An Almost Paracontact Structure on the Indicatrix Bundle of a Finsler Space

Manuela Gîrţu

Abstract

In a paper by I. Hasegawa, K. Yamaguchi and H. Shimada, [2], it was proved that the indicatrix bundle of a Finsler space F^{n} = (M, L) has a natural almost contact structure. On a different way, the same structure was found by M. Anastasiei in [1]. Adopting the approach from [1] we prove that the indicatrix bundle of F^{n} carries also an almost paracontact structure.

Mathematics Subject Classifications 2000: 53C60 Key words and phrases: tangent manifold, framed f–structure, indicatrix bundle, almost paracontact structure.

1 Introduction

Let F^{n} = (M, L) be a Finsler space. Here M is a real C^{∞} manifold of dimension n with local coordinates (x^{i}), i, j, k... = 1, ..., n. For the tangent manifold T M with the projection τ over M we take the local coordinates (x^{i} ∘ τ, y^{i}), where y^{i} are the

components of a vector from T_{p}M, in the natural basis ∂_{i} = \frac{∂}{∂x^{i}}.

The function L : T_{0}M : T M \ {0} → R_{+} is smooth, positively homogeneous of

degree 1 with respect to y^{i} and the matrix (g_{ij}(x, y) = \frac{1}{2} \frac{∂^{2}L^{2}}{∂y^{i}∂y^{j}} ) is of rank n. We

set ˙∂_{i} = \frac{∂}{∂y^{i}}.

The homogeneity of L implies

L^{2}(x, y) = g_{ij}(x, y)y^{i}y^{j} = y^{i}y_{i} for y_{i} = g_{ij}y^{j}.

The functions N_{j}^{i}(x, y) = \frac{1}{2} ˙∂_{j}(γ_{00}^{i}), for γ_{00}^{i} = γ_{jk}^{i}(x, y)y^{j}y^{k} and γ_{jk}^{i}(x, y) the ”generalized” Christoffel symbols, are the local coefficients of the nonlinear Cartan connection. See [Ch. VIII, 4] for details. One considers a new local basis {δ_{i}, ˙∂_{i}}, with δ_{i} = ∂_{i} −N_{i}^{k}(x, y) ˙∂_{k}, on T_{0}M. Its dual basis is (dx^{i}, δy^{i}) with δy^{i} = dy^{i} +N_{k}^{i}(x, y)dx^{k}. If we assume that the quadratic form g_{ij}(x, y)ξ^{i}ξ^{j}, ξ ∈ R^{n} is positive definite, then

∗ Balkan Journal of Geometry and Its Applications, Vol.7, No.2, 2002, pp. 43-48.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2002.

G_{S} = g_{ij}(x, y)dx^{i} ⊗ dx^{j} + g_{ij}(x, y)δy^{i} ⊗ δy^{j}

is a Riemannian metric on T_{0}M.

The linear operator P given in the local basis by

(1.1)  P (δ_{i}) = δ_{i}, P ( ˙∂_{i}) = − ˙∂_{i},

defines an almost product structure on T_{0}M and we have

(1.2)  G_{S}(P X, P Y) = G_{S}(X, Y), X, Y ∈ χ(T_{0}M).

Here χ(T_{0}M) is the module of vector fields on T_{0}M. The vector field C = y^{i} ˙∂_{i} is called the Liouville vector field on T_{0}M and S = y^{i}δ_{i} is the geodesic spray of F^{n}.

An almost paracontact structure on a manifold N is a set (φ, ξ, η), where φ is a tensor field of type (1, 1), ξ a vector field and η an 1-form such that

(1.3)  η(ξ) = 1, φ(ξ) = 0, η ∘ φ = 0, φ^{2} = +I − η ⊗ ξ,

where I denotes the Kronecker tensor field.

This structure generalizes as follows. One considers on a manifold N of dimension (2n + s) a tensor field f of type (1, 1). If there exists on N the vector fields (ξ_{α}) and the 1- forms (η^{α}) (α = 1, 2, ...s) such that

(1.4)  η^{α}(ξ_{β}) = δ_{β}^{α}, f(ξ_{α}) = 0, η^{α} ∘ f = 0, f^{2} = I − ∑_{α}η^{α} ⊗ ξ_{α},

then (f, (ξ_{α}), (η^{α})) is called a framed f(3, −1)- structure. The term was suggested by the equation f^{3} − I = 0. This is in some sense dual to the framed f-structure which generalizes the almost contact structure and which may be called a framed f(3, +1)- structure. For an account of such kind of structures we refer to the book [3].

In the following (Section 2) we show that the slit tangent bundle T_{0}M of a Finsler space carries a natural framed f(3, −1)- structure. The set I(M) = {(x, y) | L(x, y) = 1} is a (2n − 1)− dimensional submanifold of T_{0}M. In Section 3 we prove that the framed f(3, −1)- structure on T_{0}M induces on I(M) an almost paracontact structure. We note that it was known that I(M) carries an almost contact structure [2], [1] but only the approach from [1] allowed us to construct this almost paracontact structure.

2 A framed f(3, −1)- structure on T_{0}M

Let us put ξ_{1} := S = y^{i}δ_{i} and ξ_{2} := C = y^{i} ˙∂_{i}. By a direct calculation one finds (P is the almost product structure (1.1)). Lemma 2.1. P (ξ_{1}) = ξ_{1}, P (ξ_{2}) = −ξ_{2}. We consider the 1- forms

η^{1} = \frac{y_{i}}{L^{2}} dx^{i}, η^{2} = \frac{y_{i}}{L^{2}} δy^{i}

and we prove Lemma 2.2. η^{1} ∘ P = η^{1}, η^{2} ∘ P = −η^{2}. Proof. It is sufficient to check these equalities on the adapted basis (δ_{i}, ˙∂_{i}). We have

(η^{1} ∘ P)(δ_{i}) = η^{1}(P (δ_{i})) = η^{1}(δ_{i}) and (η^{1} ∘ P)( ˙∂_{i}) = −η^{1}( ˙∂_{i}) = 0.

Then

(η^{2} ∘ P)(δ_{i}) = η^{2}(δ_{i}) = 0 and (η^{2} ∘ P)( ˙∂_{i}) = −η^{2}( ˙∂_{i}).
□

Let be G = \frac{1}{L^{2}} G_{S} a Riemannian metric which is conformal with G_{S}.

Lemma 2.3. η^{1}(X) = G(X, ξ_{1}), η^{2}(X) = G(X, ξ_{2}), ∀X ∈ χ(T_{0}M).

Proof. It is sufficient to check these equalities on the basis (δ_{i}, ˙∂_{i}). We have: η^{1}(δ_{j}) =

= \frac{y_{j}}{L^{2}} = di\frac{1}{L^{2}} g_{jk}y^{k} and G(δ_{j}, ξ_{1}) = \frac{1}{L^{2}} G_{S}(δ_{j}, y^{k}δ_{k}) = \frac{1}{L^{2}} y^{k}G_{S}(δ_{j}, δ_{k}) = \frac{1}{L^{2}} y^{k}g_{jk}.

Further, η^{1}( ˙∂_{i}) = 0 and G( ˙∂_{j}, ξ_{1}) = \frac{1}{L^{2}} G_{S}( ˙∂_{i}, y^{k}δ_{k}) = 0. Similarly, one checks the

equation η^{2}(X) = G(X, ξ_{2}).

□

Now we define a tensor field p of type (1, 1) on T_{0}M by

(2.1)  p(X) = P (X) − η^{1}(X)ξ_{1} + η^{2}(X)ξ_{2}, X ∈ χ(T_{0}M).

This can be written in a more compact form as p = P − η^{1} ⊗ ξ_{1} + η^{2} ⊗ ξ_{2}.

Theorem 2.1. For the data (p, (ξ_{a}), (η^{a})), a = 1, 2 the following hold

(i) η^{a}(ξ_{b}) = δ_{b}^{a}, p(ξ_{a}) = 0, η^{a} ∘ p = 0,

(ii) p^{2} = I − η^{1} ⊗ ξ_{1} − η^{2} ⊗ ξ_{2}, X ∈ χ(T_{0}M),

(iii) p is of rank 2n − 2 and p^{3} − p = 0.

Proof. (i) follows easily from Lemmas 2.1, 2.2 and the formula (2.1). For (ii) we have

\begin{matrix}p^{2}(X) = p(p(X)) = P (P (X) − η^{1}(X)ξ_{1} + η^{2}(X)ξ_{2}) − η^{1}(P (X) − η^{1}(X)ξ_{1}+ \\ +η^{2}(X)ξ_{2}) + η^{2}(P (X) − η^{1}(X)ξ_{1} + η^{2}(X)ξ_{2}) = +X − η^{1}(X)ξ_{1} − η^{2}(X)ξ_{2}, \end{matrix}

the other terms vanish or cancel because of Lemmas 2.1, 2.2 and (i). Applying p to the equality (ii) and using again the Lemmas 2.1, 2.2 and (i) one gets p^{3} − p = 0. From the second equation in (i) we see that the subspace span(ξ_{1}, ξ_{2}) is contained in Ker p. Let now X = X^{i}δ_{i} + Y^{i} ˙∂_{i} ∈ Kerp. On using (2.1),

p(X) = X^{i}δ_{i}−Y^{i} ˙∂_{i}−(X^{i} \frac{y_{i}}{L^{2}} )ξ_{1}+Y^{i} \frac{y_{i}}{L^{2}} ξ_{2} = (X^{i}−\frac{(X^{k}y_{k})}{L^{2}}y^{i})δ_{i}−(Y^{i}−(Y^{k} \frac{y_{k}}{L^{2}} )y^{i}) ˙∂_{i} = 0

equivalent to

X^{i} = \frac{X^{k}y_{k}}{L^{2}} y^{i}, Y^{i} = \frac{(Y^{k}y_{k})}{L^{2}}y^{i}.

Hence X = \frac{X^{k}y_{k}}{L^{2}} ξ_{1} + \frac{Y^{k}y_{k}}{L^{2}} ξ_{2} that is X belongs to span(ξ_{1}, ξ_{2}). In other words, Ker p =span(ξ_{1}, ξ_{2}). Thus rank p = 2n − 2.

□

Theorem 2.2. The Riemannian metric G = \frac{1}{L^{2}} G_{S} satisfies

(2.2)  G(pX, pY) = G(X, Y) − η^{1}(X)η^{1}(Y) − η^{2}(X)η^{2}(Y), X, Y ∈ χ(T_{0}M).

Proof. Use (2.1) and Lemma 2.3 and Lemma 2.1 as well as G(ξ_{1}, ξ_{1}) = 1, G(ξ_{2}, ξ_{2}) = 1, G(ξ_{1}, ξ_{2}) = 0 to obtain

G(pX, pY) = G(P X, P Y) − η^{1}(Y)G(P X, ξ_{1}) + η^{2}(Y)G(P X, ξ_{2})−

−η^{1}(X)G(ξ_{1}, P Y) + η^{1}(X)η^{1}(Y) + η^{2}(X)G(ξ_{2}, P Y) + η^{2}(X)η^{2}(Y) =

= G(X, Y) − η^{1}(Y)η^{1}(P (X)) + η^{2}(Y)η^{2}(P X) − η^{1}(X)η^{1}(P Y) + η^{2}(X)η^{2}(P Y)+

+η^{1}(X)η^{1}(Y) + η^{2}(X)η^{2}(Y) = G(X, Y) − η^{1}(X)η^{1}(Y) − η^{2}(X)η^{2}(Y)

□

Remark. In the local basis (δ_{i}, ˙∂_{i}), we get

(2.3)  \begin{aligned}G(p(δ_{i}), p(δ_{j})) & = & \frac{1}{L^{2}} (g_{ij} − \frac{y_{i}y_{j}}{L^{2}} ), G(p(δ_{i}), p( ˙∂_{i})) = 0, \\ G(p( ˙∂_{i}), p( ˙∂_{j})) & = & \frac{1}{L^{2}} (g_{ij} − \frac{y_{i}y_{j}}{L^{2}} ). \end{aligned}

Let us put

(2.4)  h(X, Y) = G(pX, Y), X, Y ∈ χ(T_{0}M).

We have Theorem 2.3. The map h is a symmetric bilinear form on T_{0}M of rank 2n − 2, with the null space span(ξ_{1}, ξ_{2}). Proof. h is bilinear since G is so. As for the symmetry we have

h(Y, X) = G(pY, X) = G(pY, p^{2}X + η^{1}(X)ξ_{1} + η^{2}(X)ξ_{2}) =
= G(pY, p(pX)) + η^{1}(X)G(pY, ξ_{1}) + η^{2}(X)G(pY, ξ_{2}) =
= G(pY, p(pX)) + η^{1}(X)η^{1}(P Y) + η^{2}(X)η^{2}(P Y) =
= G(Y, pX) − η^{1}(Y)η^{1}(pX) − η^{2}(Y)η^{2}(pX) = G(Y, pX) = h(X, Y).

Then we have h(ξ_{1}, ξ_{1}) = h(ξ_{2}, ξ_{2}) = 0. Thus span(ξ_{1}, ξ_{2}) is contained in the null space of h. Conversely, if X = X^{i}δ_{i} is such that h(X, X) = 0 ⟺ G(pX, X) = 0 it

results X = \frac{X^{k}y_{k}}{L^{2}} ξ_{1} and similarly, if X = Y^{i} ˙∂_{i} is such that h(X, X) = 0, it results

X = \frac{Y^{k}y_{k}}{L^{2}} ξ_{2}. Thus the null space of h is just span(ξ_{1}, ξ_{2}) and the proof is finished. Remark. The map h is a singular pseudo-Riemannian metric on T_{0}M. Locally it looks as follows

h = \frac{1}{L^{2}} (g_{ij} − \frac{y_{i}y_{j}}{L^{2}} )dx^{i} ⊗ dx^{j} − \frac{1}{L^{2}} (g_{ij} − \frac{y_{i}y_{j}}{L^{2}} )δy^{i} ⊗ δy^{j},

with

rank (g_{ij} − \frac{y_{i}y_{j}}{L^{2}} ) = n − 1

since

(g_{ij} − \frac{y_{i}y_{j}}{L^{2}} )y^{j} = y_{i} − y_{i} = 0 (y_{j}y^{j} = L^{2}).

3 An almost paracontact structure on the indicatrix bundle of the Finsler space F^{n} = (M, L)

The indicatrix bundle of F^{n} is the submanifold

I(M) = {(x, y) ∈ T_{0}M | L(x, y) = 1}

of T_{0}M projected over M. It is well-known that ξ_{2} = y^{i} ˙∂_{i} is normal to I(M) and this is unitary with respect to G since

G(ξ_{1}, ξ_{2}) = \frac{1}{L^{2}} y^{i}y^{j}g_{ij} = 1.

We consider T_{0}M with the Riemannian metric G and then I(M) appears as a hypersurface of T_{0}M with normal vector field ξ_{2}. We restrict to I(M) all the objects introduced above and indicate this fact by putting a bar over the letters denoting those objects. We have:

• \overline{ξ}_{1} = ξ_{1} since ξ_{1} is tangent to I(M),

• \overline{η}^{2} = 0 on I(M) since η^{2}(X) = G(X, ξ_{2}) = 0 for X ∈ χ(I(M)) ,

• \overline{G} = G_{S} |_{I(M)} because L^{2} = 1 on I(M),

• \overline{p}(X) = P (X) − \overline{η}^{1}(X)ξ_{1} for X ∈ χ(I(M)) .

• The map \overline{p} is an endomorphism of the tangent bundle to I(M) since G(\overline{p}X, ξ_{2}) = 0.

We put \overline{ξ}_{1} = \overline{ξ}, \overline{η}^{1} = \overline{η} and as a consequence of the Theorem 2.1 we get Theorem 3.1. The triple (\overline{p}, \overline{ξ}, \overline{η}) defines an almost paracontact structure on I(M), that is,

(i) \overline{η}(\overline{ξ}) = 1, \overline{p}(\overline{ξ}) = 0, \overline{η} ∘ \overline{p} = 0,

(ii) \overline{p}^{2}(X) = X − \overline{η}(X)\overline{ξ}, X ∈ χ(I(M))),

(iii) \overline{p}^{3} − p = 0, rank \overline{p} = 2n − 2 = (2n − 1) − 1.

Using the restriction to I(M) and the Theorem 2.2 one infers Theorem 3.2. The Riemannian metric \overline{G} satisfies

(3.1)  \overline{G}(\overline{p}X, \overline{p}Y) = \overline{G}(X, Y) − \overline{η}(X)\overline{η}(Y), X, Y ∈ χ(I(M)).

From the last two theorems we see that the ensemble (\overline{p}, \overline{ξ}, \overline{η}, \overline{G}) defines an almost metrical paracontact structure on I(M).

References

[1] Anastasiei, M., A framed f–structure on tangent manifold of a Finsler space, Analele Univ. Bucureşti, Mat.- Inf., XLIX, 2000, 3-9.

[2] Hasegawa, I., Yamaguchi, K., Shimada, H., Sasakian structures on Finsler manifolds in P.L. Antonelli and R. Miron (eds.), Lagrange and Finsler Geometry, Kluwer Academic Publishers, 1996, p. 75–80.

[3] Mihai, I., Roşca, R., Verstraelen, L., Some aspects of the differential geometry of vector fields, padge, Katholieke Universiteit Leuven, vol.2, 1996.

[4] Miron, R., Anastasiei, M., The geometry of Lagrange spaces: theory and applications, Kluwer Academic Publ., FTPH 59, 1994.

Faculty of Sciences

University of Bacău 5500, Bacău, Romania