On the Randers Spaces of Second Order

Marcel Roman

Abstract

The geometry of Randers spaces of order one has been investigated since fifties by G. Randers [9], using Riemannian techniques, only. A Finsler approach of this Randers space was given by R. Ingarden [4]. A fully Finslerian treatment was developed by R. Miron in [6,7]. In this paper we define and study the geometrical theory of a Randers space of order two, using the theory of the bundle of accelerations of order two, T^{2}M.

Mathematics Subject Classification: 53C60 Key words: Randers spaces, nonlinear connection, N−connection

1 Introduction

The first examples of regular Lagrangians of order greater then one were given by R.Miron. These are the prolongations of order k, (k > 1) of a Riemannian, or a Finslerian metric. Using the prolongation of order two of a Riemannian metric γ_{ij} and a globally defined covector field A_{i} on the base manifold, we define the fundamental function F of a Randers space RF^{(2)} = (M, α+β). Then we determine the fundamental tensor g_{ij} of RF^{(2)} and the relation between d-tensors γ_{ij} and g_{ij}, that is similar with the k=1 case. As this Randers space is a Finsler space of order two, we use techniques of Finslerian geometries of higher order to determine the canonical nonlinear connection, the canonical metrical N−connection. We may notice here that all the geometrical objects depend on the electromagnetic tensor F_{ij} of the space RF^{(2)}.

2 The notion of Randers space of order two

Let M be an n-dimensional, smooth manifold and R^{n} = (M, γ_{ij}(x)) be a Riemannian space. We denoted by P rol^{2}R^{n} = (T̃^{2}M, G) the prolongation of order two of the

space R^{n}, [8]. The nonlinear connection N^{∘} of the space P rol^{2}R^{n}, [8], has the dual coefficients

Balkan Journal of Geometry and Its Applications, Vol.6, No.1, 2001, pp. 87-92 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

(1.1)  \{ \begin{matrix}M_{j}^{i}^{∘}_{(1)}= γ_{jh}^{i}(x)y^{(1)h}, \\ M_{j}^{i}^{∘}_{(2)}= \frac{1}{2}(Γ M_{j}^{i}^{∘}_{(1)} + M_{m}^{i}^{∘}_{(1)}M_{j}^{m}^{∘}_{(1)}), \end{matrix}

where Γ is the operator

(1.2)  Γ = y^{(1)i} \frac{∂}{∂x^{i}} + 2y^{(2)i} \frac{∂}{∂y^{(1)i}}

and γ_{jk}^{i}(x) are the Christoffel symbols of the Riemannian space R^{n}. Between the coefficients of the nonlinear connection N_{j}^{i}^{∘}_{(1)}, N_{j}^{i}^{∘}_{(2)} and the dual coefficients M_{j}^{i}^{∘}_{(1)}, M_{j}^{i}^{∘}_{(2)} we have the relations:

(1.3)  \{ \begin{matrix}N_{j}^{i}^{∘}_{(1)}=M_{j}^{i}^{∘}_{(1)}, \\ N_{j}^{i}^{∘}_{(2)}=M_{j}^{i}^{∘}_{(2)} − M_{m}^{i}^{∘}_{(1)}M_{j}^{m}^{∘}_{(1)} . \end{matrix}

Having the coefficients of the nonlinear connection N^{∘} we can write its adapted

basis, \{\frac{δ^{∘}}{δx^{i}} , \frac{δ^{∘}}{δy^{(1)i}} , \frac{δ^{∘}}{δy^{(2)i}} \}, given by

(1.4)  \{ \begin{matrix}\frac{δ^{∘}}{δx^{i}} = \frac{∂}{∂x^{i}} − N_{i}^{j}^{∘}_{(1)} \frac{∂}{∂y^{(1)j}} − N_{i}^{j}^{∘}_{(2)} \frac{∂}{∂y^{(2)j}} , \\ \frac{δ^{∘}}{δy^{(1)i}} = \frac{∂}{∂y^{(1)i}} − N_{i}^{j}^{∘}_{(1)} \frac{∂}{∂y^{(2)j}} \\ \frac{δ^{∘}}{δy^{(2)i}} = \frac{∂}{∂y^{(2)i}} . \end{matrix}

The dual basis of the previous adapted basis is given by {dx^{i}, δ^{∘} y^{(1)i}, δ^{∘} y^{(2)i}}, where:

\{ \begin{matrix}δ^{∘} y^{(1)i} = dy^{(1)i}+ M_{j}^{i}^{∘}_{(1)} dx^{j}, \\ δ^{∘} y^{(2)i} = dy^{(2)i}+ M_{j}^{i}^{∘}_{(1)} dy^{(1)j}+ M_{j}^{i}^{∘}_{(2)} dx^{j}. \end{matrix}

We consider the Liouville d-vector fields, [8]

(1.5)  \begin{matrix}z^{(1)m} = y^{(1)m}, \\ z^{(2)m} = y^{(2)m} + \frac{1}{2}γ_{ij}^{m}(x)y^{(1)i}y^{(1)j}. \end{matrix}

The following theorem is known

Theorem 2.1 The function α^{2} = γ_{ij}(x)z^{(2)i}z^{(2)j} is a differential Lagrangian. It has the properties:

1) α^{2} is global defined on T̃^{2}M = T^{2}M \ {0};

2) α^{2} is a regular Lagrangian;

3) α^{2} depends on the metric γ_{ij}, only;

4) The Lagrangian α^{2} is homogeneous of order 4 on the fibres of T^{2}M;

5) The fundamental tensor field of α^{2} is given by

(1.6)  \frac{1}{2} \frac{∂^{2}α^{2}}{∂y^{(2)i}∂y^{(2)j}} = γ_{ij}(x).

Let us consider the functions

(1.7)  β(x, y^{(1)}, y^{(2)}) = A_{i}(x)z^{(2)i}

where A_{i}(x) are the electromagnetic potentials on the base manifold M. Clearly, the function β has a physical means.

Let us consider the function F : T^{2}M → R which is given by F = α + β, i.e.

(1.8)  F (x, y^{(1)}, y^{(2)}) = \sqrt{γ_{ij}(x)z^{(2)i}z^{(2)j}} + A_{i}(x)z^{(2)i}

and the square of this function

(1.8)′  L(x, y^{(1)}, y^{(2)}) = (α + β)^{2}.

We can formulate

Theorem 2.2 a) The function L = (α + β)^{2} is a differentiable Lagrangian of order 2;

b) F is 2-homogeneous and L is 4-homogeneous on the fibers of T^{2}M;

c) The fundamental tensor field of the Lagrangian L = (α + β)^{2} is given by

(1.9)  g_{ij} = (pγ_{ij} + l_{i}l_{j}) − p l_{i}^{∘}l_{j}^{∘}

where l_{i}^{∘}= \frac{∂α}{∂y^{(2)i}} , l_{i} =l_{i}^{∘} +A_{i}, p = \frac{α + β}{α}.

Proof. a) L is of C^{∞}-class on T̃^{2}M and continuous on the null section of the projection π : T^{2}M ⟶ M because α and β have these properties.

b) It is known that z^{(2)i} is 2-homogeneous of the fibres of T̃^{2}M.

c) By a straightforward calculus we obtain the formula (1.9). □

Theorem 2.3 The pair F^{(2)n} = (M, F = α + β) is a Finsler space of order 2.

Proof. We must show the following properties:

1. F is of C^{∞}-class on T̃^{2}M and continuous on the null section;

2. F is positive on an open set, where β ≥ 0;

3. F is 2-homogeneous on the fibres of T^{2}M;

4. The Hessian of F^{2} with the elements:

g_{ij} = \frac{1}{2} \frac{∂^{2}F^{2}}{∂y^{(2)i}∂y^{(2)j}}

is positively defined.

Since the properties 1,2,3 holds in virtue of the properties of α and β we have to prove the property 4. In order to do it, we calculate the contravariant g^{ij} of the fundamental tensor field of the Lagrangian L.

The contravariant g^{ij} of g_{ij} is given by:

(1.10)  g^{ij} = \frac{1}{p}γ^{ij} − \frac{1}{p^{2}} [l^{i}^{∘}l^{j}^{∘} (1 − l̃^{2})+ l^{i}^{∘} A^{j}+ l^{j}^{∘} A^{i}],

where l̃^{2} = \frac{1}{p}γ^{ij}l_{i}l_{j}.

Moreover, det||g_{ij}|| = p^{n+1}det||γ_{ij}|| where p = \frac{α + β}{β} > 0. □

The Finsler space RF^{(2)n} = (M, F) is called the Randers space of order two.

Theorem 2.4 The nonlinear connection N^{∘} of the space P rol^{2}R^{n}, from (1.1), is a nonlinear connection for the Randers space of order 2 RF^{(2)n} determined only by the fundamental function F = α + β.

3 Nonlinear connection of the space RF^{(2)n}

Let us consider the mixed form F_{j}^{i}(x) = γ^{im}(x)F_{mj}(x) of the electromagnetic tensor

field F_{mj}(x) = \frac{∂A_{j}}{∂x^{m}} − \frac{∂A_{m}}{∂x^{j}} .

Theorem 3.1 The Randers space RF^{(2)n} = (M, α + β) has a nonlinear connection N, whose dual coefficients are given by (2.1)

\{ \begin{matrix}M_{j}^{i}_{(1)}=M_{j}^{i}^{∘}_{(1)} −F_{j}^{i}||y^{(1)}||, \\ M_{j}^{i}_{(2)}=M_{j}^{i}^{∘}_{(2)} −\frac{1}{2}Γ(F_{j}^{i}||y^{(1)}||) + \frac{1}{2}(M_{m}^{i}^{∘} F_{j}^{m}_{(1)} + M_{j}^{m}^{∘}_{(1)} F_{m}^{i})||y^{(1)}|| − \frac{1}{2}||y^{(1)}||^{2}F_{m}^{i}F_{j}^{m}. \end{matrix}

where ||y^{(1)}|| = \sqrt{γ_{ij}(x)y^{(1)i}y^{(1)j}}.

Indeed, M_{j}^{i}_{(1)} has the same rule of transformation with respect to changes of coor-

dinates on T̃^{2}M like M_{j}^{i}^{∘}_{(1)} and M_{j}^{i}_{(2)}= \frac{1}{2}(Γ M_{j}^{i}_{(1)} + M_{m}^{i}_{(1)}M_{j}^{m}_{(1)}).

The nonlinear connection (2.1) will be called canonical nonlinear connection of the Randers space RF^{(2)n}.

The relations between coefficients (N_{j}^{i}_{(1)}, N_{j}^{i}_{(2)}) and dual coefficients of the nonlinear

connection N are given by:

(2.2)  \{ \begin{matrix}N_{j}^{i}_{(1)}=M_{j}^{i}_{(1)}, \\ N_{j}^{i}_{(2)}= \frac{1}{2}(Γ M_{j}^{i}_{(1)} + M_{m}^{i}_{(1)}M_{j}^{m}_{(1)})− M_{m}^{i}_{(1)}M_{j}^{m}_{(1)}= \frac{1}{2}(Γ N_{j}^{i}_{(1)} − N_{m}^{i}_{(1)}N_{j}^{m}_{(1)}). \end{matrix}

We use this canonical nonlinear connection in study of the canonical metrical N-connection.

4 The canonical N-linear connection

Let us consider d-tensor field

(3.1) T_{i}^{j} = Γ(F_{i}^{j}||y^{(1)}||) + F_{j}^{i}||y^{(1)}|| − (M_{m}^{i}^{∘} F_{j}^{m}_{(1)} + M_{j}^{m}^{∘}_{(1)} F_{m}^{i})||y^{(1)}|| + ||y^{(1)}||^{2}F_{m}^{i}F_{j}^{m}

The adapted basis \{\frac{δ}{δx^{i}} , \frac{δ}{δy^{(1)i}} , \frac{∂}{∂y^{(2)i}} \} of the canonical nonlinear connection can

be expressed in the following form

(3.2)  \{ \begin{matrix}\frac{δ}{δx^{i}} = \frac{δ^{∘}}{δx^{i}} + F_{i}^{j}||y^{(1)}||\frac{δ^{∘}}{δy^{(1)j}} + \frac{1}{2}T_{i}^{j} \frac{∂}{∂y^{(2)j}} , \\ \frac{δ}{δy^{(1)i}} = \frac{δ^{∘}}{δy^{(1)i}} + F_{i}^{j}||y^{(1)}||\frac{∂}{∂y^{(2)j}} \\ \frac{δ}{δy^{(2)i}} = \frac{∂}{∂y^{(2)i}} . \end{matrix}

Of course, this adapted basis depend only on the fundamental function F = α + β of the Randers spaces of order two, RF^{(2)n}.

Using a very known method we can determine the canonical N-linear connection D on T̃^{2}M metrical with respect to fundamental tensor g_{ij}, which depend only on the fundamental function F of the space RF^{(2)n}.

We have:

Theorem 4.1 1) There exists an unique N-linear connection D on T̃^{2}M which verifies the following axioms:

1^{∘} The nonlinear connection N is specified by (2.2).

2^{∘} g_{ij|k} = 0, g_{ij |^{(1)}k} = 0, g_{ij |^{(2)}k} = 0,

3^{∘} T_{jk}^{i} =S_{jk}^{i}^{(1)}=S_{jk}^{i}^{(2)}= 0.

2) This connection has the coefficients given by the generalized Christoffel symbols:

(3.3)  \{ \begin{matrix}L_{jm}^{i} = \frac{1}{2}g^{is} \{\frac{δg_{sm}}{δx^{j}} + \frac{δg_{sj}}{δx^{m}} − \frac{δg_{jm}}{δx^{s}} \} , \\ C_{(1)} _{jm}^{i} = \frac{1}{2}g^{is} \{\frac{δg_{sm}}{δy^{(1)j}} + \frac{δg_{sj}}{δy^{(1)m}} − \frac{δg_{jm}}{δy^{(1)s}} \} , \\ C_{(2)} _{jm}^{i} = \frac{1}{2}g^{is} \{\frac{∂g_{sm}}{∂y^{(2)j}} + \frac{∂g_{sj}}{∂y^{(2)m}} − \frac{∂g_{jm}}{∂y^{(2)s}} \} . \end{matrix}

3) This connection depend only on Randers space RF^{(2)n}.

The metrical connection from the previous Theorem is the canonical connection of the Randers space RF^{(2)n} = (M, α + β). 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.

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