On infinitesimal holomorphically projective transformations on the tangent bundles with respect to the Sasaki metric

The purpose of the present article is to find solutions to a system of partial differential equations that characterize infinitesimal holomorphically projective transformations on the tangent bundle with the Sasaki metric and an adapted almost complex structure. Moreover, it is proved that if the tangent bundle of a Riemannian manifold admits a non-affine infinitesimal holomorphically projective transformation, then the Riemannian manifold is locally flat.


INTRODUCTION
Let M be an n-dimensional manifold and T (M) its tangent bundle.We denote by ℑ r s (M) the set of all tensor fields of type (r, s) on M. Similarly, we denote by ℑ r s (T (M)) the corresponding set on T (M).The problems of determining infinitesimal holomorphically projective transformation on M and T (M) have been considered by several authors.Ishihara [11] has introduced the notion of infinitesimal holomorphically projective transformation, and Tachibana and Ishihara [18] investigated infinitesimal holomorphically projective transformations on the Kahlerian manifolds.Hasegawa and Yamauchi [8] have proved that 1) infinitesimal holomorphically projective transformation is infinitesimal isometry on a compact Kahlerian manifold with non-positive constant scalar curvature and 2) a compact Kahlerian manifold M with constant scalar curvature is holomorphically isometric to a complex projective space with the Fubini-Study metric if M admits a non-isometric infinitesimal holomorphically projective transformation.In [9,10] they also investigated infinitesimal holomorphically projective transformations on T (M) with respect to horizontal and complete lift connections.Recently, Tarakci, Gezer, and Salimov [19] studied similar problems on T (M) with the metric II + III.In [14] new results of the holomorphically projective mappings and transformations were presented.
Starting from a Riemannian connection ∇ of g on M, Sasaki [16] has shown how to construct the Riemannian metric S g on T (M).This metric has been called diagonal lift metric, Sasaki metric or the metric I + III.The Sasaki metric has been extensively studied by many authors, including Gudmundsson and Kappos [7], Kowalski [12], Musso and Tricerri [15], and Aso [4].Kowalski [12] calculated the Levi-Civita connection S ∇ of the Sasaki metric on T (M) and its Riemannian curvature tensor S R. With this in hand, Kowalski [12], Aso [4] and Musso and Tricerri [15] derived interesting connections between the geometric properties of (M, g) and (T (M), S g).
In this paper, we shall use the method of adapted frames to determine infinitesimal holomorphically projective transformations on T (M) with the Sasaki metric.We also note that here everything will be always discussed in the C ∞ -category, and manifolds will be assumed to be connected and dimension n > 1.

Basic formulas on tangent bundles
Let T (M) be a tangent bundle of M, and π the natural projection π : T (M) → M. Let the manifold M be covered by a system of coordinate neighbourhoods (U, x i ), where (x i ), i = 1, ..., n is a local coordinate system defined in the neighbourhood U. Let (y i ) be the Cartesian coordinates in each tangent space T p (M) at P ∈ M with respect to the natural base ∂ ∂ x i , P being an arbitrary point in U whose coordinates are x i .Then we can introduce local coordinates (x i , y i ) on the open set π −1 (U) ⊂ T (M).We call them coordinates induced on π −1 (U) from (U, x i ).The projection π is represented by (x i , y i ) → (x i ).The indices I, J, ... run from 1 to 2n, the indices ī, j, ... run from n + 1 to 2n.Summation over repeated indices is always implied.
Let X = X i ∂ ∂ x i be the local expression in U of a vector field X on M. Then the horizontal lift H X and the vertical lift V X of X are given, with respect to the induced coordinates, and where Γ i jk are the coefficients of the Levi-Civita connection ∇.Explicit expression for the Lie bracket [,] of the tangent bundle T (M) is given by Dombrowski [6].The bracket products of vertical and horizontal vector fields are given by the formulas jks X j Y k ∂ ī with respect to the induced coordinates.The Sasaki metric S g on the tangent bundle T (M) over a Riemannian manifold (M, g) is defined by three equations for all vector fields X,Y ∈ ℑ 1 0 (M).It is obvious that the Sasaki metric S g is contained in the class of natural metrics.(Recall that by a natural metric on tangent bundles we shall mean a metric which satisfies conditions (3) and ( 4)).

Expressions in adapted frames
With the Riemannian connection ∇ given on M, we can introduce on each induced coordinate neighbourhood π −1 (U) of T (M) a frame field which is very useful in our computation.It is called the adapted frame on π −1 (U) and consists of the following 2n linearly independent vector fields {E λ } = {E i , E ī} on π −1 (U): 1) and ( 2), the vertical lift V X and the horizontal lift H X of X ∈ ℑ 1 0 (M) are defined as follows: H X = X a E a and V X = X a E ā with respect to the adapted frame.By straightforward calculation, we have the following: Lemma 1.The Lie brackets of the adapted frame of T (M) satisfy the following identities: where R a i jb denote the components of the curvature tensor of M [21].Lemma 2. Let Ṽ be a vector field on T (M), then [9].
If g = g ji dx j dx i is the expression of the Riemannian metric g on M, the Sasaki metric S g is expressed in the adapted local frame by [6,16] S g = g ji dx j dx i + g ji δ y j δ y i .
For the Levi-Civita connection of the Sasaki metric, we have the following: Lemma 3. Let S ∇ be a Levi-Civita connection of the Sasaki metric S g, then with respect to the adapted frame {E λ } , where Γ h ji denote the Christoffel symbols constructed with g ji on M [12,16] (for details, see [21, p. 160]).
Let us consider a tensor field J of type (1,1) on T (M) defined by Then we obtain J2 = −I.Therefore, J is an almost complex structure on T (M).This almost complex structure is called an adapted almost complex structure.It is known that J is integrable if and only if M is locally flat [6] (see also [21, p. 118]).

MAIN RESULTS
Let ∇ be an affine connection on M. A vector field V on M is called an infinitesimal projective transformation if there exists a 1-form Ω on M such that Next let (M, J, g) be an almost complex manifold with an affine connection ∇.A vector field V on M is called an infinitesimal holomorphically projective transformation if there exists a 1-form Ω on M such that , where L V is the Lie derivation with respect to V .In this case Ω is also called the associated 1-form of V .Especially, if Ω = 0, then V is an infinitesimal affine transformation.
Theorem 4. Let (M, g) be a Riemannian manifold and T (M) its tangent bundle with the Sasaki metric and the adapted almost complex structure.A vector field Ṽ is an infinitesimal holomorphically projective transformation with associated 1-form Ω on T (M) if and only if there exists ϕ, ψ Proof.We shall use the method proposed by Hasegawa and Yamauchi [9,10] in the proof and prove only the necessary condition because it is easy to prove the sufficient condition.Let Ṽ be an infinitesimal holomorphically projective transformation with the associated 1-form Ω on and Contracting i and h in (6), we have Ω j = ∂ j ψ, where ψ = − 1 n+1 ∂ ā Ṽ a .Hence ( 6) is rewritten as follows: Differentiating ( 8) partially, we have and where A h i and P h ji are certain functions which depend only on the variables (x h ).The coordinate transformation rule implies that A = (A h i ) ∈ ℑ 1 1 (M) and P = (P h ji ) ∈ ℑ 1 2 (M).Using (8), we have from which, using (9), we obtain On the other hand, using (10), we have Using ( 10), (11), and ( 12), we obtain from which Ṽ h = B h + y a A h a + 2ϕy h − y a Ψ a y h , where B h are certain functions which depend only on (x h ).The coordinate transformation rule implies that Similarly, from (7), there exists Next, from (5), we have or From ( 13), we get Comparing both sides of the above equation, we obtain from which we get the following important information: 14), we also note that B is an infinitesimal projective transformation on M.This completes the proof.
Using Theorem 4, we at last come to the following: Theorem 5. Let (M, g) be a Riemannian manifold and T (M) its tangent bundle with the Sasaki metric and the adapted almost complex structure.If T (M) admits a non-affine infinitesimal holomorphically projective transformation, then M is locally flat.

CONCLUSIONS
Tangent bundles of differentiable manifolds are of great importance in many areas of mathematics and physics.In the last decades a large number of publications have been devoted to the study of their special differential geometric properties.
Let (M, g) be a Riemannian manifold.A Riemannian metric g on the tangent bundle T (M) of M is said to be natural with respect to g on A natural metric g is constructed in such a way that the vertical and horizontal subbundles are orthogonal and the bundle map π: (T (M), g) → (M, g) is a Riemannian submersion.All the preceding metrics belong to the wide class of the so-called g-natural metrics on the tangent bundle, initially classified by Kowalski and Sekizawa [13] and fully characterized by Abbassi and Sarih [1][2][3].A well-known example of g-natural metrics is the Sasaki metric S g.Its construction is based on a natural splitting of the tangent bundle T T M of T (M) into its vertical and horizontal subbundles by means of the Levi-Civita connection ∇ on (M, g).The purpose of this article is to characterize infinitesimal holomorphically projective transformations with respect to the Sasaki metric.In Theorem 4 we give a necessary and a sufficient condition for the vector field Ṽ on (T (M), S ∇, J) to be an infinitesimal holomorphically projective transformation.This condition is represented by a set of relations involving certain tensor fields on M of type (0, 0), (1, 0), and (1, 1).We obtain these relations by giving the formula (5) in an adapted frame.Further, in Theorem 5 we show that if (T (M), S ∇, J) has a non-affine infinitesimal holomorphically projective transformation, then M and T (M) are locally flat.Another well-known g-natural Riemannian metric g CG had been defined, some years before, by Musso and Tricerri [15] who, inspired by paper [5] of Cheeger and Gromoll, called it the Cheeger-Gromoll metric.The Levi-Civita connection of g CG and its Riemannian curvature tensor are calculated by Sekizawa in [17] (for more details, see [7]).Similar problems may be investigated for the Cheeger-Gromoll metric.