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Tangent Lie algebra of a diffeomorphism group and application to holonomy theory

In this paper we introduce the notion of tangent space TG of a (not necessary smooth) subgroup G of the diffeomorphism group Diff(M) of a compact manifold M. We prove that TG is a Lie subalgebra of the Lie algebra of smooth vector fields on M. The construction can be generalized to subgroups of any (finite or infinite dimensional) Lie groups. The tangent Lie algebra TG introduced this way is a generalization of the classical Lie algebra in the smooth cases. As a working example, we discuss in detail the tangent structure of the holonomy group and fibered holonomy group of Finsler manifolds.


Introduction
Important geometric objects, structures or properties can often be investigated through algebraic structures. In many interesting cases, these algebraic structures are groups, where the group operations are smooth maps. Such groups became indispensable tools for modern geometry, analysis, and theoretical physics. Lie groups and diffeomorphism groups are the most important examples for such structures.
Considering a Lie group G L , it is well known that most of the important information about it is captured in its tangent object, the Lie algebra g L . Naturally, if G is a Lie subgroup of G L , then its Lie algebra g is a Lie subalgebra of g L . The Lie subalgebra g ⊂ g L can be used to obtain information or eventually to determine the subgroup G. In many relevant geometric situations, however, this framework is not general enough because of two factor: Firstly, G L is not a (finite dimensional) Lie group but the (infinite dimensional) diffeomorphism group Diff ∞ (M ) of some manifold M . Secondly, the subgroup G is not necessarily a Lie subgroup of Diff ∞ (M ). Nevertheless, natural questions arise: can we introduce a tangential property and tangent objects to the subgroup G in this situation? Does the set of tangent elements possess a special algebraic stricture? Can this algebraic structure be used to get information about the subgroup and thus about geometric properties? In this paper we answer these questions.
We introduce the notion of tangent vector fields to a subgroup G of the diffeomorphism group Diff ∞ (M ). Denoting by T e G the set of tangent vector fields to G at the identity, we prove that T e G is a Lie subalgebra of the Lie algebra of smooth vector fields on M (Theorem 3.4). It follows that subalgebras of T e G inherit the tangential properties, therefore the elements of a subalgebra generated by vector fields tangent to the subgroup G are tangent to G (Corollary 3.6). This property can be particularly interesting when the Lie bracket of two tangent vector fields to G generates a new direction: the tangential property will be satisfied in this new direction as well. As we show in Theorem 3.10, the group generated by the exponential image of T e G is a subgroup of the closure of G in Diff ∞ (M ) which can give important information about the group G itself, especially in the infinite dimensional cases.
We note, that a similar tangential property was already introduced in [10, Definition 2], but we also remark that the concept had two major defects: the tangent property introduced in [10] is not preserved under the bracket operation, therefore in that approach it is not true that tangent vector fields to a subgroup G generate a tangent Lie algebra to G. Secondly, [10] was not able to guaranty the existence of the tangent Lie algebra T e G associated to G. We note that with our approach we are able to overcome both deficiencies.
The main reason to investigate the tangent structure of a subgroup G of the diffeomorphism group is that it can provide valuable information about the group G itself. This method can be very effective when G is for example a symmetry group, the holonomy group, etc. We note that in many cases the determination of T e G or its subalgebras can be highly nontrivial, especially in the infinite dimensional cases. As working examples, we consider the holonomy group and the fibered holonomy group of Finsler manifolds. The holonomy group is the transformation group generated by parallel translations with respect to the canonical connection along closed curves. For Riemannian manifolds it has been extensively studied and now the complete classification is known [2,1,3,6]. In particular, it is well known, that the holonomy group of a simply connected Riemannian manifold is a closed Lie subgroup of the special orthogonal group SO(n). Despite the analogues in the construction, Finslerian holonomy groups can be much more complex and up to now, we do not know much about them: For special spaces the holonomy can be a finite dimensional Lie group (see [17] and [7]), but recent results show that there are Finsler manifolds with infinite dimensional holonomy group [12,13,14]. These latter results show the difficulties: on cannot use the well understood principal bundle machinery in the investigation because the structure group should be infinite dimensional. In [9] P. Michor proposed a general setting for the study of infinite dimensional holonomy groups and holonomy algebras which was the motivation for Z. Muzsnay and P.T. Nagy to start investigating the tangent objects to a subgroup of the diffeomorphism group [10]. In this paper we are able to step forward: using the results of Chapter 3. we are able to introduce the notion of holonomy algebra and fibered holonomy algebra for Finslerian manifolds. By improving the results of [10] we also prove that the curvature and the infinitesimal holonomy algebras (resp. their restrictions) are Lie subalgebras of the fibered holonomy algebras (resp. the holonomy algebra). We are confident that in the future, the tools described above can be used successfully in the investigation of geometric structures in general and in the holonomy theory in particular.

Preliminaries
In this chapter we introduce the basic notions and concepts of Finsler geometry which are necessary to understand in Chapter 4.1 and Chapter 4.2 the nontrivial application of the theory discussing the tangent structure of a subgroup of the diffeomorphism group. These notions are not necessary to understand Chapter 3, therefore the reader who is not particularly interested in these applications, can jump directly to the next chapter.
In this paper, M denotes a C ∞ -smooth n-dimensional manifold, X ∞ (M ) is the Lie algebra of C ∞ vector fields and Diff ∞ (M ) is the group of C ∞ diffeomorphisms of M . We will denote by T M the tangent manifold and by T M = T M \{0} the slit tangent manifold. Local coordinate charts (U, x i ) on M induce local coordinate charts (π −1 (U ), (x i , y i )) on T M , where π : T M → M is the canonical projection. The vertical distribution VT M ⊂ T T M on T M is given by VT M = Ker π * .

Finsler manifold.
A Finsler manifold is a pair (M, F), where the norm F : T M → R + is a positively 1homogeneous continuous function, which is smooth onT M and the symmetric bilinear form Geodesics of (M, F) are determined by a system of second order ordinary differential equationsẍ i + 2G i (x,ẋ) = 0, i = 1, . . . , n in a local coordinate system ( Clearly, for any X 0 ∈ T c(0) M there is a unique parallel vector field X(t) along the curve c such that X 0 = X(0). Moreover, if X(t) is a parallel vector field along c, then λX(t) is also parallel along c for any λ ≥ 0. Then the homogeneous (nonlinear) parallel translation along a curve c(t) (3) P t c : T c 0 M → T ct M is defined by the positive homogeneous map P t c : X 0 → X t given by the value X t = X(t) of the parallel vector field with initial value X(0) = X 0 . We remark that (3) preserves the Finslerian norm, therefore it can be considered as a map between the indicatrices (4) P t c : I c 0 M → I ct M. Moreover, since the parallel translation is a homogeneous map, (3) and (4) determine each other.

Holonomy.
The holonomy group Hol p (M ) of a Finsler manifold (M, F ) at a point p ∈ M is the group generated by parallel translations along piece-wise differentiable closed curves starting at p. Considering the parallel translation (4) on the indicatrix, a holonomy element is a diffeomorphism P c : I p → I p , therefore the holonomy group Hol p (M ) ⊂ Diff ∞ (I p ) is a subgroup of the diffeomorphism group of the indicatrix I p .
In the particular case, when (M, F) is a simply connected Riemann manifold, the holonomy group is a closed Lie subgroup of the special orthogonal group SO(n). Finslerian holonomy groups can, however, be much more complex: in [12,13,14] one can find examples of Finsler manifolds with infinite dimensional holonomy groups. Until now it is not known if the Finsler holonomy groups are (finite of infinite dimensional) Lie groups or not.

Horizontal lift and curvature.
The parallel translation on a Finsler manifold can also be introduced by considering the associated Ehresmann connection (cf. [18]): the horizontal distribution is determined by the horizontal lift T x M → T (x,y) T M defined in the local basis as Then the parallel translation can be geometrically obtained as P t c (X 0 ) = c h (t). We remark that the horizontal lift ϕ h t of the flow ϕ t of a vector field X ∈ X ∞ (M ) is the flow of the horizontal lift of the vector field X h ∈ X ∞ (T M ). Therefore the parallel translation along the integral curves of X can be calculated in terms of the horizontal lift of the flow: The horizontal distribution HT M is, in general, non-integrable. The obstruction to its integrability is given by the curvature tensor R = 1 2 [h, h] which is the Nijenhuis torsion of the horizontal projector [4].

Tangent Lie algebra of a subgroup of the diffeomorphism group
In this paragraph we investigate the tangential property and tangential structure of subgroups of the diffeomorphism group. Let G be a subgroup of Diff ∞ (M ) where M is a differentiable manifold. We do not suppose any special property on G, in particular, we do not suppose that G is a Lie subgroup of Diff ∞ (M ). Questions that we consider: can we introduce a tangential property and tangent object to the subgroup G? Does the set of tangent elements possess a special algebraic structure? Can this algebraic structure be used to get information about the subgroup? In this paragraph, we answer all these questions.
A smooth curve c : I → M on the manifold M has a (k−1) st -order singularity at t = 0, if its derivatives vanish up to order k−1, (k ≥ 0). It is well known that if a curve c has a (k−1) st -order singularity at 0 ∈ R then its k th order derivative c (k) (0) = X p is a tangent vector at p = c(0). In that case, the curve c is called a k th -order integral curve of the vector X p ∈ T p M . Extending this concept to vector fields, we can introduce the following there exists k ∈ N such that for any point p ∈ M the curve t → ϕ t (p) is a k th -order integral curve of X(p) ∈ T p M. This k ∈ N is called the order of the integral curve ϕ t of the vector field X.
In particular, the flow ϕ X t of X ∈ X ∞ (M ) is a 1 st -order integral curve of X. Moreover, if k > 1 and t → ϕ t is a k th −order integral curve of the vector field X then we have Using the terminology of Definition 3.1 we introduce the following of the diffeomorphism group if there exists an integral curve of X in G. The set of tangent vector fields of G is denoted by T e G.
there exists k ∈ N such that equation (7) is satisfied.
One can observe that in Definition 3.2 we do not suppose that G is a Lie subgroup of Diff ∞ (M ). Indeed, we use the differential structure of the later to formulate the smoothness condition on the curve in G. Nevertheless, we have the following Proof. To prove the theorem, we have to show that According to Definition 3.1, there exist k, l ∈ N such that ϕ t , ψ t ∈ G are integral curves of X and Y respectively. Let us suppose that ϕ t is a k th −order integral curve of X and ψ t is an l th -order integral curve of Y (k, l ≥ 1). Then • Proof of (8a). The computation is similar to that of [8]: Considering the group theoretical commutator we get a two-parameter family of diffeomorphisms such that if one of the parameters s or t is zero then (11) is the identity transformation. From (9) and (10) we also know that the first, potentially nonzero derivative is the (k + l) th order mixed derivative: where d ϕ −1 s ϕs(p) denotes the tangent map (or Jacobi operator) of ϕ −1 s at the point ϕ s (p). Since d ϕ −1 s=0 ϕs(p) = id, the above formula can be written in the form which yields (14) ∂ k (ϕ −1 t ) ∂t k t=0 = −X. Therefore we get that (13) can be written as (15) d ∂ l ϕ s ∂s l s=0 p which is the Lie bracket of the vector fields X and Y , that is From (16) we get that t → [ϕ t , ψ t ] is a (k+l) th -order integral curve of [X, Y ] in G. Therefore [X, Y ] ∈ T e G which proves (8a).
For any c 1 , Moreover, if r = denotes the least common multiple of k and l and showing that ψ t is an r th -order integral curve of X + Y in G, therefore X + Y is tangent to G.
It is clear that in the case when λ ≥ 0, one can reparametrize the integral curve of X, and using that the lower order terms are zero, we get In the case when λ < 0 one can use (14) and we get From (21) and (22) we get that λX is tangent to G, that is λX ∈ T e G, and from 11b) and 11c) we get that any linear combinations of X and Y are in T e G.
Motivated by the results of Theorem 3.4 we propose the following Definition 3.5. T 0 G is called the tangent Lie algebra of the subgroup G ⊂ Diff ∞ (M ).
As a direct consequence of Theorem 3.4 we get the following Corollary 3.6. Let G be a subgroup of Diff ∞ (M ) and S be a subset of X ∞ (M ) such that the elements of S are tangent to G. Then the Lie subalgebra S Lie of X ∞ (M ) generated by the elements of S is also tangent to G, that is (i) if X is strongly tangent to G, then X ∈ T e G.
(ii) if X ∈ T e G, then it is weakly tangent to G.
Proof. (i) If X ∈ X ∞ (M ) is a strongly tangent vector field to the subgroup G ⊂ Diff ∞ (M ), there exists a k-parameter commutator like family of diffeomorphisms φ t 1 ...t k ∈ G which is C ∞ -smooth in Diff ∞ (M ), φ t 1 ,...,t k = id M whenever one of its parameters is zero and Consequently, if we consider the map t → ϕ t where ϕ t = φ t,...,t , we get a 1-parameter family of diffeomorphisms which satisfies the conditions of Definition 3.2. Therefore, the vector field X is tangent to G.
To prove (ii), let us suppose that ϕ t is a k th order integral curve of X. Then we have (9) and one can write ϕ t (p) as which proves (ii).
Remark 3.9. One may wonder why to introduce a new tangent property when there are already two, the weak and the strong tangent property (using the terminology of Remark 3.7) introduced in the literature. As an answer we point out that, the concept in [10] has a major defect: the weak tangent property is not preserved under the bracket operation, therefore it is not true in general that weakly tangent vector fields to a subgroup G generate a weakly tangent Lie algebra to G. To overcome this difficulty, the authors introduced the strongly tangent property but the strongly tangent property was not preserved under the linear combination. It follows that [10] and the succeeding papers using these techniques were not able to guaranty the existence of the tangent Lie algebra T e G associated to G. With our approach we are able to overcome this major deficiency.
The main feature of T e G is that one can obtain information about the group G. Indeed, one has the following Theorem 3.10. Let G be a subgroup of Diff ∞ (M ) and G its topological closure with respect to the C ∞ topology. Then the group generated by the exponential image of the tangent Lie algebra T e G with respect to the exponential map exp : Proof. From the proof of Proposition 3.8 we know that for any element X ∈ T e G there exists a C 1 -differentiable 1-parameter family {ψ t } ⊂ G of diffeomorphisms of M such that ψ 0 = id M and X = ∂ψt ∂t t=0 . Then, using the argument of [15, Corollary 5.4, p. 84] on ψ t we get that ψ n t n = ψ t n • · · · • ψ t n in G, as a sequence of Diff ∞ (M ), converges uniformly in all derivatives to exp(tX). It follows that for any X ∈ T e G. Therefore, one has exp (T e G) ⊂ G and if we denote by exp(T e G) the group generated by the exponential image of T e G we get exp(T e G) ⊂ G, which proves Theorem 3.10.
The concept worked out in Definition 3.2 and Theorem 3.4 can be adapted not only for subgroups of the diffeomorphism group but for any subgroup of any (finite or infinite dimensional) Lie group: Definition 3.11. Let G L be a Lie group, e ∈ G L is the unit element of G L and g L := T e G L the Lie algebra of G L . If G ⊂ G L is a subgroup of G L , then X ∈ g L is called tangent to G if there exist a C ∞ −smooth curve ϕ : there exists k ∈ N such that t → ϕ t is a k th order integral curve of X. The set of tangent vector of G is denoted by T e G.
Then, adapting the proof of Theorem 3.4 and Theorem 3.10 we can get the following Theorem 3.12. If G is a subgroup of a Lie group G L , then T e G is a Lie subalgebra of g L . The group exp(T e G) generated by the exponential image of T e G with respect to the exponential map exp : g L → G L is a subgroup of the topological closure G of G in G L .
It is clear that in the case when G is a Lie subgroup of G L , then T e G = g is just the usual Lie subalgebra of g L associated to the Lie subgroup G. Therefore Definition 3.11 generalizes the classical notion of the Lie subalgebra associated to a Lie subgroup.

Application: the holonomy algebra
The notion of the holonomy group was already introduced in Chapter 2.3. It is well known that in the particular case when (M, F) is a Riemann manifold, the holonomy group is a compact Lie subgroup of the orthogonal group O(n) and its Lie algebra is a Lie subalgebra of o(n). It is also clear that the holonomy group of a linear connection is a subgroup of the linear group GL(n) and its Lie algebra is a Lie subalgebra of gl(n). However, the situation for a Finsler manifold or in a more general context the holonomy of a homogeneous connection can be much more complex. Examples show that in some cases the holonomy group can not be a finite dimensional Lie group [11,12,13]. Until now it is not known if the Finsler holonomy groups are (finite of infinite dimensional) Lie groups or not. Nevertheless, the theory developed in Chapter 3 allows us to consider its tangent Lie algebra, the holonomy algebra.

The fibered holonomy algebra and its Lie subalgebras.
Let (M, F) be a Finsler manifold. The notion of fibered holonomy group Hol f (M ) appeared in [10]: It is the group generated by fiber preserving diffeomorphisms Φ of the indicatrix bundle (IM, π, M ), such that for any p ∈ M the restriction Φ p = Φ| Ip is an element of the holonomy group Hol p (M ). It is obvious that   In the sequel we will investigate the two most important Lie subalgebras of hol f (M ) which can be introduced with the help of the curvature tensor (see Paragraph 2.3) of a Finsler manifold: the curvature algebra and the infinitesimal holonomy algebra.
The Lie subalgebra R of vector fields generated by curvature vector fields is called the curvature algebra.
It is easy to see that from the definition of the curvature tensor that a curvature vector field can be calculated as and from the definition we have also R ⊂ X ∞ (IM ). Moreover, we have the following To prove the first part of the proposition, we have to show that the curvature vector fields are tangent to the fibered holonomy group Hol f (M ), that is they are elements of hol f (M ). Let ξ ∈ X ∞ (IM ) be a curvature vector field and X, Y ∈ X ∞ (M ) such that ξ = R(X h , Y h ). We denote by ϕ and ψ the integral curves of X and Y respectively. Define 3t ≤ s ≤ 4t.
and β t,s := ψ −s ϕ −s ψ s ϕ s , 0 ≤ s ≤ t. Then, for every p ∈ M and fixed t the map α t (p) : s → α t,s (p) and β t (p) : s → β t,s (p) are parametrized curves: α t (p) : s → α t,s (p) is a (not necessarily closed) parallelogram and β t (p) joins the endpoints of α t (p). Indeed, for every p ∈ M and fixed t the endpoint of α t (p) coincides with the endpoint of β t (p) and consequently the curve α t (p) * β −1 t (p) defined as going along the curve α t (p) then continuing along β −1 t (p) (which is the curve β t (p) with opposed orientation) is a closed curve that starts and ends at p ∈ M . Let us consider (24) h t,p := P αt(p) * β −1 t (p) = P αt(p) • P −1 βt(p) , the parallel translation along α t (p) * β −1 t (p). We have the following Lemma 4.5. For any p ∈ M (1) h t,p ∈ Hol p (M ), (2) t → h t,p is a second order integral curve of the vector field ξ p := ξ Ip ξ p ∈ X ∞ (I p ) .
Proof. Indeed, for every p ∈ M and sufficiently small t the curve α t (p) * β −1 t (p) is a closed loop starting and ending at p, therefore the parallel transport h t,p : I p → I p is a holonomy transformation at p and we get (1) of the lemma.
To show (2) we first remark that α 0 (p) and β 0 (p) are the trivial curves (s → α 0,s (p) = β 0,s (p) ≡ p), therefore the parallel translation along them is the identity transformation and On the other hand, as we have seen in Chapter 2, the parallel transport along a curve is determined by the horizontal lift of the curve. Consequently, the parallel transport along the integral curves of the vector fields X and Y can be expressed with the flows of the horizontal lifts X h and Y h . Let us consider first the parallel transport along the curve α t (p): the parallel transport of a vector v ∈ I p along the curve α t (p) is Therefore, P αt(p) corresponds to the infinitesimal (not necessarily closed) parallelogram having as sides the integral curves of the horizontal lifts X h and Y h . From the well known properties of the Lie brackets (see for example [16, p.162]) we get that On the other hand, the parallel transport of a vector w ∈ I αt(p) along β −1 t (p) can be calculated with the help of it's horizontal lift P β −1 h v , thus, from the two equations of (26) and the two equations of (27) we get where we also used (23). To summarize, we get from (25) and (28): which means that the reparametrized map t → h t/ √ 2,p is a second order integral curve of the curvature vector field ξ p ∈ X ∞ (I p ) and proves point (2) of the lemma.
t → h t is a second order integral curve of the vector field ξ ∈ X ∞ (I). which shows that the curvature vector field ξ is tangent to Hol f (M ) and proves the first part of the proposition. Applying Corollary 3.6, we get that the Lie algebra generated by the curvature vector field is tangent to Hol f (M ) which proves the second part of the proposition.
We remark that (1) of Proposition 4.4 is an improvement of Proposition 3. and Corollary 2. of [10]. Indeed, the tangent property proved in [10] is weaker: C 1 instead of C ∞ smoothness. Moreover, [10] uses a very strong topological restriction on the manifold M supposing it is diffeomorphic to the n-dimensional euclidean space. In Proposition (4.4) we presented a natural geometric construction without any constraints on the topology of the manifold M . We have the following We remark that the first part of Proposition 4.7 is an improvement of [10, Theorem 2], because in Proposition 4.7 the strong topology condition on the manifold M is dropped.

Holonomy algebra and its Lie subalgebras.
Let (M, F ) be an n−dimensional Finsler manifold. At any points p ∈ M the indicatrix defined in (1) is an (n − 1)-dimensional compact manifold in T p M . Therefore, the diffeomorphism group Diff ∞ (I p ) is an infinite dimensional Fréchet Lie group whose Lie algebra is X ∞ (I p ), the Lie algebra of smooth vector fields on I p . As it was introduced in Chapter 2.3, the holonomy group  In the sequel we identify two important Lie subalgebras of the holonomy algebra of Finsler manifolds.
Definition 4.10. A vector field ξ p ∈ X ∞ (I p ) on the indicatrix I p ⊂ T p M is called a curvature vector field at p ∈ M if there exist tangent vectors X p , Y p ∈ T p M such that ξ p = R(X h p , Y h p ). The Lie subalgebra R p of vector fields generated by curvature vector fields at p ∈ M is called the curvature algebra at p.
The relationship between the curvature algebra R p at p ∈ M and the curvature algebra R introduced in Definition 4.3 is: that is R p is the restriction of R to the indicatrix I p . We have Proposition 4.11. The elements of the curvature algebra R p at p ∈ M are tangent to the group Hol p (M ) and the curvature algebra R p is a Lie subalgebra of the holonomy algebra hol p (M ).
The proof is a direct consequence of the computation of Proposition (4.4). Moreover, by localizing the infinitesimal holonomy algebra at a point we can obtain We note that by the construction of the infinitesimal holonomy algebra, the curvature vector fields are elements of hol * p (M ), therefore we have the sequence of the Lie algebras (31) . We also remark that the first parts of the statement of Proposition 4.11 and 4.13 are improvements of the results of [10] because the tangential property of the Lie algebra is improved: we can guaranty C ∞ -smoothness instead of C 1 -smoothness.

Concluding remarks.
Many interesting geometric results can be obtained on the holonomy structure from the Lie algebras (31) through the tangent property. Indeed, by using Theorem 3.10, one can find examples where, in contrast to the Riemannian case, the holonomy group Hol p (M ) is not a compact Lie group [11,12,13], or where the closure of the holonomy group is the infinite dimensional Lie group Diff ∞ + (I p ) of the orientation preserving diffeomorphism group of the indicatrix [5,14]. All these results were obtained by using the tangent property of the curvature algebra R p (M ) and the infinitesimal holonomy algebra hol * p (M ). The method developed in Chapter 3, however, allows us to introduce in a natural and canonical way a potentially larger Lie algebra, the holonomy algebra, which is the tangent Lie algebra of the holonomy group. This Lie algebra gives the best linear approximation of the holonomy group. The technique can be applied in other fields of geometry as well. We are convinced that the method, exploring the tangential property of a group associated with a geometric structure, can be used successfully to investigate various geometric properties.