On Holland’s frame for C-reducible Finsler space. P.L.Antonelli
∗
Department of Mathematical Sciences, University of Alberta Edmonton, Alberta, Canada, T6G 2G1 e-mail:
[email protected]
I.Bucataru
†
Faculty of Mathematics, ”Al.I.Cuza” University Iasi, 6600, Romania e-mail:
[email protected]
Abstract It has been proved that the class of C-reducible Finsler spaces, introduced by M. Matsumoto, [11] reduces to Randers and Kropina spaces, [12]. In two previous paper, [2], [3], we have determined a canonical Finsler frame for these two Finsler spaces: Randers and Kropina. As is well known, these Finsler spaces have two canonical metrics, a Riemannian one, and one that is truly Finslerian. The Finsler frame we have determined relates these metrics. Using this we can determine the maximum domain where the Finsler metric is positive definite. A Finsler connection induced by this Finsler frame and with respect to which, the Finsler frame is covariant constant, is determined. For this Finsler connection all components of curvature are zero.
2000 Mathematics Subject Classification: 53C60, 53B40. Key words and phrases: Finsler space, Finsler connection, Finsler frame. ∗ †
Partially supported by NSERC-7667 PIMS Postdoctoral Fellow
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Introduction It is well known that every two-dimensional Finsler space F 2 has a canonical, orthogonal frame. It was introduced by Berwald in [5], and is called now the Berwald frame. Using it, important geometric objects in the geometry of a Finsler space of dimension two, for example the main scalar, can be determined. For a three-dimensional Finsler space there exists an orthogonal frame, called Mo´or frame, [15]. For the n-dimensional case a Finsler frame for the class of so-called strongly non-Riemannian spaces, has been introduced [14]. This frame is called Miron frame, [9], p.179. The important class of Finsler spaces with (α, β) metrics was introduced by Matsumoto in [11]. He proved, [10] that a Finsler space with (α, β) metric of dimension n > 3 is not strongly Riemannian. So, for this class of Finsler spaces with (α, β) metric we didn’t have a canonical frame. In this paper we determine a Finsler frame, called by us Holland’s frame, for the best known Finsler spaces with (α, β)-metrics, the Randers and Kropina spaces. These two types of Finsler metrics determine the class of C-reducible Finsler spaces, [11]. We determine a Finsler connections with respect to which, the Finsler frame is h- and v-covariant constant. As this Finsler connection is flat, then for a C-reducible Finsler space there exist a canonical flat Finsler connection.
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Finsler spaces
Consider M an n-dimensional C ∞ -manifold, and denote by (T M, π, M ) its tangent bundle. If (U, ϕ = (xi )) is a local chart at p ∈ M , then the corresponding local chart at a point u ∈ π −1 (p) is (π −1 (U ), φ = (xi , y i )). The indices, i, j, k, ... run from 1 to n. The summation convention is implied. Denote by Tg M the tangent space with zero section removed. Definition 1.1 A Finsler space is a pair (M, F ) with F : T M → R (F is called the fundamental function) such that: 1o F is of C ∞ -class on Tg M and continuous on the zero section; 2o F is positive homogeneous of degree one with respect to y; 1 ∂ 2F 2 is everywhere nondegener3o The matrix with the entries gij = 2 ∂y i ∂y j ate on Tg M. 2
A tensor field of (r, s) type on T M is called a Finsler tensor field (or a d-tensor field) if under a change of the induced coordinates on T M its components transform like the components of a (r, s)-type tensor on the base manifold. It is very easy to see that gij are the components of a (0, 2)-type Finsler tensor field and is called the fundamental tensor, or the metric tensor of the Finsler space. From this we have another important Finsler tensor field, 3 2 ij Cijk = 21 ∂g = 41 ∂yi∂∂yFj ∂yk , which is called the Cartan tensor. Two other ∂y k important tensors in Finsler geometry are: Ci = Cijk g jk , the so-called torsion 2F vector, and hij = F ∂y∂i ∂y j , the angular metric tensor. Now consider aij a metric (it can be a Riemannian or non-Riemannian metric) and bi a 1-form on the base manifold M . Definition 1.2 A Finsler space (M, F ) is called with (α, β)-metric if the fundamental function F (x, y) is a positively homogeneous function L(α, β) of first degree in two variables (1.1)
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α(x, y) = (aij (x)y i y j ) 2 , and β(x, y) = bi (x)y i .
Two particular cases are very important in Finsler geometry: 1o If F (x, y) = α(x, y)+β(x, y), then the Finsler space is called a Randers space; α2 (x, y) , then the Finsler space is called a Kropina space. 2o If F (x, y) = |β(x, y)| Definition 1.3 ([11]) A Finsler space of dimension n > 2 is called Creducible if the Cartan tensor Cijk has the following form: (1.2)
Cijk = (hij Ck + hjk Ci + hki Cj )/(n + 1).
According to [12], we have the following theorem: Theorem 1.1 A Finsler space is C-reducible, if and only if the space is either a Randers or a Kropina space. The Randers and Kropina spaces are more related that the previous theorem says. Hrimiuc and Shimada have proved that these two spaces are dual via a Legendre transformation, [8]. And we did use this duality to determine a Finsler frame for a Kropina space from a Finsler frame of a Randers space.
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2
Finsler frame for a C-reducible Finsler space.
Let (aij ) be a metric (a Riemannian or a non-Riemannian metric) and bi (x) an 1-form on the base manifold M . Consider α(x, y) and β(x, y) as they have been defined in (1.1). Next all the indices will be raised and down only by the metric aij , i.e. yi = aij y j , bi = aij bj . The function F : T M → R, defined by F (x, y) = α(x, y) + β(x, y) is the fundamental function of a Finsler space, [11]. The pair (M, F ) is called a ∂2F 2 Randers space. If we consider the fundamental tensor gij = 21 ∂y i ∂y j , then the metric tensors aij and gij are related by: (2.1)
1 β F aij + (bi yj + bj yi ) − 3 yi yj + bi bj . α α α
gij =
Theorem 2.1 ([2]) For a Randers space (M, F ), consider the open subset V of T M where Fα > 0. 1o The matrix with the entries: (2.2)
Yji
r
=
s
1 1 α i F δj − y i bj + ( − 1)y i yj F F αF α
is everywhere invertible on V , its inverse is given by s
(2.2)0
(Y −1 )ij =
s
1 1 F i F δj + √ y i b j − √ ( − 1)y i yj . α α αF α αF
2o The metric tensor gij and aij are related by the following formula (2.3)
gij (x, y) = Yik (x, y)Yjl (x, y)akl (x).
The proof of the theorem can be found in [2]. The frame given by (2.2) is called the Holland frame of the Randers space. It was first P.R. Holland, who found a similar version of (2.2) for a 4-dimensional Randers spaces, associated to a Lorentz manifold, [6], [7]. From the (2.3) formula we can deduce: Corollary 2.1 The Randers metric gij has the same signature on V as the metric aij does have on M . Consequently if (aij ) is positively defined on M , then the Randers metric is also positive defined on V .
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We have some similar properties for a Kropina space (M, F ) where F (x, y) = α (x, y) . The relation between the Kropina metric gij and the metric aij is |β(x, y)| given by: 2
(2.4)
gij =
2α2 3 α2 2 2 (y b + b y ) + [a − bi bj + 2 yi yj ]. i j i j ij 2 2 β β 2β α
Theorem 2.2 ([3]) For a Kropina space (M, F ), consider V , the open subset of T M where β(x, y) 6= 0. 1o The Holland frame for the Kropina space is given by: (2.5) √ √ √ √ √ " # 2 i 1 2 i 2β 1 2 i α 2 i α 2 i i δj − b yj + ( − )y bj − 2 ( − )y yj + b bj , Yj (x, y) = β αb β αb α β αb 2bβ q
for (x, y) ∈ V . We have denoted by b = aij bi bj . 2o The metric tensor gij and aij are related by the following formula (2.6)
gij (x, y) = Yik (x, y)Yjl (x, y)akl (x).
We may remark here that (2.5) can be determined from (2.2) by considering the L-duality between Randers and Kropina spaces. From (2.6) we have a similar consequence as the Corollary 2.1, so the Kropina metric has the same signature with the metric aij .
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Finsler connections induced by a Finsler frame
In the last section we have seen that every C-reducible Finsler space (Randers and Kropina) has a canonical Finsler frame (Yji (x, y)), such that the Finsler metric (gij ) and the Riemannian metric (aij ) are related by the formula gij (x, y) = Yik (x, y)Yjl (x, y)akl (x). We shall see in this section that the existence of such a frame determines a flat Finsler connection. If (T M, π, M ) its tangent bundle of the manifold M , denote by π∗ : T T M → T M the linear map induced by the canonical projection π. The kernel of the linear mapping π∗ : T T M → T M is called the vertical distribution and is denoted by V T M. For every u ∈ T M, Ker π∗,u = Vu T M is 5
spanned by { ∂y∂ i |u }. By a nonlinear connection on T M we mean a regular n-dimensional distribution H : u ∈ T M 7→ Hu T M which is supplementary to the vertical distribution i.e. (3.1)
Tu T M = Hu T M ⊕ Vu T M, ∀ u ∈ T M.
A basis for Tu T M adapted to the direct sum (3.1) has the form { δxδ i |u = ∂ | − Nij (u) ∂y∂ j |u , ∂y∂ i |u }. The dual basis of this is (dxi , δy i = dy i + Nji dxj ). ∂xi u These are the Berwald bases. The vector field mapping J : χ(T M ) → χ(T M ), defined by: J = ∂y∂ i ⊗dxi , is globally defined on T M and is called the almost tangent structure. Definition 3.1 A linear connection D on T M is called a Finsler connection (or an N -linear connection) if: 1◦ D preserves by parallelism the horizontal distribution HT M ; 2◦ The almost tangent structure J is absolute parallel with respect to D. For a Finsler connection D it is immediate that D preserves also the vertical distribution. In the Berwald basis ( δxδ i , ∂y∂ i ) adapted to the decomposition (3.1), a Finsler connection can be expressed as: (3.2)
D
D
δ δxi ∂ ∂y i
δ δxj
= Fjik δxδ k ; D
δ δxj
= Cjik δxδ k ; D
δ δxi ∂ ∂y i
∂ ∂y j
= Fjik ∂y∂ k ;
∂ ∂y j
= Cjik ∂y∂ k .
Observe that under a change of induced coordinates on T M the functions Fjik transform like the coefficients of a linear connection on M and Cjik is a (1, 2)-type Finsler tensor field. For a Finsler connection D we may define the horizontal and the vertical covariant derivative of a Finsler tensor field, that is also a Finsler tensor field. For example, if Tji are the components of an (1, 1) Finsler tensor field then h and v-covariant derivatives are given by: (3.3)
i Tj|k
=
δTji δxk
i r + Frk Tjr − Fjk Tri and
Tji |k =
∂Tji ∂y k
i r + Crk Tjr − Cjk Tri .
For a Finsler connection D one considers typically T (X, Y ) = DX Y − DY X − [X, Y ] and R(X, Y )Z = DX DY Z − DY DX Z − D[X,Y ] Z 6
the torsion and the curvature. It is well known [4], [13] that in the Berwald basis there are only five components of torsion and three components of curvature. The five components of torsion are:
(3.4)
hT ( δxδ i , δxδ j ) =: Tijk δxδ k = (Fjik − Fijk ) δxδ k ; δN k
k ∂ = ( δxji − vT ( δxδ i , δxδ j ) =: Rij ∂y k
δNjk ) ∂ ; δxi ∂y k
hT ( ∂y∂ i , δxδ j ) = Cjik δxδ k ;
(h)h−torsion (v)h−torsion (h)hv−torsion
∂N k
vT ( ∂y∂ i , δxδ j ) =: Pijk ∂y∂ k = ( ∂yji − Fijk ) ∂y∂ k ;
(v)hv−torsion
vT ( ∂y∂ i , ∂y∂ j ) =: Sijk ∂y∂ k = (Cjik − Cijk ) ∂y∂ k
(v)v−torsion.
The three components of curvature are given by:
(3.5)
i Rjkh =
i = Pjkh
i Sjkr
=
i δFjk δxh
−
i ∂Fjk h ∂y
m i i Pkh + Cjm − Cjk|h
i ∂Cjk ∂y r
−
i δFjh δxk
i ∂Cjr ∂y k
m i m i i m + Fjk Fmh − Fjh Fmk + Cjm Rkh
m i m i Cmk Cmr − Cjr + Cjk
For a Finsler connection D we have the following Ricci identity
(3.6)
i i X|k|r − X|r|k
i i m m = X m Rmkr − X|m Tkr − X i |m Rkr
X|ki |r − X i |r|k
m i m i − X i |m Pkr − X|m Ckr = X m Pmkr
i m X i |k |r − X i |r |k = X m Smkr − X i |m Skr .
Now for the Finsler Frame determined by (2.2) or (2.5) we fix the covariant index. In order to express this we shall use greek letters for a fixed index. So the Finsler frame will be denoted by now as (Yαi (x, y)), α ∈ {1, ..., n}. Its index (with the contravariant index fixed) is denoted by (Yjα (x, y)). So we have (3.7)
Yαi Yjβ = δαβ , Yαi Yjα = δji .
Theorem 3.1 For a C-reducible Finsler space there exists a Finsler connections D for which the Finsler frame (Yαi ) is h- and v-covariant constant. 7
i Proof From (3.3) we have that Yα|k = 0 is equivalent with
δYαi i + Yαm Fmk = 0. δxk Using (3.7) we can find the horizontal coefficients of the Finsler connection, we are looking for as: (3.8)
i = −Yjα Fjk
α δYαi i δYj = Y . α δxk δxk
Similarly, we have that Yαi |k = 0 is equivalent with ∂Yαi i = 0. + Yαm Cmk ∂y k Using (3.7) we can find the vertical coefficients of the Finsler connection, we are looking for as: (3.8)0
i Cjk = −Yjα
α ∂Yαi i ∂Yj = Y . α ∂y k ∂y k
i i So, the Finsler connection D with the local coefficients Fjk , and Cjk given by (3.8) and (3.8)’ satisfies the theorem. We call this Finsler connection, the Crystallographic connection of the Finsler frame, [1]. Proposition 3.1 All three components of the curvature of the Crystallographic connection vanish, so, the Crystallographic connection is flat. i Proof As Yα|j = 0, and Yαi |j = 0, from the Ricci identities (3.6) we have i i i = 0. Using (3.7) we have that = 0, and Yαm Smkr = 0, Yαm Pmkr Yαm Rmkr i i i Rjkl = Pjkl = Sjkl = 0. Remarks It is well known that for a two-dimensional Finsler space there exists an orthogonal Finsler frame. This is called the Berwald frame and is given by li = F1 y i , mi = C1 C i , where C i is the torsion vector and C is the Finslerian length of C i . Denote by Y1i = li , and Y2i = mi , then Yαi , α ∈ {1, 2} is the Berwald frame. As the Berwald frame is h-covariant constant with respect to the Cartan connection we can see that the horizontal components of the Cartan connection of a two-dimensional Finsler case are given by (3.8). Acknowledgements. The second author would like to thank to Professor P.L. Antonelli for his support during the PDF at the University of Alberta.
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