Distributions on Spray Spaces

Mihai Anastasiei

Abstract A spray S is a 2-homogeneous second order differential equation (SODE) on a manifold M and the pair (M, S) is called a spray space. The curves solutions of this SODE are called geodesics. A distribution D on M is said to be geodesically invariant (g.i.) if for any geodesics whose initial tangent vector is in D it follows that all its tangent vectors belong to D. In §2 we give a characterisation of g.i. distributions on a spray space (M, S) using the Berwald connection induced by S. In particular, one regains the results of A.D. Lewis for linear connections in ([3]). Applications to certain distributions on tangent manifold T M are also considered.

Mathematics Subject Classification: 53C60 Key words: sprays, geodesics, distributions.

1 Spray spaces

Let M be a real, smooth C^{∞} i.e., finite dimensional manifold and τ : T M → M its tangent bundle. Let (U, (x^{i})) be a local chart on M. The indices i, j, k, ... will run from 1 to n = dim M and the Einstein convention on summation will be implied. One associates to v ∈ τ^{−1}(U) the coordinates (x^{i}(τ(u)) and (y^{i}) provided by

v_{τ(v)} = y^{i}∂_{i}, ∂_{i} := \frac{∂}{∂x^{i}}

and T M becomes a smooth orientable manifold.

A spray on M is a vector field

S = y^{i}∂_{i} − 2G^{i}(x, y) ˙∂_{i}, ˙∂_{k} = \frac{∂}{∂y^{k}} ,

whitch satisfies the condition

(1.1)  G^{i}(x, λy) = λ^{2}G^{i}(x, y), λ > 0.

The functions G^{i}(x, y) are called the local coefficients of the spray. If the condition (1.1) is not satisfied, then S is called a semispray. The pair (M, S) is called a spray space.

Balkan Journal of Geometry and Its Applications, Vol.6, No.1, 2001, pp. 1-6 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Example 1.1. Every manifold endowed with a linear connection ∇ is a spray space. If Γ_{jk}^{i}(x) are the local coefficients of ∇ then

G^{i}(x, y) = \frac{1}{2}Γ_{jk}^{i}(x)y^{j}y^{k}.

Example 1.2. Let (M, L) be a Lagrange space. Here L : T M ⟶ R is a regular

Lagrangian, i.e., the matrix g_{ij} = \frac{1}{2} ˙∂_{i} ˙∂_{j}L is invertible. Let g^{jk} be its inverse. Then

G^{i}(x, y) = \frac{1}{4}g^{ik}(( ˙∂_{k}∂_{h}L)y^{h} − ∂_{k}L))

is in general a semispray. For a general class of Lagrange spaces for which the above semispray reduces to a spray we refer to [2].

Example 1.3. Let (M, F) be a Finsler space. Taking L = \frac{1}{2}F^{2} it becomes a Lagrange

space and the G^{i}(x, y) from Example 2 is always a spray. Thus any Finsler space is a spray space. Let γ_{ij}(x, y) be the local components of the Finsler metric of (M, F). Then F^{2} = γ_{ij}(x, y)y^{i}y^{j}. If one sets

γ_{jk}^{i}(x, y) = \frac{1}{2}γ^{ih}(∂_{j}γ_{hk} + ∂_{k}γ_{jh} − ∂_{h}γ_{jk}),

then the local coefficients of the spray are

G^{i}(x, y) = γ_{jk}^{i}(x, y)y^{j}y^{k}.

Example 1.4. Every Riemannian manifold is a spray space since it is a particular Finsler space. The local coefficients of its spray are given as G^{i}(x, y) = γ_{jk}^{i}(x)y^{j}y^{k}, where now γ_{jk}^{i}(x) are the Christoffel symbols.

A curve

c : t → c(t) = (x^{i}(t)) ∈ M, t ∈ I ⊆ R

is said to be a geodesic of S if its lift to T M , given as (x^{i}(t), \frac{dx^{i}}{dt} (t)) is an integral

curve of S. The curve c is a geodesic of S if and only if the functions (x^{i}(t)) are solutions of the equations

(1.2)  \frac{d^{2}x^{i}}{dt^{2}} + 2G^{i} (x, \frac{dx}{dt}) = 0.

2 Distributions on a spray space

We consider a distribution D of rank r on M. This means a smooth mapping p → D_{p} ⊂ T_{p}M, for p ∈ M and D_{p} a subspace of dimension r in T_{p}M. The union D of all D_{p} gives a submanifold of dimension n + r in the tangent manifold T M. We denote by X (D) the set of sections of D, i.e., p → X_{p} ∈ D_{p}.

A distribution D is geodesically invariant (g.i) if for every geodesic c : [a, b] → M such that c′(a) ∈ D_{c(a)} it follows that c′(t) ∈ D_{c(t)} for each t ∈ (a, b). If D is integrable and g.i. one says that D is totally geodesic.

Theorem 2.1. The distribution D is g.i. if and only if S is tangent to D. Proof. The spray S is tangent to D if and only if the integral curves of S, when start from D, remain in D. This fact is clearly equivalent with the condition that D is geodesically invariant. □ Theorem 2.2. X ∈ X (D) if and only if X^{v} is tangent to D. Here X^{v} denotes the vertical lift of X. Proof. Let (U, φ) be a local chart on M and a basis X_{1}, ..., X_{n} for X (U) such that (X_{1}(p), ..., X_{r}(p)) is a basis for D_{p}, p ∈ U. Then

X_{i}(p) = A_{i}^{j}(p) \frac{∂}{∂x^{j}} |_{p} with rank(A_{i}^{j}(p)) = n.

Let (B_{j}^{k}) be the inverse of the matrix (A_{i}^{j}). Thus

\frac{∂}{∂x^{j}} |_{p} = B_{j}^{k}(p)X_{k}(p).

Let be X ∈ X (U). Then

X = X^{i} \frac{∂}{∂x^{i}} = X^{i}B_{i}^{k}X_{k}

and it follows that X ∈ X (D) if and only if B_{i}^{a}X^{i} = 0, a = r + 1, ..., n. We have

u = y^{i} \frac{∂}{∂x^{i}} = y^{i}B_{i}^{k}X_{k} for u ∈ T M.

Thus we may use z^{k} = y^{i}B_{i}^{k} as new coordinates on T M together with (x^{i}). It results that

\frac{∂}{∂y^{i}} = B_{i}^{k} \frac{∂}{∂z^{k}} .

The vertical lift of X, that is X^{v} = X^{i} \frac{∂}{∂y^{i}} takes the form X^{v} = X^{i}B_{i}^{k} \frac{∂}{∂z^{k}} which is

tangent to D ⊂ T M if and only if B_{i}^{a}X^{i} = 0, a = r + 1, ..., n. □

It is well-known that the spray S induces the so-called Berwald connection. This can be thought as a linear connection in the vertical bundle over T M or as a linear connection on T M. In the former case it is locally given in the form

(2.1)  \begin{matrix}D̂_{∂_{i}} ˙∂_{j} = ˙∂_{j}(N_{i}^{k}) ˙∂_{k}, \\ D̂ ˙∂_{i} ˙∂_{i} = 0, \end{matrix}

where

N_{i}^{k} = ˙∂_{i}G^{k}.

Now in the set X (M) a symmetric product is defined as follows:

(2.2)  < X : Y >= D̂_{X}Y^{v} + D̂_{Y} X^{v}.

A direct calculation in local coordinates gives Theorem 2.3. For every X, Y ∈ X (M) one has

[X^{v}, [S, Y^{v}]] =< X : Y > .

Using the above theorems the following main result is obtained Theorem 2.4. A distribution D is g.i. if and only if from X, Y ∈ X (D) it follows

< X : Y >∈ X (D) for every X, Y ∈ X (M).

Let now (M, ∇) be a spray space whose spray is defined by a linear (with torsion) connection ∇. Then it is easy to see that D_{X}Y = (∇_{X}Y)^{v}.

This fact in combination with Theorem 2.2 shows Corollary 2.1. Let D be a distribution on the manifold M endowed with a linear connection ∇. Then D is g.i. if and only if from X, Y ∈ X (D) it results

∇_{X}Y + ∇_{Y} X ∈ X (D)

for every X, Y ∈ X (M).

The result stated in this Corollary was proven by A.D. Lewis in [3].

3 Applications

1. Let (M, S) be a spray space and let the Berwald connection D̂ viewed as a linear

connection on T M. In the frame (δ_{i} = ∂_{i} = N_{i}^{k} ˙∂_{k}, ˙∂_{k} = \frac{∂}{∂y^{k}} ) this is given as

follows:

D̂_{δ_{i}}δ_{j} = \frac{∂^{2}G^{k}}{∂y^{j}∂y^{i}} δ_{k},

D̂_{δ_{i}} ˙∂_{j} = \frac{∂^{2}G^{k}}{∂y^{j}∂y^{i}} ˙∂_{k}

D̂ ˙∂_{i}δ_{j} = 0, D̂ ˙∂_{i} ˙∂_{j} = 0.

On T M there exists the so-called Liouville vector field C = y^{i} ˙∂_{i}. We consider on T M endowed with the linear connection D the distribution {S, C}. By a direct calculation one finds

\begin{matrix}[C, S] = S, \\ D̂_{C}S = S, \\ D̂_{S}C = 0. \end{matrix}

Then Corollary 2.1 applies and one gets that the distribution {S, C} is g.i. and integrable. Thus it is totally geodesic.

2. Let us consider again the manifold T M endowed with the Berwald connection D̂ and the vertical distribution on T M. This is integrable. From D̂ ˙∂_{i} ˙∂_{j} = 0 and Corollary 2.1 it follows it is also g.i. The geodesics are straight lines in the leaves T_{p}M. The leaves are totally geodesic. The same fact, except that the geodesics are no more stright lines in the leaves T_{p}M, holds when D̂ is replaced with a d-connection i.e. with a connection with preserves by parallelism the vertical distribution as well as the horizontal distribution locally spanned by δ_{i} = ∂_{i} − N_{i}^{k} ˙∂_{k}. For the definition of d -connection see [4].

3. Let F^{n} = (M, F) be a Finsler space. Then TM is endowed with a Sasaki type metric

\begin{matrix}G(x, y) = g_{ij}dx^{i} ⊗ dx^{j} + g_{ij}δy^{i} ⊗ δy^{j}, \\ δy^{i} = dy^{i} + N_{k}^{i}dx^{k}. \end{matrix}

Let ∇ be the Levi-Civita connection of G. We notice only a part of the local form of this connection following the paper [1].

\begin{matrix}∇_{δ_{k}}δ_{i} = F_{jk}^{i}δ_{i} + A_{jk}^{a} ˙∂_{a}, \\ ∇ ˙∂_{b} ˙∂_{c} = C_{cb}^{a} ˙∂_{a} + B_{cb}^{i}δ_{i}, \end{matrix}

with

F_{jk}^{i} = \frac{1}{2}g^{ih}(δ_{j}g_{hk} + δ_{k}g_{hj} − δ_{h}g_{jk}),

A_{jk}^{a} = \frac{1}{2}(−R_{jk}^{a} − g^{ab} ˙∂_{b}g_{jk}),

C_{bc}^{a} = \frac{1}{2}g^{ad}( ˙∂_{b}g_{cd} + ˙∂_{d}g_{bd} − ˙∂_{d}g_{bc}),

B_{ab}^{k} = −\frac{1}{2}g^{kj}g_{ab∥j},

where ∥ denotes the h-covariant derivative with respect to the Berwald connection.

Using these formulae one obtains the following results:

• The vertical distribution is totally geodesic only when F^{n} reduces to a Riemannian space ;

• The horizontal distribution is g.i. if and only if F^{n} reduces to a Riemannian space.

We notice that when F^{n} reduces to a Riemannian space, the projection τ becomes a Riemannian submersion and from the theory of submersions the last two statements are derived, as well. Moreover, the horizontal distribution is integrable if and only if the Riemannian space is flat.

Acknowledgements. A version of this paper was presented at the Third Conference of Balkan Society of Geometers, Workshop on Electromagnetic Flows and Dynamics, July 31 - August 3, 2000, University POLITEHNICA of Bucharest, Romania.

References

[1] M.Anastasiei, H.Shimada, Deformations of Finsler Metrics, in vol. Finslerian Geometries ed. by P.L. Antonelli, Kluwer Academic Publishers, FTPH 109, 2000, pp. 53-65.

[2] P.L.Antonelli, D.Hrimiuc, A new class of spray generating Lagrangians, in vol. Lagrange and Finsler geometry, Applications in Physics and Biology, P.L. Antonelli and R. Miron eds., Kluwer, pp. 81-92.

[3] D.A.Lewis, A symmetric product for vector fields and its geometric meaning, to appear.

[4] R.Miron, M.Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications, Kluwer, Dordrecht, 1994.

Faculty of Mathematics University ”Al.I. Cuza” Iaşi 6600, Iaşi, Romania