ON SOME INFINITESIMAL AUTOMORPHISMS OF RIEMANNIAN FOLIATION

In Riemannian foliation, a transverse affine vector field preserves the curvature and its covariant derivatives. In this paper we solve the converse problem. Actually, we show that an infinitesimal automorphism of a Riemannian foliation which preserves the curvature and its covariant derivatives induces a transverse almost homothetic vector field. If in addition the manifold is closed and the foliation is irreducible harmonic , then a such infinitesimal automorphism induces a transverse killing vector field. Subjclass : 57R30; 53C12.


Introduction
It is well-known that an affine vector field on Riemannian manifold preserves the curvature tensor and its covariant differentials.Using the definition of a transverse affine vector field on a Riemannian manifold endowed with a Riemannian foliation, we can easily show in the same way that an affine infinitesimal automorphism of the foliation preserves the curvature and its covariant derivatives.
In [10] the authors discussed the inverse problem and they used the decomposition of de Rham to prove that a vector field which preserves the curvature and its covariant derivatives is homothetic.
In this paper we use the basic connection [11] and the Blumenthal decomposition of a Riemannian foliation [3] to extend the Nomizu-Yano theorem to the Riemannian foliation case building on their proof idea.We show that an infinitesimal automorphism of Riemannian analytic foliation which preserves the connection and its covariant derivatives is a transverse almost homothetic.
The following theorems are the main results of this work.
Theorem 1.Let F be an irreducible analytic harmonic g M -Riemannian foliation of codimension ≥ 2 on a closed analytic Riemannian manifold (M, g M ), and let X be an infinitesimal automorphism of F such that Θ(X)∇ m R = 0 for all m ∈ N.
Theorem 2. Let F be an analytic g M -Riemannian foliation without Euclidean part on an analytic Riemannian manifold (M, g M ), and let X be an infinitesimal automorphism of F such that Θ(X)∇ m R = 0 for all m ∈ N.
The paper is organized as follows.In section 2 we recall some definitions and we give some examples.In section 3 we introduce a transverse tensor field of type (1,2) measuring the deviation of transverse vector fields to be transverse affine and we provide some preliminary results.Section 4 is devoted to some integral formulas.Moreover, we prove that on a closed Riemannian manifold endowed with harmonic Riemannian foliation any transverse affine vector field is a transverse Killing.In section 5 we prove some preliminaries theorems.The proofs of the main theorems are given in section 6.

Adapted connection
Let (M, g M ) be a Riemannian connected manifold of dimension n with a foliation F of codimension q.The foliation F is given by an integrable subbundle E of the tangent bundle T M over M .Let E ⊥ denote the orthogonal complement bundle of E, and let Q indicate the normal bundle T M/E.The bundle T M splits orthogonally as T M = E ⊕ E ⊥ with the map σ : Q −→ E ⊥ ⊂ T M splitting the following exact sequence: Let ∇ M be the Riemannian connection on (M, g M ).We can define an adapted connection ∇ in Q by putting , where Γ (L) denotes the space of all sections of the bundle L.
The torsion T ∇ of ∇ is given by for all X, Y ∈ Γ(T M), and the curvature R ∇ of ∇ is defined by for all X, Y, Z ∈ Γ(T M).

Q-tensor field, Lie-differentiation and covariant differential
We recall [7] that a Q-tensor field K of type (0, r) We denote by V(F) the space of all infinitesimal automorphisms of F. For X ∈ V(F), the Lie-differentiation Θ(X) with respect to X is defined by for all s ∈ Γ(Q), for any Q-tensor field K of type (0, r) or (1, r), and any s 1 , ..., s r ∈ Γ(Q).
Let K be a Q-tensor field of type (0, r) or (1, r).The covariant derivative of K is defined by The tensor K is called holonomy invariant (or parallel along the leaves of F) if Θ(X)K = ∇ X K = 0 for all X ∈ Γ(E).
On the other hand, An elimination process leads then to the formula (2.1). ii ] and ∇ X K = 0, we obtain is said to be transverse homothetic (resp., transverse Killing).The section π (X) is said to be transverse almost homothetic if the normal bundle Q can be decomposed into a direct sum is the restriction of the metric g Q to the subbundle Q i and c i is a real constant.Finally, π (X) is referred to us a transverse affine section if X preserves the connection ∇ in Q, that is, Θ(X)∇ = 0.

Riemannian foliation and holonomy group
From now on, we suppose F to be a g M -Riemannian foliation (i.e., the metric g M is bundle-like in the sense of Reinhart).So the induced metric g Q is holonomy invariant.
Recall from [11] that where

Proof.
Suppose that for m ≥ 1 the Q-tensor fields K, ..., Accordingly, ∇ m K is also holonomy invariant.2 As an immediate consequence of Proposition 2, we get the following.
The proof of the next Bianchi identities are routine and therefore omitted.
Proposition 3. The curvature tensor R satisfies the first Bianchi identity and satisfies the second Bianchi identity Let x ∈ M and let C(x) be the space of all loops at x.For each τ ∈ C(x), the parallel transport along τ is an isometry of Q x .The set of all such isometries of Q x is the holonomy group Ψ(x) of ∇ with reference point x.Let C 0 (x) be the subset of C(x) consisting of loops which are homotopic to zero.The subgroup of Ψ(x) consisting of the parallel transport along τ ∈ C 0 (x) is the restricted holonomy group Ψ 0 (x) of ∇ with reference point x.We say that F is irreducible (reducible) if the action of Ψ(x) on Q x is irreducible (reducible) see [3].It is obvious that if F is irreducible then the normal bundle Q does not have a connection invariant proper subbundle.We say that F is without Euclidian part if Ψ(x) has no non zero fixed vector.
As in [7], we may get quickly that if M is analytic and F is an analytic Riemannian foliation, then the restricted holonomy group Ψ 0 (x) is completely determined by the values of all successive covariant differentials ∇ m R, m = 0, 1, 2, ..., at the point x.
Example 1. (i) If M is closed and F is a Lie R q -foliation of codimension q ≥ 2, then F is defined by independent closed one-formes ω 1 , ..., ω q [6].So F has Euclidian part and hence it is reducible.
(ii) If M is closed with π 1 (M ) abelian and F is a one-dimensional Euclidean foliation (i.e.Riemannian tranversally affine), then F is reducible [4].
In the sequel we assume that M is a closed manifold and F is a codimension two Euclidean foliation.In this case, we have the following equivalences For the following results, see [1].
(iii) If all the leaves of F are simply connected, then F is reducible.
-foliation with trivial normal bundle, that is defined by independent one-forms ω 1 , ω 2 satisfying Then F is irreductible.
(viii) We say that F is Riemannian tranversally almost parallelisable foliation [C] if there is a G-reduction P of the normal frame bundle F (Q) compatible with the foliation, where G is a discrete Lie subgroup of the orthogonal group O(q, R).If M is a connected component of P, then the bundle projection p : M −→ M is a connected covering space with G as the group of deck transformations such that e F = p −1 (F) is tranversally parallelisable (e-foliation) [5].Moreover, if F is two codimentional, then e F is a Lie R 2 -foliation.On the other hand F is tranversally almost parallelisable, if and only if the basic connection

Some computational results
Let X ∈ V(F), Y ∈ Γ(T M) and s ∈ Γ(Q), so we have Then the operator measures the deviation of ν = π(X) of being transverse affine (i.e.X preserves the connection ∇).
for all Y ∈ Γ(E), see [9], then K is a semi-basic form in the sense that i Y K = 0. Consequently, for s ∈ Γ(Q), the operator is a well defined endomorphism on Γ(Q), that is K is a Q−tensor field of type (1, 2) on M .On the other hand, since T ∇ = 0, then we have where A X is a Q-tensor field of type (1,1) on M defined by Proposition 4. We have the following properties, i) A X is holonomy invariant, ii) K is holonomy invariant.
Proposition 5.The Q-tensor field K has the following properties.
i) for any s, t ∈ Γ(Q) and Y ∈ Γ(M ) we have ii) for any s, t ∈ Γ(Q) we have iii) if ν is a transverse conformal, then for s ∈ Γ(Q) we have iv) for all s ∈ Γ(Q) we have Proof.i) Let s, t ∈ Γ(Q), according to (3.4) and (2.3) we have Consequently, from (2.4) we obtain because X ∈ V (F).It follows from (3.5), (3.15) and the first Bianchi identity (2.5) that By taking the covariant differential ∇ Y of (3.16) and using (3.16) again, we obtain the relation (3.10).ii) Let s, t ∈ Γ(Q), we take the covariant differential of (3.5) and use (3.4) to obtain Since A X is holonomy invariant, then we have On the other side let ν ∈ Γ(Q), so So by virtu of (3.17), (3.18), (3.19), (3.3) and the second Bianchi identity (2.6) we get Hence we have the formula (3.11).

Transverse affine vector field and harmonic Riemannian foliation
In this section we generalize some classical results on Riemannian manifolds to the Riemannian foliation case.

Harmonic foliation
For unexplained notation and terminology, we refer the reader to [8.11].
Let (E i ) 1≤i≤n be a local orthonormal frame of T M such that E i ∈ Γ(E) for 0 ≤ i ≤ p, and E i ∈ Γ(E ⊥ ) for p + 1 ≤ i ≤ n, where p + q = n.For 1 ≤ i ≤ q let e i = π(E p+i ), so σ(e i ) = E p+i and the family (e i ) 0≤i≤q is a local orthonormal frame of Γ(Q).Let X ∈ Γ(T M) and s ∈ Γ(Q), the classical divergence operator with respect to the connection ∇ M is defined Similarly, the transverse divergence operator div ∇ with respect to ∇ is defined by The tension field τ of the foliation F is defined by It's easily seen that the following equation holds for all s ∈ Γ(Q).Proposition 6.Let X ∈ V(F) and ν = π(X), then div ∇ ν is a basic function.The following equation

holds.
Proof.i) Let Y ∈ Γ(E), first we note the following two points, 1) since ∇ Y ν = π([Y, X]) = 0 and i Y R ∇ = 0, for all 1 ≤ i ≤ q, we have Consequently Consequently, the transverse divergence is a basic function.
ii) Let f be a basic function, the relation (4.2) follows from 2 Definition 1.The foliation F is harmonic or minimal if all the leaves of F are minimal submanifolds.

Some integral formulas
First, we give some notations that are needed in the sequel.Let X ∈ V(F) and put ν = π(X), σ(e i ),σ(e i ) , (4.3) and for all s ∈ Γ(Q).Then (i) ξ is parallel along the leaves, and (ii) the relations ii) Now we prove the relation (4.7).Indeed, Therefore the relations (4.4) and (4.5) can be reformulated as follows g Q (A X (e i ), e j )g Q (A X (e j ), e i ).
On the other hand, for 1 ≤ i, j ≤ q, we have (Θ(X)g Q )(e i , e j ) = g Q (A X (e i ), e j ) + g Q (e i , A X (e j )).
So the relation (4.6) becomes and we are done.2 Let s, t ∈ Γ(Q).The Ricci curvature Ric with respect to ∇ is the symmetric bilinear form on Q given by As R and g Q are holonomy, Ric is also holonomy invariant.Proposition 9. Let X ∈ V(F) and ν = π(X).The equations Proof.i) We prove the relation (4.9).First we notice that ii) In view of (3.14) and (3.5), we may write and the proof is complete.2 Now we arrive to the main result of this section Theorem 3. Let F be an harmonic g M -Riemannian foliation on a closed Riemannian manifold (M, g M ), and let X be an infinitesimal automorphism of F. If π(X) is transverse affine then π(X) is transverse Killing.

Proof.
From the first integral formula (4.11) and the relation (4.13) it follows that R M (div ∇ ν) 2 d M = 0. Hence div ∇ ν = 0. Now by the second integral formula (4.12) and the relation (4.14), we obtain Whence Θ(X)g Q = 0 and π(X) is thus transverse Killing. 2 We end this section by the following result Proposition 12. Let F be a g M -Riemannian foliation on a closed Riemannian manifold (M, g M ), and let X be an infinitesimal automorphism of F. If π(X) is transverse homothetic, then π(X) is transverse affine.

Proof.
The local flow ϕ t generated by X maps leaves into leaves.Let Φ t be the induced flow on T M. Then Φ t sends E to itself, and thus induces a local flow e Φ t of bundle maps of Q over ϕ t , i.e., making the diagram Φ * t g Q = ae tc g Q for all t and for some constants a > 0 and c ∈ R. By the uniqueness theorem for the metric and torsion free connection of a Riemannian foliation [11], the connection associated to g Q and ae tc g Q are the same for all t.This proves that X preserves the connection. 2

Preliminaries Theorems
Theorem 4. Let F be an irreductible g M -Riemannian foliation of codimension ≥ 3 on a Riemannian manifold, and let X be an infinitesimal automorphism of F such that π(X) is a transverse conformal vector field and Θ(X)R = 0. Then π(X) is transverse homothetic.
Proof.There exists a basic function f on M such that Θ(X)g Q = f.gQ .Let ω = df • σ.We claim that ω is parallel, i.e. ∇ω = 0. To this end, we argue in two steps.
1) The form ω is parallel along the leaves, that is where we use the fact that f is basic.

Proofs of the main Theorems
Obviously, Theorem 1 follows directly from Theorems 6,3 and Proposition 12.The proof of Theorem 2 is now in order.
Proof.Let x ∈ M .In view of [3], we have the direct sum of mutually orthogonal subspaces invariant under Ψ(x), where (Q 0 ) x is the set of vectors in Q x which are fixed by Ψ(x) and where (Q 1 ) x , ..., (Q k ) x are all irreducible.Since F is without Euclidean part, dim(Q 0 ) x = 0.For each i = 1, .., k, let F i be the foliation of M which is integral to the distribution where d σ(Q i ) indicates that σ(Q i ) is omitted.Each F i is an irreducible g M -Riemannian foliation.Indeed, in [3] the authors show that the metric g Q is a direct sum g Q = L 1≤i≤k g Q i and the restriction of the connection ∇ to each Q i is the unique torsion free metric connection associated to g Q i .In other words, for i = 1, ..., k the metric g Q i is holonomy invariant with respect to the foliation F i .Now we show that for i = 1, ..., k, the vector field X is an infinitesimal automorphism of the foliation F i , that is X ∈ V(F i ).Let x ∈ M .We know by (5.5) that the endomorphism (A X ) x lies in the normalizor of the holonomy algebra G x .Thus the 1-parameter group of linear transformation exp tA X of Q x lies in the normalizor of the holonomy group Ψ(x).By virtue of the uniqueness of the decomposition it follows that, for each t and 1 ≤ j ≤ k, (exp tA X )(Q i ) x coincides with some (Q l ) x .By continuity, we see that for every t.
This implies that A X (Q j ) x ⊂ (Q j ) x .But Q j is invariant under the parallel transport, that is ∇ X (Q j ) ⊂ Q j .Then (3.3) leads to Θ(X)(Q j ) ⊂ Q j and hence Θ(X)(L i ) ⊂ L i .Now let ∇ i be the adapted connection to respect the Riemannian foliation F i .The tensor curvature R ∇ i of ∇ i induces a Q i -tensor field R i .