F (a 1, a 2, . . . , a n )−structure Lagrangian manifolds

F (a ₁ , a ₂ , . . . , a _n )−structure Lagrangian manifolds

Lovejoy S. Das, Jovanka Nik´ıc and Ram Nivas

Abstract. In this paper, authors have shown that if an almost prod- uct structure P on the tangent space of a 2n – dimensional Lagrangian manifold E is defined and the F (a1, a2, . . ., an) – structure on the verti- cal tangent space Tv(E) is given, then it is possible to define the similar structure on the horizontal subspace TH(E) and also on the tangent space T (E) of E. Linear connections on the Lagrangian F (a1, a2, . . .an) – struc- ture manifold E are also discussed. Certain other interesting results like geodesics in E are also studied.

M.S.C. 2000: 53D12, 53D35.

Key words: Lagrangian manifold, distribution, parallelism and anti-parallelism, geodesics.

1 Introduction

Let M be an n – dimensional and E be a 2n – dimensional differentiable manifold and let η = (E, π, M ) be the vector bundle with π(E) = M . Suppose U is a coordinate neighborhood in M with local coordinates (x¹, x², . . .xⁿ). The induced coordinates in π⁻¹(U ) are (xⁱ, y^α), 1 ≤ i ≤ n, 1 ≤ α ≤ n [6]. The canonical basis for tangent space Tu(E) at u ∈ π⁻¹(U ) is n

∂

∂xⁱ,_∂y^∂α

o

or simply {∂i, ∂α} where ∂i = _∂x^∂i etc. If (x^h, y^α1) be coordinates of a point in the intersecting region π⁻¹(U ) ∩ π⁻¹(U 0), we can write

xⁱ¹ = xⁱ¹(xⁱ) (1.1.1a)

y^α1= ∂x^α1

∂x^αy^α (1.1.1b)

It is easy to prove for another canonical basis in the intersecting region.

∂i¹ = ∂xⁱ

∂xⁱ¹∂i

(1.1.2a)

D

c

° Balkan Society of Geometers, Geometry Balkan Press 2006.

∂α¹ = ∂y^α

∂y^α1∂α

(1.1.2b)

We denote by T (E) the tangent space of E spanned by {∂i, ∂α} and its subspaces by TV(E) and TH(E) spanned by {∂α} and {∂i} respectively. Obviously

T (E) = TV(E) ⊕ TH(E) (1.1.3)

and

dimTV(E) = dimTH(E) = n

Let us suppose that the Riemannian material structure on T (E) is given by [4]

G = gij(xⁱ, y^α)dxⁱ⊗ dx^j+ gab(xⁱ, y^α) δy^a⊗ δy^b (1.1.4)

where

gij(xⁱ, y^α) = gij(xⁱ) and gab=1

2 ∂a∂bL(xⁱ, y^α)

where L(xⁱ, y^α) the Lagrange function. We call such a manifold as Lagrangian manifold [4].

If X ∈ T (E), we can write

X = Xⁱ∂_i+ X^α∂_α (1.1.5)

The automorphism P : χ (T (E)) → χ (T (E)) defined by

P X = Xⁱ∂i+ X^α∂α

(1.1.6)

is a natural almost product structure on T (E) i.e. P²= I, I unit tensor field. If υ and h are the projection morphisms of T (E) onto TV(E) and TH(E) respectively, then

P0h = υ0P (1.1.7)

2 The F (a

, a

, . . . a

) – structure

If on the vertical space TV(E), there exists a non–null tensor field Fυ of type (1, 1) satisfying

anF_υⁿ+ an−1F_υⁿ⁻¹. . . + a2F_υ²+ a1Fυ= 0 (2.2.1)

where a1, a2. . .anare real or complex constants, we say that TV(E)admits F (a1, a2. . .an)–

structure [3]. In this case rank (Fυ) = r which is constant every where. Let us call Fυ as Lagrange vertical structure on TV(E)

Theorem 2.1 If Lagrange vertical structure Fυis defined on the vertical space TV(E), it is possible to define similar structure on the horizontal subspace TH(E) with the help of the almost product structure of T (E).

Proof. [Proof] Let us put

Fh= P FυP (2.2.2)

then Fhis a tensor field of type (1, 1) on TH(E).

Also

F_h²= (P FυP )(P FυP ) = P F_υ²P as P is an almost product structure on T (E).

Similarly F_h³= P F_υ³P and so on. Thus, we have

anF_hⁿ+ an−1F_hⁿ⁻¹+ . . . + a2F_h²+ a1Fh

= P (anF_υⁿ+ an−1F_υⁿ⁻¹+ . . . + a2F_υ²+ a1Fυ)P (2.2.3)

= 0 by virtue of (2.2.1).

Thus, Fhgives F (a1, a2, . . ., an)–structure on TH(E).

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Theorem 2.2 If Lagrange vertical F (a1, a2, . . ., an) – structure Fυ of rank r be de- fined on TV(E), the similar type of structure can be defined on the enveloping space T (E) with the help of projection morphism of T (E).

Proof. [Proof] Since Lagrange structure Fυ is defined on TV(E), the Lagrange hori- zontal structure Fhis induced on TH(E) by theorem (2.1). If υ and h are projection morphisms of TV(E) and TH(E) on T (E), let us put

F = Fhh + Fυυ (2.2.4)

As hυ = υh = 0 and h²= h, υ²= υ, we have F²= F_h²h + F_υ²υ Similarly F³= F_h³h + F_υ³υ and so on.

Thus

anFⁿ+ an−1Fⁿ⁻¹+ . . . + a2F²+ a1F

= (anF_hⁿ+ an−1Fⁿ⁻¹+ . . . + a2F_h²+ a1Fh)h +(a_nF_υⁿ+ a_n−1F_υⁿ⁻¹+ . . . + a₂F_υ²+ a₁F_υ)υ

= 0

by (2.2.1) and (2.2.3).

Hence

anFⁿ+ an−1Fⁿ⁻¹+ . . . + a2F²+ a1F = 0.

Since rank(Fυ) = rank(Fh) = r, hence rank(F ) = 2r.

On T (E) with F (a1, a2, . . ., an)–structure of rank 2r, let us define operators

` = −anFⁿ⁻¹+ an−1Fⁿ⁻²+ ... + a2F a1

(2.2.5) and

m = I +anFⁿ⁻¹+ an−1Fⁿ⁻²+ ... + a2F a1

Then it is easy to show that

`<sup>2</sup>= `, m²= m, ` + m = I, `m = m` = 0.

Hence the operators ‘`’ and ‘m’ when applied to the tangent space are comple-

mentary projection operators [5].

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3 Parallelism of distributions

Let E be 2n–dimensional Lagrangian manifold. For F (a1, a2. . .an)–structure on T (E), let L and M be the complementary distributions corresponding to complementary projection operators ‘`’ and ‘m’. Let ¯∇ and ˜∇ be defined as follows

∇¯XY = `∇X(`Y ) + m(∇XmY ) (3.3.1)

and

∇˜_XY = `∇<sub>`X</sub>(`Y ) + m∇<sub>mX</sub>(mY ) + `[mX, `Y ] + m[`X, mY ] (3.3.2)

It can be shown easily that ¯∇ and ˜∇ are linear connections on E.

Definition 3.1 The distribution L is called ∇–parallel if for all X ∈ L, Y ∈ T (E) the vector field ∇YX ∈ L.

Definition 3.2 The distribution L will be said ∇–half parallel if for all X ∈ L,Y ∈ T (E),(∆F )(X, Y ) ∈ L where

(∆F )(X, Y ) = F ∇XY − F ∇YX − ∇F XY + ∇Y(F X) (3.3.3)

Definition 3.3 We call the distribution L as ∇–anti half parallel if for all X ∈ L, Y ∈ T (E), (∆F )(X, Y ) ∈ M .

We now prove the following theorems.

Theorem 3.4 On the F (a1, a2. . .an)–structure manifold, the distributions L and M are ¯∇ as well as ˜∇ parallel.

Proof. [Proof] Since `m = m` = 0, hence from (3.3.1) and (3.3.2), we have m ¯∇XY = m∇X(mY )

If Y ∈ L, mY = 0 so m ¯∇XY = 0 Therefore

∇¯XY ∈ L. Hence for Y ∈ L, X ∈ T (E)

⇒ ¯∇XY ∈ L. So L is ¯∇–parallel.

Similarly for X ∈ T (E), Y ∈ L

∇˜XY = m∇mxmY + m[`X, mY ] = 0 as mY = 0.

So ˜∇XY ∈ L. Hence L is ˜∇–parallel.

In a similar manner, ¯∇ and ˜∇ parallelism of M can also be proved.

¤

Theorem 3.5 On the F (a1, a2. . .an)–structure manifold, the distributions L and M are ∇–parallel if and only if ∇ and ¯∇ are equal.

Proof. [Proof] If L, M are ∇–parallel then ∀X,Y ∈ T (E)m∇X(`Y ) = 0 and `∇X(mY ) = 0.

Therefore, since ` + m = I,

∇X(`Y ) = `∇X(`Y ) and ∇XmY = m∇X(mY ) So

∇_XY = `∇<sub>X</sub>(`Y ) + m∇_X(mY ) = ¯∇_XY Hence ∇ = ¯∇.

The converse of the theorem can be proved easily.

¤

Theorem 3.6 On the F (a1, a2. . .an)–structure manifold, E, the distribution M is ¯∇ –anti half parallel if for all X ∈ M , Y ∈ T (E)

m ¯∇Y(F X) = m∇F XmY.

Proof. [Proof] Since F m = mF = 0, hence in view of the equation (3.3.3) for connec- tion ¯∇

m(∆F )(X, Y ) = m ¯∇YF X − m ¯∇F XY (3.3.4)

In view of the equation (3.3.1), we have

∇¯F XY = `∇F X(`Y ) + m∇F X(mY ) m ¯∇F XY = m∇F X(mY ) as `m = 0, m²= m

m(∆F )(X, Y ) = m ¯∇Y(F X) − m∇F X(mY ) As (∆F )(X, Y ) ∈ L so m(∆F )(X, Y ) = 0. Thus

m ¯∇Y(F X) = m∇F X(mY ),

which proves the proposition.

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4 Geodesics on the Lagrangian manifold

Let γ be a curve in E with tangent T . Then γ is called geodesic with respect to connection ∇ if ∇TT = 0.

Theorem 4.1 A curve γ will be geodesic with respect to connection ¯∇ if the vector fields

∇_TT − ∇_T(mT ) ∈ M and ∇_T(mT ) ∈ L.

Proof. [Proof] Since γ is geodesic with respect to connection ¯∇, hence ¯∇TT = 0. On making use of the equation (3.3.1), the above equation assumes the following form

`∇T(`T ) + m∇T(mT ) = 0.

Since ` + m = I, we can write the above equation as

`∇T(I − m)T + m∇T(mT ) = 0 or

`∇TT − `∇T(mT ) + m∇T(mT ) = 0.

Therefore

`(∇TT − ∇T(mT )) = 0

and m∇T(mT ) = 0. Hence ∇TT − ∇T(mT ) ∈ M and ∇T(mT ) ∈ L, which proves

the proposition.

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Theorem 4.2 The (1, 1) tensor field ‘`’ and ‘m’ are always covariantly constants with respect to connection ¯∇.

Proof. [Proof] ∀X, Y ∈ T (E), we have

( ¯∇X`)(Y ) = ¯∇X(`Y ) − ` ¯∇XY.

(4.4.1)

Making use of (3.3.1), we get

( ¯∇X`)(Y ) = `∇X(`<sup>2</sup>Y ) + m∇X(m`Y ) − `{`∇X`Y + m∇XmY } Since `²= `, m<sup>2</sup>= m, `m = m` = 0, we get

( ¯∇X`)(Y ) = `∇X(`Y ) − `∇X`Y = 0.

So, ‘`’ is covariantly constant. The fact that ‘m’ is covariantly constant can be proved

analogously.

¤

Acknowledgement. The authors are thankful to Professor Vladimir Balan for providing suggestions for the improvement of this paper.

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Authors’ addresses:

Lovejoy S. Das

Department of Mathematics, Kent State University, Tuscarawas, New Philadelphia, Ohio 44663, USA.

email: ldas@kent.edu Jovanka Nik´ıc

Faculty of Technical Sciences, Trg D. Obradovi´ca 6, 2100 Novi Sad, Yugoslavia.

Ram Nivas

Department of Mathematics & Astronomy, Lucknow University, Lucknow 226007, India.

email: rnivas@sify.com