Many geometric concepts can be defined by a suitable algebraic formalism. This point of view has interest because one can compare different geometric structures having similar algebraic expressions. In this paper, we will consider biparacomplex structures on a smooth 4n-dimensional manifold. An almost biparacomplex structure on a smooth manifold consists of two almost product structures φ1 and φ2 which satisfy [6, 8]

The Nijenhuis tensor Nα of φα, for α=1 or 2, is defined by

It is well known that the structure φα is integrable if and only if the corresponding Nijenhuis tensor Nα vanishes, Nα=0. Also, the classical definition of integrability of structures can be given in this way: a paracomplex structure φ is integrable if there exists a torsion free connection parallelizing φ [9]. If a torsion free connection parallelizes φ, then Nφ=0.

An anti-biparaHermitian metric is a Riemannian metric which is compatible with the (almost) biparacomplex structure φα in the sense that the metric g is pure with respect to each φα. We called such a structure (almost) anti-biparaHermitian. If φα is parallel with respect to the Levi-Civita connection for α=1 and 2, then the manifold is called anti-biparaKaehlerian manifold. The existence of anti-biparaKaehlerian structures on 4n-% dimensional Riemannian manifolds allow one to construct new Riemannian metrics on the second tangent bundle over 4n-dimensional Riemannian manifolds. This paper aims to construct a new metric on the second tangent bundle over an anti-biparaKaehlerian manifold and study its geometry.

Let (Mn,g) be an n-dimensional Riemannian manifold and (TM,π,M) be its tangent bundle. A local chart (U,xi)i=1...n on Mn induces a local chart (π−1(U),xi,ui)i=1...n on TM. Denote by Γijk the Christoffel symbols of g and by ∇ the Levi-Civita connection of g.

We have two complementary distributions on TM, the vertical distribution V and the horizontal distribution H. Let X=Xi∂∂xi be a local vector field on Mn. The vertical and the horizontal lifts of X are defined by

(see [11]).

An anti-paraKaehlerian manifold is a triple (Mn,g,φ) such as Mn is a manifold of even dimension (n=2k) and φ is an integrable almost product structure (φ2=I and ∇φ=0) verifying

or equivalently

for all vector fields X,Y (for more details on the integrability, see [9]).

Definition 2.1. Let (Mn,g,φ) be an anti-paraKaehlerian manifold. A (φ,δ)-deformed Sasaki metric on TM is defined by

• 1. gφ,δ(XH,YH)=ε2g(X,Y)∘π,

• 2. gφ,δ(XH,YV)=0,

• 3. gφ,δ|(x,u)(XV,YV)=gx(X,Y)+δ2gx(X,φ(u))gx(Y,φ(u)),

where X, Y are vector fields on Mn, (x,u)∈TM, ε∈{1,2} and δ is a constant.

• 1. If δ=0 and ε=2 then gφ,δ is the Sasaki metric [11],

• 2. If δ=1 and ε=2 then gφ,δ is the Berger type deformed Sasaki metric [1].

Lemma 2.2. Let (Mn,g,φ) be an anti-paraKaehlerian manifold. For all x∈Mn and u=ui∂∂xi∈TxM, we have the following

• 1. XH(g(u,u))(x,u)=0,

• 2. XH(g(Y,u))(x,u)=g(∇XY,u)x,

• 3. XV(g(u,u))(x,u)=2g(X,u)x,

• 4. XV(g(Y,u))(x,u)=g(X,Y)x,

• 5. XV(g(Y,φ(u)))=g(Y,φ(X),

• 6. XH(g(Y,φ(u)))=g(∇XY),φ(u).

Proposition 2.3. Let (Mn,g,φ) be an anti-paraKaehlerian manifold. Then we have

for all vector fields X,Y on Mn [9].

Let (Mn,g) be a Riemannian manifold and ∇ its Levi-Civita connection. The second tangent bundle is the natural bundle of 2-jets of differentiable curves, defined by

Theorem 3.1. ([3]) If TM⊕TM denotes the Whitney sum, then

is a diffeomorphism of natural bundles.

In the induced coordinate, we have

Definition 3.2. ([3]) Let T2M be the second tangent bundle endowed with the vectorial structure induced by the diffeomorphism S. For any section σ∈Γ(T2M), we define two vector fields on Mn by

where P1 and P2 denote the first and the second projection from TM⊕TM onto TM.

Definition 3.3. ([3]) Let (Mn,g) be a Riemannian manifold and X∈Γ(TM) be a vector field on Mn. For λ=0,1,2, the λ-lifts of X to T2M are defined by

Theorem 3.4. Let (Mn,g) be a Riemannian manifold. If R denotes the Riemannian curvature tensor of (Mn,g), then on T2M we have

• 1. [X(0),Y(0)]=[X,Y](0)−(Rx(X,Y)u)(1)−(Rx(X,Y)w)(2),

• 2. [X(0),Y(i)]=(∇XY)(i),

• 3. [X(i),Y(j)]=0,

where (x,u,w)=S(p) and i,j=1,2 [3].

Let the quadruple (Mn,g,φ1,φ2) be an anti-biparaKaehlerian manifold such that n=4k and φ1(x)≠φ2(x) everywhere on Mn.

Definition 3.5. Let (Mn,g,φ1,φ2) be an anti-biparaKaehlerian manifold. We define a Berger type deformed Sasaki metric gBS on the second tangent bundle by

From Definition 3.5 and the formula (3.3), we obtain the following proposition.

Proposition 3.6. Let (Mn,g,φ1,φ2) be an anti-biparaKaehlerian manifold. If p∈T2M, then for all vector fields X, Y on Mn and i,j∈{0,1,2} (i≠j), we obtain

• 1. gBS(X(0),Y(0))p=g(X,Y)x,

• 2. gBS(X(i),Y(j))p=0,

• 3. gBS(X(1),Y(1))p=g(X,Y)+δ2g(X,φ1(u))g(Y,φ1(u)))x,

• 4. gBS(X(2),Y(2))p=g(X,Y)+η2g(X,φ2(w))g(Y,φ2(w))x,

where S(p)=(x,u,w)∈TxMn⊕TxMn.

Theorem 3.7. Let (Mn,g,φ1,φ2) be an anti-biparaKaehlerian manifold and (T2M,g) be its second tangent bundle equipped with the Berger type deformed Sasaki metric. If ∇˜ denotes the Levi-Civita connection of (T2M,gBS), then for p∈T2M and for all vector fields X,Y on Mn we have

• 1. (∇˜X(0)Y(0))p=(∇XY)(0)−12(R(X,Y)u)(1)−12(R(X,Y)w)(2),

• 2. (∇˜X(0)Y(1))p=(∇XY)(1)+12(R(u,Y)X)(0),

• 3. (∇˜X(0)Y(2))p=(∇XY)(2)+12(R(w,Y)X)(0),

• 4. (∇˜X(1)Y(0))p=12(R(u,X)Y))(0),

• 5. (∇˜X(2)Y(0))p=12(R(w,X)Y))(0),

• 6. (∇˜X(1)Y(1))p=δ2λg(X,φ1(Y))(φ1(u))(1),

• 7. (∇˜X(2)Y(2))p=η2βg(X,φ2(Y))(φ2(w))(2),

• 8. (∇˜X(1)Y(2))p=(∇˜X(2)Y(1))p=0,

where S(p)=(x,u,w), λ=1+δ2|u|2, β=1+η2|w|2, ∇ and R denote the Levi-Civita connection and the Riemannian curvature tensor of (Mn,g), respectively.

Proof. Using the Proposition 3.6, the Lemma 2.2 and the Koszul formula, the Theorem 3.7 follows.

Let Mn,g,φ1,φ2 be an anti-biparaKaehlerian manifold and (T2M,gBS) its second tangent bundle equipped with the Berger type deformed Sasaki metric, F:TM→TM

be a smooth bundle endomorphism of TM. The vector fields F(1), F(2) and F(0) on T2M are defined by

Locally, we have

Proposition 4.1. Let Mn,g,φ1,φ2 be an anti-biparaKaehlerian manifold and (T2M,gBS) its second tangent bundle equipped with the Berger type deformed Sasaki metric. Then we have the following formulas

• 1. (∇˜X(0)F(0))p=((∇XF)(y))(0)−12((Rx(X,Fy)u))(1)−12((Rx(X,Fy)w))(2),

• 2. (∇˜X(0)F(1))p=((∇XF)(u))(1)+12((Rx(u,Fu)X))(0),

• 3. (∇˜X(0)F(2))p=((∇XF)(w))(2)+12((Rx(w,Fw)X))(0),

• 4. (∇˜X(1)F(0))p=((FX))(0)+12((Rx(u,X)Fu)))(0),

• 5. (∇˜X(2)F(0))p=((FX))(0)+12((Rx(w,X)Fw)))(0),

• 6. (∇˜X(1)F(1))p=δ2λg(X,φ1(Fu))(φ1(u))(1)+(FX)(1),

• 7. (∇˜X(2)F(2))p=η2βg(X,φ1(Fw))(φ2(w))(2)+(FX)(2),

• 8. (∇˜X(1)F(2))p=(∇˜X(2)F(1))p=0

for any vector field X on Mn, S(p)=(u,w) and y∈{u,w}.

Proof. The results come directly from the Theorem 3.7.

Theorem 4.2. Let Mn,g,φ1,φ2 be an anti-biparaKaehlerian manifold and (T2M,gBS) its second tangent bundle equipped with the Berger type deformed Sasaki metric. Then we have the following formulas

for all vector fields X, Y,Z on Mn.

Proof. The results come directly from the Theorem 3.7 and the Proposition 4.1.

Let (e1,...,en) (resp (e¯1,...,e¯n)) be an orthonormal frame of TxM where e1=φ1u‖u‖ (resp e¯1=φ2w‖w‖). Then

is an orthonormal frame of TpT2M, where p=S−1(x,u,w).

Let K˜ be the sectional curvature of (T2M,gBS) defined by

for orthonormal vector fields X˜,Y˜ on T2M. From the Theorem 4.2, standard calculations give the following result.

Proposition 4.3. Let Mn,g,φ1,φ2 be an anti-biparaKaehlerian manifold and (T2M,gBS) its second tangent bundle equipped with the Berger type deformed Sasaki metric. We have the following formulas

• 1. K˜(Ei,Ej)=K(ei,ej)−34‖R(ei,ej)u‖2−34‖R(ei,ej)w‖2,

• 2. K˜(Ei,En+1)=K˜(Ei,E2n+1)=0,

• 3. K˜(Ei,En+k)=14‖R(u,ek)ei‖2,

• 4. K˜(Ei,E2n+k)=14‖R(w,e¯ k)ei‖2,

• 5. K˜(En+1,En+k)=δ4λ2(λ−1)[g(φ1(u),u)g(φ1(ek),ek)−g(ek,u)2],

• 6. K˜(E2n+1,E2n+k)=η4β2(β−1)[g(φ2(w),w)g(φ2(e¯ k),e¯ k)−g(e¯ k,w)2],

• 7. K˜(En+l,En+k)=δ2λ[g(φ1(el),el)g(φ1(ek),ek)−g(ek,φ1(el))2],

• 8. K˜(E2n+l,E2n+k)=η2β[g(φ2(e¯ l),e¯ l)g(φ2(e¯ k),e¯ k)−g(e¯ k,φ2(e¯ l))2].

The relationship between the scalar curvature r˜ of (T2M,gBS) and the scalar curvature r of (Mn,g) is given in the following theorem.

Theorem 4.4. Let Mn,g,φ1,φ2 be an anti-biparaKaehlerian manifold and (T2M,gBS) its second tangent bundle equipped with the Berger type deformed Sasaki metric. The corresponding scalar curvature r˜ is given by

where A=∑ i=2ng(φ1(ei),ei) and B=∑ i=2ng(φ2(e¯i),e¯i).

Theorem 4.5. Let Mn,g,φ1,φ2 be a flat anti-biparaKaehlerian manifold and (T2M,gBS) its second tangent bundle equipped with the Berger type deformed Sasaki metric. The corresponding scalar curvature r˜ is given by

where A=∑ i=2ng(φ1(ei),ei) and B=∑ i=2ng(φ2(e¯i),e¯i).

Lemma 5.1. ([10])} Let (Mn,g) be a Riemannian manifold. If X,Y are vector fields and (x,u)∈TM such that Xx=u, then we have

Lemma 5.2. Let (Mn,g) be a Riemannian manifold. If Z∈Γ(TM), σ∈Γ(T2M) and p=σ(x). Then we have

Proof. Using the Lemma 5.1, we obtain

Lemma 5.3. Let (Mn,g,φ1,φ2) be an anti-biparaKaehlerian manifold and (T2M,gBS) be its second tangent bundle equipped with the Berger type deformed Sasaki metric and x:I→Mn be a curve on Mn. If C:t∈I→C(t)=S−1(x(t),y(t),z(t)) is a curve in T2M such that y(t),z(t) are vector fields along x(t) (i.e., y(t),z(t)∈Tx(t)M), then we have

where x˙=dxdt and C˙=dCdt.

Proof. Locally, if Y,Z are vector fields such that Y(x(t))=y(t) and Z(x(t))=z(t), then from the Lemma 5.2 we obtain

where σ=S−1((Y,Z)).

Subsequently, we denote x′=x˙, x′′=∇ x˙x˙, y′=∇x˙y and y′′=∇x˙∇x˙y.

Theorem 5.4. Let (Mn,g,φ1,φ2) be an anti-biparaKaehlerian manifold and (T2M,gBS) be its second tangent bundle equipped with the Berger type deformed Sasaki metric. If C(t)=S−1(x(t),y(t),z(t)) is a curve on T2M such that y(t),z(t) are vector fields along x(t), then