕ(η)dJ E(Jξ) (4.2) dJ ϕ=0 = ϕω(Jξ, η) = (ϕ g)(jξ, jη),
hence LeX c g = ϕg. (iii) ⇐⇒ (iv)
◦
If LX c θ = ϕ θ, then for any vector field ξ on T M, (4.3)
(iii)
(4.1)
e e (LeX c θ)(jξ) = (LX c θ)(ξ) = (ϕ θ)(ξ) = ϕ θ(jξ),
e The converse may be checked in the same way. whence LeX c θe = ϕ θ.
We note that relation (v), as a characterization of conformal vector fields on a Finsler manifold, was announced first by J. Grifone [10]. Corollary 12. Let (M, F ) be a Finsler manifold. For a vector field X on M, the following conditions are equivalent: (i) X is a homothetic vector field, i.e., LeX c g = α g, where α is a real number;
(ii) the energy function is an eigenfunction of X c with eigenvalue α, i.e., X c E = α E; (iii) LX c θ = α θ; e (iv) LeX c θe = α θ;
(v) LX c ω = α ω.
In conditions (iii) − (v) α is a real number. With the choice α := 0 we obtain criteria that a vector field X on M be a Killing vector field of (M, F ). Proposition 13. Let (M, F ) be a Finsler manifold. If a vector field X on M is both affine and conformal, then X c is a conformal vector ◦
field on the Riemannian manifold (T M, G), i.e., LX c G = ϕ G, where ◦
ϕ ∈ C 0 (T M) ∩ C 1 (T M) and G is the Sasaki extension of the metric tensor of (M, F ). ◦
Conversely, if X c is a conformal vector field of (T M, G), then X is a conformal vector field on the Finsler manifold (M, F ). 18
Proof. Suppose first that X is both an affine and a conformal vector field on (M, F ). Applying (4.6), (2.7) and Proposition 3/(vii), for any vector fields ◦
ξ, η on T M we have (LX c G)(ξ, η) = LX c (G(ξ, η)) − G(LX c ξ, η) − G(ξ, LX c η) = LX c (g(jξ, jη)) + LX c (g(Vξ, Vη)) − g(jLX c ξ, jη) − g(VLX c ξ, Vη) − g(jξ, jLX c η) − g(Vξ, VLX c η) = LeX c (g(jξ, jη)) + LeX c (g(Vξ, Vη)) − g(LeX c (jξ), jη) − g(LeX c (Vξ), Vη) − g(jξ, LeX c (jη)) − g(Vξ, LeX c (Vη)) = (LeX c g)(jξ, jη)
+ (LeX c g)(Vξ, Vη) = ϕg(jξ, jη) + ϕg(Vξ, Vη) = ϕG(ξ, η). ◦
This proves that X c is a conformal vector field on (T M, G). Conversely, under this condition we find that 2ϕ E = ϕ g(δ, δ) = ϕ g(VC, VC) = ϕ G(C, C) = (LX c G)(C, C) = X c (G(C, C)) − G([X c , C], C) − G(C, [X c, C]) = X c (G(C, C)) = X c g(δ, δ) = 2X c E, so, by Proposition 11, X is a conformal vector field on (M, F ). Proposition 14. Any homothetic vector field on a Finsler manifold is an affine vector field. Proof. Let (M, F ) be a Finsler manifold, and let S be the canonical spray for (M, F ). Suppose that X is a homothetic vector field of (M, F ). Then, by Corollary 12, there is a real number α such that X c E = α E, or, equivalently, LX c ω = α ω, so we have (4.5)
LX c dE = d(X c E) = α dE = −α iS ω = −iS (α ω) = −iS (LX c ω) = −LX c iS ω + i[X c ,S] ω = LX c dE + i[X c ,S] ω. Thus i[X c ,S] ω = 0, and hence – by the non-degeneracy of ω – [X c , S] = 0. This means that X is a Lie symmetry of the canonical spray of (M, F ). Lemma 15. If X is a conformal vector field on an n-dimensional Finsler manifold, then (with respect to the Dazord volume form) div X c = n ϕ, where ϕ is the conformal factor of X. Proof. Choose a local frame (Xi )ni=1 for T M over an open subset U of M. Then the family (Xiv , Xic )ni=1 is a local frame for T T M over τ −1 (U). It may be shown by a little lengthy inductive argument that (LX c ω)(X1v , X1c , ..., Xnv , Xnc ) = n ϕ ω(X1v , X1c , ..., Xnv , Xnc ), which implies our claim. 19
Proposition 16. If a vector field is both a projective and a conformal vector field on a Finsler manifold, then it is a homothetic vector field. Proof. Let (M, F ) be an n-dimensional Finsler manifold. Suppose that a vector field X on M is both projective and conformal. Then, on the one hand, [X c , S] = ψ C, ψ ∈ C 0 (T M) ∩ C 1 (T M), where S is the canonical spray of (M, F ). On the other hand, by Proposition 11, X c E = f v E, f ∈ C ∞ (M). Thus we get 2ψE = ψ(CE) = [X c , S]E = X c (SE) − S(X c E) = −S(f v E) = −(Sf v )E − f v (SE) = −f c E, taking into account that S is horizontal with respect to the canonical connection of (M, F ) and hence SE = 21 SF 2 = F (SF ) = 0 (see 4.3), applying furthermore the relation Sf v = f c (f ∈ C ∞ (M)), whose verification is routine. It follows that 1 ψ = − f c. 2 Now we determine the divergence (with respect to the Dazord volume form) of both sides of the relation [X c , S] = − 12 f c C. Applying the wellknown rules for calculation (see, e.g., [1], §6.5 or [11], XV,§1) we find that div[X c , S] = X c div S − S div X c
Lemmas
=
8, 15
−S(nf v ) = −nf c
and
1 1 Lemma 5 1 div(− f c C) = − (Cf c + f c div C) = − (n + 1)f c . 2 2 2 c c So (n − 1)f = 0, where n ≥ 2 (1.1 (a)), whence f = 0. This implies by the connectedness of M that f is a constant function, and therefore the conformal factor of X is constant. We note that this result is an infinitesimal version of Theorem 2 in [17]. Proposition 17. Let (M,F) be a Finsler manifold. Suppose that a vector field X on M preserves the Dazord volume form of (M, F ), i.e., LX c σ = 0. If, in addition, (i) X is a projective vector field, then X is affine; (ii) X is a conformal vector field, then X is isometric. 20
Proof. First we note that our condition LX c σ = 0 implies that div X c = 0. (i) Suppose that X is also a projective vector field, i.e., ◦
ψ ∈ C 0 (T M) ∩ C 1 (T M).
[X c , S] = ψ C, ◦
Observe that over T M the function ψ satisfies the relation C ψ = ψ. Indeed, by the Jacobi identity 0 = [C, [X c , S]] + [X c , [S, C]] + [S, [C, X c ]] = [C, [X c , S]] − [X c , S], hence [X c , S] = [C, [X c , S]] = [C, ψ C] = (Cψ)C, therefore (Cψ)C = ψ C, and so C ψ = ψ. Now, as in the previous proof, we calculate the divergence of both sides of the relation [X c , S] = ψ C. Since div X c = div S = 0, we have div[X c , S] = X c div S − S div X c = 0. On the other hand, by our above remark, div(ψ C) = ψ div C + C ψ = (n + 1)ψ. So it follows that ψ = 0, hence [X c , S] = 0. Thus X is an affine vector field on (M, F ). (ii) Now suppose that (div X c = 0 and) X is also a conformal vector field. Then, by Proposition 11, X c E = f v E, f ∈ C ∞ (M). Since n fv
Lemma
=
15
cond.
div X c = 0,
it follows that X c E = 0. Thus, by Corollary 12, X is an isometric vector field on (M, F ).
Acknowledgements This research was carried out in the framework of the Cooperation of the Czech and Hungarian Government (. . . ). The first author was supported also by Hungarian Scientific Research Fund OTKA No. NK 81402. The authors wish to express their gratitude to Bernadett Aradi, Dávid Cs. Kertész and Rezső L. Lovas for their useful comments and technical help during the preparation of the manuscript. 21
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József Szilasi Institute of Mathematics, University of Debrecen H-4010 Debrecen, P. O. Box 12, Hungary E-mail:
[email protected]
Anna Tóth Institute of Mathematics, University of Debrecen H-4010 Debrecen, P. O. Box 12, Hungary E-mail:
[email protected]
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