On Lie algebras associated with a spray

The Lie algebra of infinitesimal isometries of a Riemannian manifold contains at most two commutative ideals. One coming from the horizontal nullity space of the Nijenhuis tensor of the canonical connection, the other coming from the constant vectors fields independent of the Riemannian metric.


Introduction
Let M be a paracompact differentiable manifold of dimension n ≥ 2 and of class C ∞ , S a spray on M .To study the Lie algebra of projectable vector fields which commute with S, which will be denoted by A S cf.[7], we have associated with S the vector 1−form J defining the tangent structure on M .The vector 1−form Γ = [J, S] is considered as a connection in the sense of [4].In [1], we have shown that the elements of A S , belonging to the horizontal nullity space of the curvature R of Γ, form a commutative ideal of A S .In [2], we found that some constant elements of A S can constitute a commutative ideal.In the present study, we show that there are two possible commutative ideals.A Lie algebra of infinitesimal isometries A g of a Riemannian manifold is semi-simple if and only if the horizontal nullity space of the Nijenhuis tensor of Γ is zero and that the derived ideal [A g , A g ] coincides with A g .To illustrate our results, we give some examples of A S and A g .14 Manelo Anona and Hasina Ratovoarimanana

Preliminaries
Let K and L be two vector 1−form on a manifold M , χ(M ) the set of vector fields on M .The bracket [K, L] cf.[3] is written for all X, Y ∈ χ(M ).
The bracket The exterior derivation d L is defined by where i L the inner product with respect to L.
Let Γ be a connection in the sense of [4].The vector 1−form Γ is an almost product structure (Γ 2 = I, I being the identity vector 1−form).Noting The vector 1−form h is the horizontal projector corresponding to the eigenvalue +1, and v the vertical projector for the eigenvalue −1.The curvature of Γ is defined by The nullity space of the curvature R is: In local natural coordinates on an open set U of M , (x i , y j ) are the coordinates on T U , a spray S is written Let χ(M ) denote the complete lift on the tangent bundle T M of χ(M ) on M .The projectable elements of A S are in χ(M ).According to Jacobi's identity cf.[3], we can write with the vector 1−form J defining the tangent structure of M , On Lie algebras associated with a spray 15 By hypothesis we have [X, S] = 0 and, [J, X] = 0 according to a result of [6], we obtain Thus the connection Γ is linear and without torsion according to [4].For a connection Γ = [J, S], the coefficients of Γ become Γ j i = ∂G j ∂y i and the projectors horizontal and vertical are for each i, j, k, l ∈ {1, . . ., n} As the functions G k are homogeneous of degree 2, the coefficients Γ k ij = ∂ 2 G k ∂y i ∂y j do not depend on y i , i ∈ {1, . . ., n}.We then have , the R k l,ij (x) depend only on the coordinates of the manifold M .
3 The horizontal elements of A Γ Proposition 3.1.The elements of A Γ are projectable vector fields.
Proof.A vector field X ∈ A Γ means [X, Γ] = 0.By definition, the horizontal projector h If Y is a vertical vector field, we find hY = 0. the above relation becomes h[X, Y ] = 0 for any vertical vector field, i.e.X is a projectable vector field.
We will denote in the following by H • the set of horizontal and projectable vector fields.

Proposition 3.2 ([1]
).Let X be a projectable vector field.The following two relationships are equivalent: According to propositions 3.2 and 2 of [1], we have For the existence of such an element of A Γ h , we must have h(X) = X, that is to say The system of equations to be solved becomes The compatibility condition of such a system of equations according to the Frobenius theorem is is a module over the smooth functions of M and involutive.On the integral sub-manifold defined by A h Γ , the system (1) admits solutions in the number of the dimension of the said sub-manifold.

Lie algebras of infinitesimal isometries
Let E be an energy function, a function T M = T M − {0} into R + , with E(0) = 0, of class C ∞ on T M , of class C 2 on the null section, and homogeneous of degree two such that the 2− form Ω = dd J E being of maximum rank.The canonical spray S is defined by the derivation i S is the inner product with respect to S. The 2− form Ω defines a Riemannian metric g on the vertical bundle: for all X, Y ∈ χ(T M ).With a natural local coordinate system on an open set U of M , (x i , y j ) are the system of coordinates on T U , the function E is written where g ij (x 1 , . . ., x n ) are positive functions such that the symmetric matrix whose (i, j)-th entry is Definition 4.1.A vector field X on a Riemannian manifold (M, E) is called an infinitesimal automorphism of the symplectic form Ω if L X Ω = 0, where L X is the Lie derivative with respect to X.
We notice that the canonical spray S of (M, E) is an infinitesimal automorphism of the symplectic form Ω. The set of infinitesimal automorphisms of Ω forms a Lie algebra.We denote this Lie algebra by A g , even formed by the projectable vector fields, it is in general of infinite dimension cf.[1].

Proposition 4.2 ([1]
).Let A g defined as A g ∩ χ(M ), we have a) X ∈ A g if and only if X is a projectable vector field such that X ∈ A g and c) The elements of A g are Killing fields of the projectable vectors of the metric g belonging to A Γ .The dimension of A g is at most equal to n(n+1) 2 .
Proposition 4.3.On a Riemannian manifold (M, E), the horizontal nullity space of the curvature R is generated as a module by the projectable vector fields belonging to this nullity space and, orthogonal to the image space ImR of the curvature R.
Proof.If the potential R • = i S R is zero, then the curvature R is zero, in this case the horizontal space Imh is the horizontal nullity space of the curvature R, isomorphic to χ(U ), U being an open set of M .
In the following, we assume that R • = 0.The vertical vector field JX is orthogonal to the image ImR of the curvature if and and only if the curvature of the connection D of Cartan R(S, X)Y = 0 ∀Y ∈ χ(T M ) cf. [1].We obtain R(X, Y As R is a semi-basic vector 2−form, the above relation is only possible if X = S or X ∈ hN R , then X is generated as a module by projectable vector fields in hN R .
Theorem 4.4.The Lie algebra A g is semi-simple if and only if the horizontal nullity space of the Nijenhuis tensor of Γ is zero and, the derived ideal [A g , A g ] coincides with A g .
Proof.If the Lie algebra A g is semi-simple, any commutative ideal of A g is zero by definition.According to propositions 3.3 and 4.3 the horizontal nullity space of the Nijenhuis tensor of Γ is zero.The derived ideal [A g , A g ] coincides with A g according to a classical result.Conversely, if X ∈ A Γ , we have [X, h] = 0.According to the identity of Jacobi cf.
If the horizontal nullity space of the curvature R is zero, the semi-basic vector 2−form R is non-degenerate.The only possible case for equation ( 2) is that the commutative ideal of A g is at most formed by constant vector fields ∂ ∂x i , i ∈ {1, . . ., n} such that ∂G k ∂x i = 0 for all k ∈ {1, . . ., n} from [2].These constant vector fields can only form an ideal of affine vector fields independent of the other elements of A g .The derived ideal [A g , A g ] never coincides with A g .Hence the result.
Example 4.5.We take M = R 3 and the energy function is written: The canonical spray of E is written: The non-zero coefficients of Γ are The horizontal fields are generated as a module by The horizontal nullity space of the curvature is zero.The Lie algebra A Γ is generated as Lie algebra by: The Lie algebra A Γ = A g is simple.
Example 4.6.We take M = R 4 and the energy function is written: The canonical spray of E is written: The non-zero coefficients of Γ are The horizontal fields are generated as a module by The horizontal nullity space of the curvature is zero.The Lie algebra A Γ is generated as Lie algebra by: 5 An example of a Lie algebra A S of maximal rank n 2 + n with a Lie algebra A g of maximal rank n 2 +n 2 of different natures.
We take M = R 3 and the energy function is The canonical spray of E is written: The non-zero coefficients Γ j i of Γ are The basis of the horizontal space of Γ is written The curvature R is zero.The Lie algebra A S is generated as Lie algebra by: All derivations of A S are inner and, {e 3 , e 7 , e 12 } forms the commutative ideal of A S and include in the horizontal nullity space of the curvature.
The Levi decomposition of A S is  The Levi decomposition of A g is such that A g s = {g 1 , g 3 , g 5 } ∼ = so(3) is simple , A g i = {g 2 , g 4 , g 6 } is solvable.Hence, the Lie algebra A g is not semi-simple.

[
X, Γ] = 0, with Γ = [J, S].Let C denote the Liouville field on the tangent bundle T M , the homogeneity of X ([C, X] = 0) leads us to study A S taking S such that [C, S] = S, taking into account [C, J] = −J.

g 1 = e x 2 −x 1 2 ( ∂ ∂x 1 −(y 1 − y 2 ) 2 ∂ ∂y 1 ) − e x 1 D(g 2 ) = g 2 D 2 0Table 4 :
, A g ] = A g .The derivations of A g are inner except for the derivation (g 4 ) = g 4 D(g 6 ) = g 6 which is outer and, {g 2 , g 4 g 6 } forms the commutative ideal of A g and include in the horizontal nullity On Lie algebras associated with a spray 23 [•, •] g 1 g 2 g 3 Multiplication table of A g space of the curvature.[A g , A g ] = A g .

Table 1 :
Multiplication table of A Γ

Table 2 :
y 1 ) 2 + e x 2 (y 2 ) 2 + e x 3 (y 3 ) 2 ) Multiplication table of A Γ in Example 4.6 e 2 , e 4 , e 5 , e 6 − e 11 , e 8 , e 9 , e 10 } is simple and A S r = {e 1 + e 6 + e 11 , e 3 , e 7 , e 12 } is solvable.Hence, the Lie algebra A S is not semi-simple.The Lie algebra A g is generated as Lie algebra by