DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 293–298

F (3, 1)-STRUCTURE ON THE LAGRANGIAN SPACE AND INVARIANT SUBSPACES

JOVANKA NIKIĆ

Abstract. If almost product structure P on the tangent space T (E) = T_{V} (E) ⊕ T_{H}(E) of Lagrangian 2n dimensional manifold E is defined, and if f_{v}(3, 1)-structure on T_{V} (E) is defined, then f_{h}(3, 1)-structure on T_{H}(E) are defined in the natural way. We can define F (3, 1)-structure on T (E). The condition is given for the reduction of the structural group of such manifolds. Two cases of invariant submanifolds are considered. In the first case we get an induced almost complex structure, and in the second case we obtain an induced f(3, 1)-structure on the invariant submanifold.

1. Introduction

Let M be an n-dimensional and E 2n-dimensional differentiable manifold and let η = (E, π, M) be vector bundles and πE = M. The differential structures (U, ϕ, R^{2n}) are vector charts of the vector bundles η. Hence the canonical coordinates on π^{−1}(U) are (x^{1}, …, x^{n}, y^{1}, …, y^{n}) = (x^{i}, y^{a}), i = 1, 2, …, n ; a = 1, …, n. Transformation maps on E are x^{i′} = x^{i′}(x^{1}, x^{2}, …, x^{n}),

y^{a′} = M_{a}^{a′}(x^{1}, …, x^{n})y^{a} and rank[\frac{∂x^{i′}}{∂x^{i}} ] = rank[\frac{∂y^{a′}}{∂y^{a}} ] = rankM_{a}^{a′} = n. The inverse transformations are

x^{i} = x^{i}(x^{1′}, x^{2′}, …, x^{n′}), y^{a} = M_{a′}^{a}(x^{i′}, …, x^{n′})y^{a′}, where M_{a′}^{a}M_{b}^{a′} = δ_{b}^{a}.

The local natural bases of the tangent space T (E) are {∂_{i}, ∂_{a}}

∂_{a} = \frac{∂}{∂y^{a}} = M_{a}^{a′}(x^{i})∂_{a′}, ∂_{i} = \frac{∂}{∂x^{i}} = \frac{∂x^{i′}}{∂x^{i}} ∂_{i′} + (∂_{i}M_{b}^{a′}(x^{i}))y^{b}∂_{a′}.

The nonlinear connection on E is distribution N : u ∈ E → N_{u} ⊂ T_{u}(E) which is supplementary to the distribution V,

T_{u}(E) = N_{u} ⊕ V_{u}, ∀_{u} ∈ E. (1)

They are localy determined by δ_{i} = ∂_{i} − N_{i}^{a}∂_{a}. The local bases adapted to decompositions in (1.1) is {δ_{i}, ∂_{a}}.

1991 Mathematics Subject Classification. 53B40, 53C60.

Key words and phrases. f-structures, invariant subspace.

This paper is in final form and no version of it will be submitted for publication elsewhere.

It is easy to prove [4] that on {δ_{i}, ∂_{a}}

δ_{i′} = δ_{i} \frac{∂x^{i}}{∂x^{i′}} , ∂_{a′} = \frac{∂y^{a}}{∂y^{a′}} ∂_{a}.

The subspace of T (E) spaned by {δ_{i}} will be denoted by T_{H}(E) and the subspace spaned by {∂_{a}} will be denoted by T_{V} (E), T (E) = T_{H}(E) ⊕ T_{V} (E), dim T_{H}(E) = n = dim T_{V} (E).

Definition 1.1. If the Riemannian metrical structure on T (E) is given by G = g_{ij}(x^{i}, y^{a})dx^{i} ⊗ dx^{j} + g_{ab}(x^{i}, y^{a})δy^{a} ⊗ δy^{b} where g_{ij}(x^{i}, y^{a}) = g_{ij}(x^{i}), g_{ab} = \frac{1}{2}∂_{a}∂_{b}L(x^{i}, y^{a}) and L(x^{i}, y^{a}) is a Lagrange function, then such a space we call Lagrangian space.

Let X ∈ T (E), then X = X^{i}δ_{i} + X̄^{a}∂_{a} and the automorphism P : X (T (E)) → X (T (E)) defined by P X = X̄^{i}δ_{i} + X^{a}∂_{a} is the natural almost product structure on T (E) i.e, P^{2} = I. If we denote by v and h the projection morphism of T (E) to T_{V} (E) and T_{H}(E) respectively, we have P ∘ h = v ∘ P.

2. f(3, 1)-structures

Definition 2.1. We call Lagrange vertical f_{v}(3, 1)-structure of rank r on T_{V} (E) a non-null tensor field f_{v} of type (1,1) and of class C^{∞} such that f_{v}^{3} + f_{v} = 0, and rank f_{v} = r, where r is constant everywhere.

Definition 2.2. We call Lagrange horizontal f_{h}(3, 1)-structure on T_{H}(E) a non-null tensor field f_{h} on T_{H}(E) of type (1,1) of class C^{∞} satisfying f_{h}^{3} + f_{h} = 0, rank f_{h} = r, where r is constant everywhere.

An F (3, 1)-structure on T (E) is a non-null tensor field F of type \binom{11}{11} such that F^{3} + F = 0, rank F = 2r =const.

For our study it is very convenient to consider f_{v} or f_{h} as morphism of vector bundles [2], [3]

f_{v} : X (T_{V} (E)) → X (T_{V} (E)), f_{h} : X (T_{H}(E)) → X (T_{H}(E)).

Let f_{v} be a Lagrange vertical f_{v}(3, 1)-structure of rank r. We define the morphisms l_{v} = −f_{v}^{2} and m_{v} = f_{v}^{2} + I_{T_{V} (E)}, where I_{T_{V} (E)} denotes the identity morphism on T_{V} (E).

It is clear that l_{v} + m_{v} = I. Also we have l_{v}m_{v} = m_{v}l_{v} = −f_{v}^{4} − f_{v}^{2} = −f_{v}(f_{v}^{3} + f_{v}^{1}) = 0, m_{v}^{2} = m_{v}, l_{v}^{2} = l_{v}.

Hence the morphisms l_{v}, m_{v} applied to the X (T_{V} (E)) are complementaly projection morphisms, then there exist complementary distributions L_{v} and M_{v} corresponding to the projection morphisms l_{v} and m_{v} respectively such that dim L_{v} = r and dim M_{v} = n − r. It is easily to see that

(1)  l_{v}f_{v} = f_{v}l_{v} = f_{v}, m_{v}f_{v} = f_{v}m_{v} = 0,
f_{v}^{2}l_{v} = l_{v}f_{v}^{2} = −I, f_{v}^{2}m_{v} = 0,

Proposition 2.1. If a Lagrange f_{v}(3, 1)-structure of rank r defined on T_{V} (E), then the horizontal f_{h}(3, 1)-structure of rank r is defined on T_{H}(E) by the natural almost product structure of T (E).

F (3, 1)-STRUCTURE ON THE LAGRANGIAN SPACE 295

Proof. If we put

f_{h}X = P f_{v}P X, ∀X ∈ T_{H}(E) (2)

it is easy to see that f_{h}^{3}X = P f_{v}^{3}P X and f_{h}^{3} + f_{h} = 0, and rank f_{h} = r.

Proposition 2.2. If a Lagrange f_{v}(3, 1)-structure of rank r is defined on T_{V} (E), then an F (3, 1)-structure is defined on T (E) by the natural almost product structure of T (E).

Proof. We put

F = f_{h}h + f_{v}v, (3)

where f_{h}, is defined by (2.2), and h, v are the projection morphisms of T (E) to T_{H}(E) and T_{V} (E). Then it is easy to check that F^{2} = f_{h}^{2}h+f_{v}^{2}v, F^{3} = f_{h}^{3}h+f_{v}^{3}v. Thus F^{3} + F = 0. It is clear that rank F = 2r.

If l_{h}, m_{h} are complementary projection morphisms of the horizontal f_{h}(3, 1)- structure, which is defined by the natural almost product structure of T (E), we have

l_{h}X = −f_{h}^{2}X = −P f_{v}^{2}P X = P l_{v}P X, ∀X ∈ T_{H}(E)
m_{h}X = (f_{h}^{2} + I_{T_{H}(E)})X = P f_{v}^{2}P X + P I_{T_{V} (E)}P X = P m_{v}P X,
m_{h}X = P m_{v}P X, ∀X ∈ T_{H}(E).

If l, m are complementary projection morphism of the F (3, 1)-structure on T (E), then we have

(4)  l = −F^{2} = −f_{h}^{2}h − f_{v}^{2}v = l_{h}h + l_{v}v
m = F^{2} + I_{T (E)} = f_{h}^{2}h + f_{v}^{2}v + I_{T_{H}(E)}h + I_{T_{V} (E)}v = m_{h}h + m_{v}v.

Thus, if there is given a Lagrange f_{v}(3, 1)-structure on T_{V} (E) of rank r, then there exist complementary distributions L_{h}, M_{h} of T_{H}(E), corresponding to the morphisms l_{h}, m_{h} such that L_{h} = P L_{v} , M_{h} = P M_{v}. Thus we have the decompositions

T (E) = T_{H}(E) ⊕ T_{V} (E) = P L_{v} ⊕ P M_{v} ⊕ L_{v} ⊕ M_{v}. (5)

If L, M denote complementary distributions corresponding to the morphisms l, m respectively then from (2.4) and (2.5) we have L = P L_{v} ⊕ L_{v}, M = P M_{v} ⊕ M_{v}.

Let ḡ_{v} is a pseudo-Riemannian metric tensor, which is symmetric, bilinear and non-degenerate on T_{V} (E)

ḡ_{v} : X (T_{V} (E)) × X (T_{V} (E)) → F(T (E)).

(For examples ḡ_{v} can be the vertical part of Lagrange metric structure). The mapping a_{v} : X (T_{V} (E)) × X (T_{V} (E)) → F(T (E)) which is defined by

a_{v}(X, Y) = \frac{1}{2}[ḡ(l_{v}X, l_{v}Y) + ḡ(m_{v}X, m_{v}Y)] ∀X, Y ∈ X (T_{V} (E))

is a pseudo-Riemannian structure on T (E) such that a_{v}(X, Y) = 0, ∀X ∈ X (L_{v}), Y ∈ X (M_{v}).

Theorem 2.1. If a Lagrange f_{v}(3, 1)-structure of rank r is defined on T_{V} (E) then there exist a pseudo-Riemannian structure of T_{V} (E) with respect to which the complementary distributions L_{v} and M_{v} are orthogonal and f_{v} is an isometry on T_{V} (E).

Proof. If we put g_{v}(X, Y) = \frac{1}{2}[a_{v}(X, Y) + a_{v}(f_{v}X, f_{v}Y)] it is easy to see that g_{v}(X, Y) = 0 ∀X ∈ X (L_{v}), Y ∈ X (M_{v}). Using (2.1) we get g_{v}(f_{v}X, f_{v}Y) = \frac{1}{2}[a_{v}(f_{v}X, f_{v}Y) + a_{v}(X, Y)]. Thus f_{v} is an isometry with respect to g_{v}.

Let X ∈ X (L_{v}) then f_{v}X, f_{v}^{2}X ∈ X (L_{v}) and g_{v}(X, f_{v}X) = g_{v}(f_{v}X, f_{v}^{2}X) = −g_{v}(f_{v}X, X).

Consequently g_{v}(X, f_{v}X) = g_{v}(f_{v}X, f_{v}^{2}X) = 0 and r = 2m.

Thus we can chose in r = 2m mutualy orthogonal unit vector fields X (L_{v}) such that

\begin{matrix}f_{v}(X_{a}) = X_{α+m} & α = 1, 2, …, m, \\ f_{v}(X_{α}) = −X_{−m+α}, & α = m + 1, …, 2m. \end{matrix}

An adapted frame of the Lagrange f_{v}(3, 1)-structure on T_{V} (E) is the orthogonal frame R = {X_{α}, X_{β}}, where X_{β} is an orthogonal frame of X (M_{v}).

Let R̄ = {X̄_{α}, X̄_{β}} be another adapted frame of the Lagrange f_{v}(3, 1)-structure, and R̄ = AR, then orthogonal matrix A is an element of the group U_{(m)}×O_{(n−2m)}.

Theorem 2.2. A necessary and suficient condition for T_{V} (E) to admit Lagrange f_{v}(3, 1)-structure of rank r is that r = 2m and the structure group of the tangent bundle of the manifold can be reduced to the group U_{(m)} × O_{(n−2m)}.

We can define a metric structure g_{h} on T_{H}(E): g_{h}(X, Y) = g_{v}(P X, P Y), ∀X, Y ∈ X (T_{H}(E)). Using (2.5) the distributions L_{h}, M_{h} are orthogonal with respect to g_{h} and the horizontal f_{h}(3, 1)-structure which is define by f_{h}X = P f_{v}P X, ∀X ∈ X (T_{H}(E)) is an isometry on T_{H}(E) with respect to g_{h}.

We can also define a metric tensor G on T (E).

G(X, Y) = g_{h}(X, Y)h + g_{v}(X, Y)v. (6)

The distributions L, M are orthogonal with respect to G and the F (3, 1)- structure which is defined by F X = f_{h}h + f_{v}v, X ∈ T (E) is an isometry on T (E) with respect to G.

Proposition 2.3. If {X_{α}, X_{β}} is an adapted frame of a given Lagrange f_{v}(3, 1)- structure f_{v} on T_{V} (E) with respect to g_{v}, then the frame {P X_{α}, P X_{β}} is an adapted frame of the horizontal f_{h}(3, 1)-structure with respect to g_{h}.

It is clear that the frame {P X_{α}, P X_{β}, X_{α}, X_{β}} is adapted frame to the decomposition T (E) = L_{h} ⊕ M_{h} ⊕ L_{v} ⊕ M_{v}.

Theorem 2.3. If a Lagrange f_{v}(3, 1)-structure is defined on T_{V} (E) with pseudo-Riemannian structure g_{v}, then the structure group of the tangent bundle on T (E) can be reduced to U_{(m)} × O_{(n−2m)} × U_{(m)} × O_{(n−2m)}.

3. Invariant Submanifolds

Let N^{p} be an p-dimensional C^{∞} manifold imbedded, differentially as a submanifold in a 2n-dimensional C^{∞} manifold E^{2n} with metric structure G and F (3, 1)- structure, as in Chapter 2. Let Φ be an imbedding map Φ : N^{p} → E^{2n}, and Φ_{∗}(≡ B) the Jacobian map of Φ i.e. B : T (N^{p}) → T (E). Denoting by T (N , E) the set of all vectors tangent to the submanifold Φ(N^{p}). It is known that B : T (N^{p}) → T (N , E) is an isomorphism. Take the C^{∞} vector field X̄ = BX̃ and defined along Φ(N^{p}) which is tangential to Φ(N^{p}) and let X be the local C^{∞}-extension of X̄. We define the induced metric g̃ on N^{m} as follows

g̃(X̃, Ỹ) = G(BX̃, BỸ) for all X̃, Ỹ ∈ T (N^{p}), (1)

where G is the Riemannian metric in E. It can be easily verified that g̃ is Riemanian metric in N^{p}.

Definition 3.1. N^{p} is said to be an invariant submanifold of E if the tangent space T_{u}(Φ(N^{p})) of Φ(N^{p}) is invariant by the linear mapping F at each point of Φ(N^{p}) so that for each X̃ ∈ T (N^{p}) we have F (BX̃) = BX̃′ for some X̃′ ∈ T (N^{p}). If we define a (1,1) tensor field f̃ in N^{p} by f̃(X̃) = X̃′, then we have

F (BX̃) = B(f̃X̃). (2)

In the first case we assume that the distribution M is never tangential to Φ(N^{p}) i.e. no vector field of type mX, where m is defined in (2.4) and X ∈ T (Φ(N^{p})), is tangential to Φ(N^{p}). It shows that any vector field of type mX is independent of any vector field of the form BX̃, X̃ ∈ T (N^{p}).

Applying F to (3.2) we have

F^{2}(BX̃) = B(f̃^{2}X̃). (3)

Now we shall show that vector fields of type BX̃ belong to the distribution L in this case. If we suppose that m(BX̃) ≠ 0, then m(BX̃) = (I_{T (E)} + F^{2})(BX̃) = BX̃ + F^{2}(BX̃) = BX̃ + B(f̃^{2}X̃) = B(X̃ + f̃^{2}X̃), which, contrary to our assumption, shows that m(BX̃) is tangential to Φ(N^{p}). Therefore m(BX̃) = 0.

Now since l = −F^{2}, from (3.3) and (2.4) we have B(f̃^{2}X̃) = F^{2}(BX̃) = −l(BX̃) = −(I_{T (E)} − m)(BX̃) = −BX̃ + m(BX̃), B(f̃^{2}X̃) = B(−X̃), which in view of B being an isomorphism gives f̃^{2}(X̃) = −X̃. Thus we have the following.

Theorem 3.1. An invariant submanifold N^{p} imbedded in a F (3, 1)-structure manifold E^{2n} such that distribution M is never tangential to Φ(N^{p}) is an almost complex manifold.

Theorem 3.2. Let N^{p} be an invariant submanifold imbedded in an F (3, 1)- structure manifold E^{2n} such that the distribution M is never tangential to Φ(N^{p}). If G denotes a metric tensor on E^{2n} given by (2.6), then the induced metric g̃ in N^{p} defined by (3.1) is Hermitian.

Proof. G(X, Y) = G(F X, F Y) because F (3, 1)-structure is an isometry on T (E) with respect to G.

g̃(f̃X̃, f̃Ỹ) = G(Bf̃X̃, Bf̃Ỹ) = G(F BX̃, F BỸ) =
= G(BX̃, BỸ) = g̃(X̃, Ỹ).

In the second case we assume that the distribution M is always tangential to Φ(N^{p}). It follows, therefore, that m(BX̃) = BX̃^{0}, where X̃ ∈ T (N^{p}) for some X̃^{0} ∈ T (N^{p}).

Let us define a (1,1) tensor field m̃ on T (N^{p}) such that m̃X̃ = X̃^{0}.

We can write m(BX̃) = B(m̃X̃). Define a (1,1) tensor field l̃in N^{p} by l̃= −f̃^{2}. Then B(l̃X̃) = B(−f̃^{2}X̃) = −B(f̃^{2}X̃) = −F^{2}(BX̃) = l(BX̃). Thus we have

B(l̃X̃) = l(BX̃).

Theorem 3.3. The (1,1) tensor fields l̃ and m̃ on T (N^{p}) satisfying

l̃+ m̃ = I, l̃m̃ = m̃l̃= 0, l̃^{2} = l̃, m̃^{2} = m̃. (4)

Proof. We have l + m = I_{T (E)}, i.e. (l + m)(BX̃) = BX̃.

B(l̃X̃) + B(m̃X̃) = BX̃ or B(l̃+ m̃)(X̃) = BX̃. Therefore l̃+ m̃ = Ĩ_{T (N^{p})}. Similary we can prove other relations.

The relation (3.4) show that l̃and m̃ are complementary projection operators in N^{p} given by m̃ = I_{T (N^{p})} + f̃^{2}, l̃= −f̃^{2}. Moreover, we get B(f̃^{3}X̃) = F^{3}(BX̃) = −F (BX̃) = −B(f̃X̃). Thus we have f̃^{3} + f̃ = 0, which shows that in this case f̃ defines an f̃(3, 1)-structure on N^{p} which we call induced f̃(3, 1)-structure. Further, we have g̃(f̃X̃, f̃Ỹ) = G(Bf̃X̃, Bf̃Ỹ) = G(F BX̃, F BỸ) = G(BX, BY) = g̃(X̃, Ỹ). Thus we have the following

Theorem 3.4. An invariant submanifold N^{p} imbedded in an F (3, 1)-structure manifold E^{2n} such that the distribution M is always tangential to Φ(N^{p}) is an f̃(3, 1)-structure manifold which admits a similar metric g̃ satisfying g̃(X̃, Ỹ) = g̃(f̃X̃, f̃Ỹ).

References

1. M.ATANASIEU, Modes in Finsler and Lagrange geometry, Proc. IV^{th} Nat. Sem. on Finsler and Lagrange geometry, Brasov, (1986), 43-56.

2. F.GOULI ANDREOU, On a structure defined by a tensor field f of type (1,1) satisfying f^{5} + f = 0, Tensor, N.S. 36(1982), 79-84.

3. S.ISHIHARA, K.YANO, On integrability conditions of a structure satisfying f^{3} + f = 0, Quart-J. Math. 15(1964), 217-222.

4. J.NIKIĆ, I.ČOMIĆ, f(2 · 2^{k} + 1, −1)-structure in (k + 1)-Lagrangian Space, to be published in Review of Research Faculty of Science, Mathematics Series.

5. K.YANO, S.ISHIHARA, Invariant submanifolds of almost contact manifolds, Kodai Math. Sem. Rep. 21(1969) 350-364.

Faculty of Technical Sciences Trg D. Obradovića 6, 21000 Novi Sad, YUGOSLAVIA