MATEMATIQKI VESNIK 49 (1997), 45–50

UDK 514.763.6

originalni nauqni rad research paper

LINEAR CONNECTIONS COMPATIBLE WITH THE F (3, 1)-STRUCTURE ON THE LAGRANGIAN SPACE

Jovanka Nikić

Abstract. In this paper the F -structure, satisfying F^{3} + F = 0 on the Lagrangian space, is examined. The construction of this structure is given as the prolongation of f_{v}-structure defined on T_{V} (E) using the almost product or almost complex structure on T (E). Moreover, the metric tensor G, with respect to which F is an isometry, is constructed as wekk as the connection compatible with such structures.

1. Introduction

Let M be an n-dimensional and E 2n-dimensional differentiable manifold and let η = (E, π, M) be vector bundle with 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} = M_{a}^{a′}(x^{i})y^{a}

rank [\frac{∂x^{i′}}{∂x^{i}} ] = n, rank [\frac{∂y^{a′}}{∂y^{a}} ] = rank M_{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.  (1.1)

AMS Subject Classification: 53 B,40, 53 C 60

Communicated at the 11. Yugoslav Geometrical Seminar, Divčibare 1996.

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}}.

It is easy to prove [5] 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 metric 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 will be called a 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 morphisms 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} on T_{V} (E) of type (1,1) and of class C^{∞} such that f_{v}^{3} + f_{v} = 0, where rank f_{v} = r, and r is constant everywhere.

Definition 2.2. We call Lagrange horizontal f_{h}(3, 1)-structure of rank r 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 morphisms 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}.

Linear connections compatible with the F (3, 1)-structure 47

Hence the morphisms l_{v}, m_{v} applied to the X (T_{V} (E)) are complementary 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 easy to see that

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.  (2.1)

Proposition 2.1. If a Lagrange f_{v}(3, 1)-structure of rank r is 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).

Proof. If we put

f_{h}X = P f_{v}P X, ∀X ∈ T_{H}(E),  (2.2)

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,  (2.3)

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), respectively. 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, ∀X ∈ T_{H}(E)

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

l = −F^{2} = −f_{h}^{2}h − f_{v}^{2}v = l_{h}h + l_{v}v  (2.4)
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}.  (2.5)

Thus we have the decompositions T (E) = T_{H}(E)⊕T_{V} (E) = P L_{v} ⊕P M_{v} ⊕L_{v} ⊕M_{v}. 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} be 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 example ḡ_{v} can be the vertical part of the 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 exists 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.

We can define a maping g_{h}:

g_{h}(X, Y) = g_{v}(P X, P Y), ∀X, Y ∈ X (T_{H}(E)),

where g_{h} is a metric structure on 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 defined 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.  (2.6)

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.

3. Linear connections compatible with F (3, 1)-structure

It is well known that an arbitrary distribution D is parallel with respect to a linear connection ∇, if for any tangent field Y, ∇_{Y} is a transformation of D [4].

Definition 3.1. An f_{v}(3, 1)-connection on T_{V} (E) (or a linear connection compatible with a f_{v}(3, 1)-structure) is a linear connection ∇ on T_{V} (E) with the property that distributions L_{v} and M_{v} are parallel with respect to ∇.

In a similar way ∇′ is an f_{h}(3, 1)-connection if distributions L_{h} and M_{h} are parallel with respect to ∇′, and ˜∇ is an F (3, 1)-connection if distributions L and M are parallel with respect to ˜∇.

Theorem 3.1. Let l_{v}, m_{v} be the complementary projection morphisms of f_{v}(3, 1)-structure.

A linear connection on T_{V} (E) is an f_{v}(3, 1)-connection if and only if

∇_{X}l_{v} = 0 ∀X ∈ X T (E).

Proof. Since l_{v} is a morphism on T_{V} (E),

(∇_{X}l_{v})(Y) = ∇_{X}l_{v}Y − l_{v}∇_{X}Y, ∀X ∈ X T (E), Y ∈ X T_{V} (E).  (3.1)

If ∇_{X}l_{v} = 0, then from l_{v} + m_{v} = I, we have

∇_{X}m_{v}Y = 0, ∇_{Y} l_{v}Y = l_{v}∇_{X}Y, ∇_{X}m_{v}Y = m_{v}∇_{X}Y.

Since m_{v}l_{v} = l_{v}m_{v} = 0, we have

m_{v}∇_{X}Y = 0, ∀Y ∈ X T (L_{v}), X ∈ X T (E),

and

l_{v}∇_{X}Y = 0, ∀Y ∈ X T (M_{v}), X ∈ X T (E).

Thus ∇_{X}Y ∈ X T (L_{v}) for every Y ∈ X T (L_{v}) and ∇_{X}Y ∈ X T (M_{v}) for every Y ∈ X T (M_{v}).

Conversely, using the decomposition Y = l_{v}Y + m_{v}Y and (3.1) we get

(∇_{X}l_{v})(Y) = ∇_{X}l_{v}Y − l_{v}∇_{X}l_{v}Y − l_{v}∇_{X}m_{v}Y.

Since ∇_{X} is an f_{v}(3, 1)-connection ∇_{X}m_{v}Y ∈ X T (M_{v}). Consequently l_{v}∇_{Y} m_{v}Y = 0, (∇_{X}l_{v})(Y) = ∇_{X}l_{v}Y − l_{v}∇_{X}l_{v}Y = 0, because l_{v}^{2} = l_{v}.

Thus: ∇_{X}l_{v} = 0, ∀X ∈ X T (E).

In a similar way we have:

Proposition 3.1. A linear connection ∇′_{X} on T_{H}(E) is an f_{h}(3, 1)-connection iff ∇′_{X}l_{h} = 0, ∀X ∈ X T (E).

Proposition 3.2. A linear connection ˜∇_{X} on T (E) is an F (3, 1)-connection iff ˜∇_{X}l = 0, ∀X ∈ X T (E).

Theorem 3.2. If ¯∇_{X} is an arbitrary linear connection on T_{V} (E) then the operator

∇_{X} = f_{v} ¯∇_{X}f_{v}, ∀X ∈ X T (E)

is an f_{v}(3, 1)-connection.

Proof. Applying the theorem 3.1 we have

(∇_{X}l_{v})Y = f_{v}∇_{X}f_{v}l_{v}Y − l_{v}f_{v}∇_{X}f_{v}Y, ∀Y ∈ X T_{V} (E).

Since f_{v}l_{v} = l_{v}f_{v} = f_{v}, we have ∇_{X}l_{v} = 0, ∀X ∈ X T (E), i.e. ∇_{X} is an f_{v}(3, 1)-connection.

Let ∇_{X} be a linear connection on T_{V} (E). We define the linear connection ∇′_{X} on the T_{H}(E) by

∇′_{X}Y = P ∇_{X}P Y, ∀X ∈ X T (E), Y ∈ X T_{H}(E).  (3.2)

Theorem 3.3. If ∇_{X} is an f_{v}(3, 1)-connection on T_{V} (E) then ∇′_{X} defined by (3.2) is a linear connection compatible with the horizontal f_{h}(3, 1)-structure.

Proof. Using (3.1), (3.2) we have

(∇′_{X}l_{h})Y = ∇′_{X}l_{h}Y − l_{h}∇′_{X}Y = P ∇_{X}P P l_{v}P Y − P l_{v}P P ∇_{X}P Y
= P (∇_{X}l_{v}P Y − l_{v}∇_{X}P Y), ∀Y ∈ X T_{H}(E).

According to theorem 3.1, ∇_{X}l_{v} = 0, thus ∇′_{X}l_{h} = 0, i.e. horizontal connection ∇′_{X} is compatible with the f_{h}(3, 1)-structure.

Next, we define linear connection ˜∇_{X} on T (E) by

˜∇_{X}Y = ∇′_{X}hY + ∇_{X}vY, ∀X, Y ∈ X T (E).  (3.3)

Theorem 3.4. If ∇_{X} is an f_{v}(3, 1)-connection on T_{V} (E) then ˜∇_{X} defined by (3.2) is a linear connection compatible with the F (3, 1)-structure.

Proof. We shall prove that ( ˜∇_{X}l)Y = 0, ∀X, Y ∈ X T (E).

( ˜∇_{X}l)Y = ∇′_{h}hlY + ∇_{X}vlY − l∇′_{X}hY − l∇_{X}vY

= ∇′_{X}h(l_{h}h + l_{v}v)Y + ∇_{X}v(l_{h}h + l_{v}v)Y − (l_{h}h + l_{v}v)∇′_{X}hY − (l_{h}h + l_{v}v)∇_{X}vY.

Consequently ( ˜∇_{X}l)(Y) = (∇′_{X}l_{h})(hY) + (∇_{X}l_{v})vY, ∀X, Y ∈ X T (E). According to theorems 3.3 and 3.1 we have ˜∇_{X}l = 0, ∀X ∈ X T (E) which

proves the theorem.

REFERENCES

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

[4] M. Matsumoto, Connections, metrics and almost complex structures of tangent bundles, Ibid. 5 (1966), 251–278.

[5] J. Nikić, f(3, 1)-structure on the Lagrangian Space and Invariant Subspaces, Differential geometry and applications, Proc. Conf. 1995. Brno, Masaryk University.

(received 15.01.1997.)

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