arXiv:0805.1336v4 [math.DG] 5 Aug 2009
Extended Absolute Parallelism Geometry Nabil. L. Youssef and A. M. Sid-Ahmed Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
[email protected],
[email protected] [email protected],
[email protected]
Abstract. In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle T M of a manifold M. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument x, but also depend on the directional argument y. Moreover, many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this different framework. We refer to such a geometry as an Extended Absolute Parallelism (EAP-) geometry. The building blocks of the EAP-geometry are a nonlinear connection assumed given a priori and 2n linearly independent vector fields (of special form) defined globally on T M defining the parallelization. Four different d-connections are used to explore the properties of this geometry. Simple and compact formulae for the curvature tensors and the W-tensors of the four defined d-connections are obtained, expressed in terms of the torsion and the contortion tensors of the EAP-space. Further conditions are imposed on the canonical d-connection assuming that it is of Cartan type (resp. Berwald type). Important consequences of these assumptions are investigated. Finally, a special form of the canonical d-connection is studied under which the classical AP-geometry is recovered naturally from the EAP-geometry. Physical aspects of some of the geometric objects investigated are pointed out and possible physical implications of the EAP-space are discussed, including an outline of a generalized field theory on the tangent bundle T M of M. 1 Keywords: Parallelizable manifold, Absolute Parallelism, Extended Absolute Parallelism, Metric d-connection, Canonical d-connection, W -tensor, Field equations, Cartan type, Berwald type. 2000 AMS Subject Classification. 53B40, 53A40, 53B50.
1
ArXiv number: 0805.1336
1
0. Introduction The geometry of parallelizable manifolds or the Absolute Parallelism geometry (APgeometry) ([4], [11], [12], [14], [17]) has many advantages in comparison to Riemannian geometry. Unlike Riemannian geometry, which has ten degrees of freedom (corresponding to the metric components for n = 4), AP-geometry has sixteen degrees of freedom (corresponding to the number of components of the four vector fields defining the parallelization). This makes AP-geometry a potential candidate for describing physical phenomena other than gravity. Moreover, as opposed to Riemannian geometry, which admits only one symmetric linear connection, AP-geometry admits at least four natural (built-in) linear connections, two of which are non-symmetric and three of which have non-vanishing curvature tensors. Last, but not least, associated with an AP-space there is a Riemannian structure defined in a natural way. Thus, AP-geometry contains within its geometric structure all the mathematical machinery of Riemannian geometry. Accordingly, a comparison can be made between the results obtained in the context of AP-geometry and general relativity, which is based on Riemannian geometry. The geometry of the tangent bundle (T M, π, M) of a smooth manifold M is very rich. It contains a lot of geometric objects of theoretical interest and of a great importance in the construction of various geometric models which have proved very useful in different physical theories. Examples of such theories are the general theory of relativity, particle physics, relativistic optics and others. In this paper, we study AP-geometry in a context different from the classical one. Instead of dealing with geometric objects defined on the manifold M, as in the case of classical AP-space, we are dealing with geometric objects defined on the tangent bundle T M of M. Accordingly, all geometric objects considered are, in general, not only functions of the positional argument x, but also depend on the directional argument y. The paper is organized in the following manner. In section 1, following the introduction, we give a brief account of the basic concepts and definitions that will be needed in the sequel. The definitions of a d-connection, d-tensor field, torsion, curvature, hv-metric and metric d-connection on T M are recalled. We end this section by the construction of a (unique) metric d-connection on T M which we refer to as the natural metric d-connection. In section 2, we introduce the Extended Absolute Parallelism (EAP-) geometry by assuming that T M is parallelizable [3] and equipped with a nonlinear connection. The canonical d-connection is then defined, expressed in terms of the natural metric d-connection. In analogy to the classical AP-geometry, two other d-connections are introduced: the dual and the symmetric d-connections. We end this part with a comparison between the classical AP-geometry and the EAP-geometry. In section 3, we carry out the task of computing the different curvature tensors of the four defined d-connections. They are expressed, in a relatively compact form, in terms of the torsion and the contortion tensors of the EAP-space. All admissable contractions of these curvature tensors are also obtained. In section 4, we introduce and investigate the different W -tensors corresponding to the different d-connections defined in the EAPspace, which are again expressed in terms of the torsion and the contortion tensors. In sections 5 and 6, we assume that the canonical d-connection is of Cartan and Berwald type respectively. Some interesting results are obtained, the most important of which is 2
that, in the Cartan type case, the given nonlinear connection is not independent of the vector fields forming the parallelization, but can be expressed in terms of their vertical counterparts. In section 7, we further assume that the canonical d-connection is both of Cartan and Berwald type. We show that, under this assumption, the classical APgeometry is recovered, in a natural way, from the EAP-geometry. In section 8, we end this paper with some concluding remarks which reveal possible physical applications of the EAP-space; among them is an outline of a generalized field theory on the tangent bundle T M of M, based on Euler-Lagrange equations [8] applied to a suitable scalar Lagrangian.
1. Fundamental Preliminaries In this section we give a brief account of the basic concepts and definitions that will be needed in the sequel. Most of the material covered here may be found in [8], [9] with some slight modifications. Let M be a paracompact manifold of dimension n of class C ∞ . Let π : T M → M be its tangent bundle. If (U, xµ ) is a local chart on M, then (π −1 (U), (xµ , y a )) is the corresponding local chart on T M. The coordinate transformation on T M is given by: ′
′
′
′
xµ = xµ (xν ), y a = paa y a , a′
′
a′
′
µ = 1, ..., n; a = 1, ..., n; paa = ∂y = ∂x and det(paa ) 6= 0. The paracompactness of ∂y a ∂xa M ensures the existence of a nonlinear connection N on T M with coefficients Nαa (x, y). The transformation formula for the coefficients Nαa is given by ′
′
′
′
Nαa′ = paa pαα′ Nαa + paa pac′α′ y c , where pac′ α′ =
∂pa c′
∂xα′
(1.1)
. The nonlinear connection leads to the direct sum decomposition Tu (T M) = Hu (T M) ⊕ Vu (T M), ∀u ∈ T M \ {0}.
(1.2)
Here, Vu (T M) is the vertical space at u with local basis ∂˙a := ∂y∂a , whereas Hu (T M) is the horizontal space at u associated with N supplementary to the vertical space Vu (T M). The canonical basis of Hu (T M) is given by δµ := ∂µ − Nµa ∂˙a ,
(1.3)
where ∂µ := ∂x∂µ . Now, let (dxα , δy a ) be the basis of Tu∗ (T M) dual to the adapted basis (δα , ∂˙a ) of Tu (T M). Then δy a := dy a + Nαa dxα (1.4) and
dxα (δβ ) = δβα , dxα (∂˙a ) = 0;
δy a (δβ ) = 0, δy a (∂˙b ) = δba .
(1.5)
Any vector field X ∈ X(T M) is uniquely decomposed in the form X = hX + vX, where h and v are respectively the horizontal and the vertical projectors associated with the decomposition (1.2). In the adapted frame (δν , ∂˙a ), hX = X α δα and vX = X a ∂˙a . 3
Definition 1.1. A nonlinear connection Nµa is said to be homogeneous if it is positively homogeneous of degree 1 in the directional argument y. Definition 1.2. A d-connection on T M is a linear connection on T M which preserves by parallelism the horizontal and vertical distribution: if Y is a horizontal (vertical) vector field, then DX Y is a horizontal (vertical) vector field, for all X ∈ X(T M). Consequently, a d-connection D on T M has only four coefficients. The coefficients α a of a d-connection D = (Γαµν , Γabν , Cµc , Cbc ) are defined by Dδν ∂˙b =: Γabν ∂˙a ;
Dδν δµ =: Γαµν δα ,
α D∂˙c δµ =: Cµc δα ,
a ˙ D∂˙c ∂˙b =: Cbc ∂a .
(1.6)
The transformation formulae of a d-connection are given by: ′
′
′
Γαµ′ ν ′ = pαα pµµ′ pνν ′ Γαµν + pαǫ pǫµ′ ν ′ , ′
′
′
′
′
′
Γab′ µ′ = paa pbb′ pµµ′ Γabµ + pac pcb′ µ′ ;
′
α , Cµα′ c′ = pαα pµµ′ pcc′ Cµc
a . Cba′ c′ = paa pbb′ pcc′ Cbc
A comment on notation: Both Greek indices {α, β, µ, ...} and Latin indices {a, b, c, ...}, as previously mentioned, take values from the same set {1, ..., n}. It should be noted, however, that Greek indices are used to denote horizontal counterpart, whereas Latin indices are used to denote vertical counterpart. Einstein convention is applied on both types of indices. Definition 1.3. A d-tensor field T on T M of type (p, r; q, s) is a tensor field on T M which can be locally expressed in the form ...up+r T = Tvu11...v ∂u1 ⊗ ... ⊗ ∂up+r ⊗ dxv1 ⊗ ... ⊗ dxvq+s , q+s
where ui ∈ {αi , ai }, vj ∈ {βj , bj }, ∂ui ∈ {δαi , ∂˙ai }, dxvj ∈ {dxβj , δy bj }, i = 1, ..., p + r; j = 1, ..., q + s, so that the number of αi ’s = p, the number of ai ’s = r, the number of βj ’s = q and the number of bj ’s = s. αa Let T = Tβb δα ⊗ ∂˙a ⊗dxβ ⊗δy b be a d-tensor field of type (1, 1; 1, 1). Let X ∈ X(T M) be such that X = hX + vX = X µ δµ + X c ∂˙c . Then, by the properties of a d-connection, we have D hX T := DhX T = (X µ T αa βb|µ )δα ⊗ ∂˙a ⊗ dxβ ⊗ δy b,
where αa ǫa α αd a αa d + Tβb Γǫµ + Tβb Γdµ − Tǫbαa Γǫβµ − Tβd Γbµ . T αa βb|µ := δµ Tβb
Similarily, where
(1.7)
D vX T := DvX T = (X c T αa βb||c )δα ⊗ ∂˙a ⊗ dxβ ⊗ δy b, αa ǫa α αd a ǫ αa d + Tβb Cǫc + Tβb Cdc − Tǫbαa Cβc − Tβd Cbc . T αa βb||c := ∂˙c Tβb
(1.8)
It is evident that (1.7) and (1.8) can be written for any d-tensor field of arbitrary type. 4
Definition 1.4. The two operators D hX (denoted locally by |) and D vX (denoted locally by ||) are called respectively the horizontal (h-) and vertical (v-) covariant derivatives associated with the d-connection D. Definition 1.5. The torsion T of a d-connection D on T M is defined by T(X, Y ) := DX Y − DY X − [X, Y ]; ∀X, Y ∈ X(T M).
(1.9)
For getting the local expression for T, we first recall that a ˙ [δµ , δν ] = Rµν ∂a ;
[δµ , ∂˙b ] = (∂˙b Nµa )∂˙a ,
where a Rµν := δν Nµa − δµ Nνa
(1.10)
is the curvature of the nonlinear connection. By a direct substitution in formula (1.9), we obtain Proposition 1.6. In the adapted basis (δα , ∂˙a ), the torsion tensor T of a d-connection α a D = (Γαµν , Γabµ , Cµc , Cbc ) is charaterized by the following d-tensor fields with the local a α a α , Tbca ) defined by: , Pµc coefficients (Λµν , Rµν , Cµc a ˙ vT(δν , δµ ) =: Rµν ∂a
hT(δν , δµ ) =: Λαµν δα , α hT(∂˙c , δµ ) =: Cµc δα ,
where Λαµν := Γαµν − Γανµ ,
a ˙ vT(∂˙c , δµ ) =: Pµc ∂a , a Pµc := ∂˙c Nµa − Γacµ ,
vT(∂˙c , ∂˙b ) =: Tbca ∂˙a , a a Tbca := Cbc − Ccb .
(1.11)
a a a , Tbca ). , Pµc , Cµc Throughout the paper we shall use the notation T = (Λαµν , Rµν a a a Corollary 1.7. The torsion tensor T = (Λαµν , Rµν , Cµc , Pµc , Tbca ) of a d-connection D vanishes if a α a a Γαµν = Γανµ , Rµν = Cµc = 0, ∂˙c Nµa = Γacµ , Cbc = Ccb .
Definition 1.8. The curvature tensor R of a d-connection D is given by R(X, Y )Z := DX DY Z − DY DX Z − D[X,
Y ] Z;
∀X, Y, Z ∈ X(T M).
By definition of a d-connection, it follows that R(X, Y )Z is determined by eight dtensor fields, six of which are independent due to the fact that R(X, Y ) = −R(Y, X). We set α a R(δµ , δν )δβ =: Rβµν δα ; R(δµ , δν )∂˙b =: Rbµν ∂˙a , α δα ; R(∂˙c , δν )δβ =: Pβνc
a ˙ ∂a , R(∂˙c , δν )∂˙b =: Pbνc
α R(∂˙b , ∂˙c )δβ =: Sβbc δα ;
a ˙ R(∂˙c , ∂˙d )∂˙b =: Sbcd ∂a .
α a α a α a Throughout the paper we shall use the notation R = (Rβµν , Rbµν , Pβνc , Pbνc , Sβbc , Sbcd ).
5
α a Theorem 1.9. The curvature R of a d-connection D = (Γαµν , Γabµ , Cµc , Cbc ) is charaterized by the d-tensor fields with local coefficients: α d α Rνµ , (a) Rβµν = δµ Γαβν − δν Γαβµ + Γǫβν Γαǫµ − Γǫβµ Γαǫν + Cβd a a d (b) Rbµν = δµ Γabν − δν Γabµ + Γcbν Γacµ − Γcbµ Γacν + Cbd Rνµ , α α α d (c) Pβνc = ∂˙c Γαβν − Cβc|ν + Cβd Pνc , a a a d (d) Pbνc = ∂˙c Γabν − Cbc|ν + Cbd Pνc , α α α ǫ α ǫ α (e) Sβbc = ∂˙b Cβc − ∂˙c Cβb + Cβc Cǫb − Cβb Cǫc , a a a e a e a (f) Sbcd = ∂˙c Cbd − ∂˙d Cbc + Cbd Cec − Cbc Ced . α a α a α a Corollary 1.10. The curvature tensor R = (Rβµν , Rbµν , Pβνc , Pbνc , Sβbc , Sbcd ) of a dconnection D vanishes iff α a α a α a Rβµν = Rbµν = Pβνc = Pbνc = Sβbc = Sbcd = 0.
Definition 1.11. An hv-metric on T M is a covariant d-tensor field G := hG + vG on T M, where hG := gαβ dxα ⊗ dxβ , vG := gab δy a ⊗ δy b such that: gαβ = gβα , det(gαβ ) 6= 0;
gab = gba , det(gab ) 6= 0.
(1.12)
The inverses of (gαβ ) and (gab ), denoted by (g αβ ) and (g ab ) repectively, are given by gαǫ g ǫβ = δβα ,
gae g eb = δba .
(1.13)
Definition 1.12. A d-connection D on T M is said to be metric or compatible with the metric G if DX G = 0, ∀X ∈ X(T M). In the adapted frame (δα , ∂˙a ), the above condition can be expressed locally in the form: gαβ|µ = gαβ||c = gab|µ = gab||c = 0. (1.14) We have the following Theorem [8]: ◦
◦
◦
◦
◦
α a Theorem 1.13. There exists a unique metrical d-connection D = ( Γαµν , Γabν , Cµc , Cbc ) on T M with the properties that ◦
◦
◦
◦
(a) Λαµν = Γαµν − Γανµ = 0,
◦
◦
T abc = C abc − C acb = 0.
◦
(b) Γabν := ∂˙b Nνa + 21 g ac (δν gbc − gdc ∂˙b Nνd − gbd ∂˙c Nνd ), ◦
◦
C αµc :=
1 2
g αǫ ∂˙c gµǫ .
◦
In this case, the coefficients Γαµν and C abc are necessarily of the form ◦
Γαµν :=
1 αǫ g (δµ gǫν + δν gǫµ − δǫ gµν ), 2
◦
a Cbc :=
6
1 ad ˙ g (∂b gdc + ∂˙c gdb − ∂˙d gbc ). 2
◦
◦
◦
◦
◦
Definition 1.14. The d-connection D = ( Γαµν , Γabν , C αµc , C abc ) defined in Theorem 1.13 will be referred to as the natural metric d-connection. The h- and v-covariant ◦ derivatives with respect to the natural metric d-connection D will be denoted by o| and ||o respectively.
2. Extended Absolute Parallelism Geometry (EAP-geometry) In this section, we study AP-geometry in a context different from the classical one. Instead of dealing with geometric objects defined on the manifold M, we will be dealing with geometric objects defined on the tangent bundle T M of M. Many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this different framework. Moreover, the basic geometric objects of the new geometry acquire a richer structure compared to the corresponding basic geometric objects of the classical APgeometry (See Table 2). As in the previous section, M is assumed to be a smooth paracompact manifold of dimension n. This insures the existence of a nonlinear connection on T M so that the decomposition (1.2) induced by the nonlinear connection holds. We assume that λ, i = 1, ..., n, are n vector fields defined globally on T M. In the i adapted basis (δα , ∂˙a ), we have λ = hλ + vλ = λα δα + λa ∂˙a . We further assume i i i i i that the n horizontal vector fields hλ and the n vertical vector fields vλ are linearly i i independent. This implies, in particular, that the n vector fields λ, themselves, are i linearly independent. Moreover, we have λα λβ = δβα , i
i
λα λα = δij ; i
λa λb = δba ,
j
i
i
λa λa = δij , i
j
(2.1)
where ( λα ) and ( λa ) denote the inverse matrices of ( λα ) and ( λa ) respectively. i
i
i
i
We refer to the above space, which we denote by (T M, λ), as an Extended Absolute Parellelism (EAP-) geometry which is characterized by the existence of 2n linearly independent vector fields defined globally on T M. The Latin indices {i, j} will be used for numbering the n vector fields (mesh indices). Einstein convention is applied on the mesh indices (which will always be written in lower position) as well as the component indices. In the sequel, to simplify notations, we will use the symbol λ without the subscript i to denote any one of the vector fields λi (i = 1, ..., n). The index i will appear only when summation is performed. Let us define gab := λi a λi b .
gαβ := λi α λi β ,
(2.2)
Then, clearly, G = gαβ dxα ⊗ dxβ + gab δy a ⊗ δy b is an hv-metric on T M. Moreover, in view of (2.1), the inverse of the matrices (gαβ ) and (gab ) are given by (g αβ ) and (g ab ) respectively, where g ab = λi a λi b .
g αβ = λi α λi β , 7
(2.3)
◦
◦
◦
◦
◦
Now, let D = ( Γαµν , Γabν , C αµc , C abc ) be the natural metric d-connection defined by Theorem 1.13, where gµν and gab are the metric tensors given by (2.2). α a Theorem 2.1. There exists a unique d-connection D = (Γαµν , Γabν , Cµc , Cbc ) such that
λα |µ = λα ||c = λa |µ = λa ||c = 0,
(2.4)
where | and || are the h- and v-covariant derivatives with respect to D. Consequently D is a metric d-connection. It is given by ◦
◦
Γαµν := Γαµν + λα λµ o| ν , i
Γabν := Γabν + λa λb o| ν ;
i
i
◦
i
◦
α Cµc := C αµc + λi α λi µ ||o c ,
a Cbc := C abc + λi a λi b ||o c .
(2.5) (2.6)
Relation (2.4) will be called the AP-condition (as in the classical AP-geometry). Proof. First, it is clear that D is a d-connection on T M. We next prove that λα |ν = 0. We have ◦
λα |ν = δν λα + λµ Γαµν = δν λα + λµ ( Γαµν + λα λµ o| ν ) i
α
= (δν λ +
◦
Γαµν λµ )
j
µ
− ( λ λµ ) λ i
j
αo
j
|
=λ
ν
j
αo |
ν
− λα o| ν = 0.
The rest is proved in a similar manner. α a Definition 2.2. The d-connection D = (Γαµν , Γabν , Cµc , Cbc ) defined in Theorem 2.1 will be referred to as the canonical d-connection of the EAP-space.
In analogy to the classical AP-space, the torsion tensor of the canonical d-connection will be called the torsion tensor of the EAP-space. Theorem 2.3. The canonical d-connection D can be expressed explicitely in terms of the λ’s only in the form: Γαµν = λα (δν λµ ),
Γabν = λa (δν λb );
(2.7)
α Cµc = λα (∂˙c λµ ),
a Cbc = λa (∂˙c λb ).
(2.8)
i
i
i
i
i
i
i
i
Proof. Since λα |ν = 0, it follows that δν λα = −λǫ Γαǫν . Multiplying by λµ , we get i λµ (δν λα ) = − Γαµν so that, by (2.1), Γαµν = λα (δν λµ ). The other formulae are derived in i i i i a similar manner. By Theorem 2.1 and Theorem 2.3, we have ◦
Corollary 2.4. The natural metric d-connection D can be expressed explicitely in terms of the λ’s only in the form ◦
◦
Γαµν = λα (δν λµ − λµ o| ν ), i
i
Γabν = λa (δν λb − λb o| ν );
i
i
◦
◦
C αµc = λi α (∂˙c λi µ − λi µ ||o c ),
i
i
C abc = λi a (∂˙c λi b − λi b ||o c ). 8
(2.9) (2.10)
Definition 2.5. The contortion tensor of an EAP-space is defined by ◦
C(X, Y ) := DY X − D Y X; ∀X, Y ∈ X(T M), ◦
where D is the cannonical d-connection and D is the natural metric d-connection. In the adapted basis (δµ , ∂˙a ), the contortion tensor is characterized by the following d-tensor fields: α C(δµ , δν ) =: γµν δα , ◦
α γµν := Γαµν − Γαµν ,
α C(δµ , ∂˙c ) =: γµc δα ; ◦
a ˙ C(∂˙b , δµ ) =: γbµ ∂a , ◦
a γbµ := Γabµ − Γabµ ;
α α γµc := Cµc − C αµc ,
a ˙ C(∂˙b , ∂˙c ) =: γbc ∂a ; ◦
a a γbc := Cbc − C abc . (2.11)
α a α a Throughout the paper we shall use the notation C = (γµν , γbµ , γµc , γbc ).
By definition of the canonical d-connection and (2.11), the contortion tensor can be expressed explicitely in terms of the λ’s only in the form: α a α = λi α λi µ ||o c , = λi a λi b o| µ , γµc γµν = λi α λi µ o| ν , γbµ
a γbc = λi a λi b ||o c .
(2.12)
ǫ c ǫ d Proposition 2.6. Let γαµν := gαǫ γµν , γabµ := gac γbµ , γαµc := gαǫ γµc , γabc := gad γbc . Then each of the above defined d-tensor fields is skew-symmetric in the first pair of α a α a indices. Consequently, γαν = γaµ = γαc = γac = 0.
Proof. We have γαµν + γµαν = λα λµ o| ν + λµ λα o| ν = ( λα λµ ) o| ν = gαµ o| ν = 0. i
i
i
i
i
i
The rest is proved analogously. A simple calculation gives a α a α a α a Proposition 2.7. Let T = (Λαµν , Rµν , Cµc , Pµb , Tbca ) and C = (γµν , γbµ , γµc , γbc ) be the torsion and the contortion tensors of the EAP-space respectively. Then the following relations hold: α α Λαµν = γµν − γνµ ,
◦
a a = −γbµ + P aµb , Pµb
◦
α α Cµc = γµc + C αµc ,
a a − γcb . Tbca = γbc
(2.13)
Consequently, α a a Λαµα = γµα =: Cµ , Tba = γba =: Cb .
(2.14)
Remark 2.8. It can be shown, in analogy to the classical AP-space [2], that 1 γαµν = (Λαµν + Λνµα + Λµνα ), 2
1 γabc = (Tabc + Tcba + Tbca ); 2
where Λαµν := gαǫ Λǫµν and Tabc := gad Tbcd . α a By (2.13) and (2.15), Λαµν (resp. Tbca ) vanishes iff γµν (resp. γbc ) vanishes. α a Definition 2.9. Let D = (Γαµν , Γabµ , Cµc , Cbc ) be the canonical d-connection.
9
(2.15)
α e = (Γ eαµν , Γ ea , C eµc ea ) is defined by (a) The dual d-connection D , C bµ bc
eαµν := Γανµ , Γ
e abµ := Γabµ ; Γ
α α eµc C := Cµc ,
a a ebc C := Ccb .
(2.16)
b = (Γ bα , Γ ba , C bα , C ba ) is defined by (b) The symmertic d-connection D µν µc bµ bc bα := 1 (Γα + Γα ), Γ µν νµ 2 µν
ba := Γa ; Γ bµ bµ
bα := C α , C µc µc
ba := 1 (C a + C a ). (2.17) C bc cb 2 bc
e and D b by We shall denote the horizontal (vertical) covariant derivative of D e e b b “ | ”(“ || ”) and “ | ”(“ || ”) respectively. It follows immediately from the above definition that λa e|µ = λa b|µ = λa |µ = 0;
λα e||c = λα b||c = λα ||c = 0, 1 α λ e; 2 |µ As easily checked, we also have λα e|µ = λβ Λαµβ ,
λα b|µ =
λa e||c = λb Tcba ,
λa b||c =
1 a λ e. 2 ||c
(2.18) (2.19)
Proposition 2.10. The covariant derivatives of the metric G with respect to the dual e and D b are given respectively by: and symmetric d-connections D gαβe|µ = Λαβµ + Λβαµ , gαβ e||c = gabe|µ = 0, gabe||c = Tabc + Tbac ; 1 g e, gαβ b||c = gabb|µ = 0, 2 αβ |µ e and D b are non-metric connections. Consequently, D gαβb|µ =
gabb||c =
1 g e. 2 ab||c
(2.20) (2.21)
We end this section with the following tables. Table 1: Fundamental connections of the EAP-space Connection
Coefficients
Natural
a α a ( Γα µν , Γbν , C µc , C bc )
| ||
a a (0, Rµν , Cα µc , P µc , 0)
metric
Canonical
a α a (Γα µν , Γbν , Cµc , Cbc )
| ||
a α a a (Λα µ ν , Rµν , Cµc , Pµc , Tbc )
metric
Dual
a α a (Γα νµ , Γbν , Cµc , Ccb )
a α a a (−Λα µν , Rµν , Cµc , Pµc , −Tbc )
non-metric
Symmetric
a α a (Γα (µν) , Γbν , Cµc , C(bc) )
a α a (0, Rµν , Cµc , Pµc , 0)
non-metric
◦
◦
◦
◦
Covariant derivative ◦
e|
◦
e ||
b| b ||
Torsion
◦
Metricity
◦
The next table gives a comparison between the classical AP-space and the EAP-space. We shall refer to the Riemannian connection in the classical AP-space and the natural metric d-connection in the EAP-space simply as the metric connection. Moreover, we consider only the metric and the canonical connections in both spaces. We also set Labν := 12 g ac (δν gbc − gdc ∂˙b Nνd − gbd ∂˙c Nνd ). 10
Table 2: Comparison between classical AP-geometry and EAP-geometry Classical AP-geometry
EAP-geometry
Underlying space
M
TM
Building blocks
λα (x)
Nµa (x, y), λ(x, y) = (λα (x, y), λa (x, y))
Metric
gµν = λµ λν i
G = (gµν , gab );
i
gµν = λµ λν , gab = λa λb i
◦
Metric
Γα µν =
1 2
◦
i
◦
i
◦
◦
◦
Γα µν =
1 2
g αǫ (δµ gνǫ + δν gµǫ − δǫ gµν ),
◦
◦
Γabν = ∂˙b Nνa + Labν ; ◦
C abc = α Γα µν = λ (∂ν λµ ) i
1 2
Cα µc =
1 2
g αǫ ∂˙c gµǫ ,
g ad (∂˙b gcd + ∂˙c gbd − ∂˙d gbc )
a α a D = (Γα µν , Γbν , Cµc , Cbc );
i
connection
α Γα µν = λ (δν λµ );
Γabν = λa (δν λb ),
α Cµc = λα (∂˙c λµ );
a Cbc = λa (∂˙c λb )
i
i
AP-condition
◦
a α a D = ( Γα µν , Γbν , C µc , C bc );
g αǫ (∂µ gνǫ + ∂ν gµǫ − ∂ǫ gµν )
connection
Canonical
i
λα |µ = 0
i
i
i
i
i
i
λα |µ = λα ||c = 0, λa |µ = λa ||c = 0
Torsion
α α Λα µν = Γµν − Γνµ
a α a a T = (Λα µν , Rµν , Cµc , Pµc , Tbc ); α α Λα µν = Γµν − Γνµ ;
a Rµν = δν Nµa − δµ Nνa ,
a Pµc = ∂˙c Nµa − Γacµ ; ◦
Contorsion
α α γµν = Γα µν − Γµν
α a α a C = (γµν , γbν , γµc , γbc ); ◦
α α γµν = Γα µν − Γνµ ; ◦
α α γµc = Cµc − Cα µc ;
Basic vector
a a a Tbc = Cbc − Ccb
α Cµ = Λα µα = γµα
◦
a γbν = Γabν − Γabν , ◦
a a γbc = Cbc − C abc
B = (Cµ , Ca ); α Cµ = Λα µα = γµα ;
11
d d Ca = Tad = γad
3. Curvature tensors in the EAP-space ◦
e Let (T M, λ) be an EAP space. Let D be the cannonical d-connection. Let D, D b and D be the natural metric d-connection, the dual d-connection and the symmetric dconnection respectively. The curvature tensors of the four d-connections will be denoted ◦ e and R. b In this section, we carry out the task of calculating the respectively by R, R, R curvature tensors together with their contractions. Lemma 3.1. The following commutation formulae hold: α d (a) λα |ν|µ − λα |µ|ν = Rβµν λβ − Λβνµ λα |β − Rνµ λα ||d α β α d α (b) λα |ν||c − λα ||c|ν = Pβνc λβ − Cνc λ |β − Pνc λ ||d α (c) λα ||b||c − λα ||c||b = Sβcb λβ − Tbcd λα ||d a d (d) λa |ν|µ − λa |µ|ν = Rbµν λb − Λβνµ λa |β − Rνµ λa ||d a β a d a (e) λa |ν||c − λa ||c|ν = Pbνc λb − Cνc λ |β − Pνc λ ||d a (f) λa ||b||c − λa ||c||b = Sdcb λd − Tbcd λa ||d ,
In view of (2.1), Corollary 1.10 and Theorem 2.1, Lemma 3.1 directly implies that a α a α a α ) of the , Sbcd , Sβbc , Pbνc , Pβνc , Rbµν Theorem 3.2. The curvature tensor R = (Rβµν canonical d-connection vanishes identically.
Corollary 3.3. The following identities hold: a α Λαβµ|α = (Cβ|µ − Cµ|β ) + Cǫ Λǫβµ + Sβµα Rµβ Cαa
(3.1)
d Tbc||d = (Cb||c − Cc||b) + Cd Tbcd .
(3.2)
Proof. The Bianchi identities [9] applied to the canonical d-connection D give a α Sβ,µ,ν (Λαβµ|ν + Λǫµν Λαβǫ + Rβµ Cνa )=0
(3.3)
a e a Sb,c,d (Tbc||d + Tcd Tbe ) = 0,
(3.4)
and where the notation Sβ,µ,ν denotes a cyclic permutation on the indices β, µ, ν and summation. (3.1) and (3.2) are obtained by setting α = ν and a = d in (3.3) and (3.4) respectively. By applying the comutation formula (c) of Lemma 3.1 with respect to the dual and symmetric d-connections respectively, taking into account (2.1) and (2.18), we obtain α α Seβbc = Sbβbc = 0.
α Cµc
This could be also deduced from Theorem 1.9 (e) and Theorem 3.2, noting that eα = C bα . =C µc
µc
12
In Theorems 3.4, 3.5 and 3.6 below, concerning the curvature tensors of the d◦ e and D, b we will make use of Theorem 3.2, namely that the curvature connections D, D tensors of the canonical d-connection vanish identically. ◦
Theorem 3.4. The curvature tensors of the natural metric d-connection D can be expressed in the form: ◦
α α ǫ α ǫ α α ǫ α d (a) Rαβµν = (γβµ|ν − γβν|µ ) + (γβν γǫµ − γβµ γǫν ) − γβǫ Λνµ − γβd Rνµ , ◦
a a d a d a a ǫ a d (b) Rabµν = (γbµ|ν − γbν|µ ) + (γbν γdµ − γbµ γdν ) − γbǫ Λνµ − γbd Rνµ , ◦
α α ǫ α ǫ α α ǫ α d (c) P αβνc = (γβc|ν − γβν||c ) + (γβν γǫc − γβc γǫν ) − γβǫ Cνc − γβd Pνc , ◦
a a d a d a a ǫ a d − γbν||c ) + (γbν γdc − γbc γdν ) − γbǫ Cνc − γbd Pνc , (d) P abνc = (γbc|ν ◦
α α α ǫ α ǫ α (e) S αβbc = (γβb||c − γβc||b ) + (γβc γǫb − γβb γǫc ) − γβd Tcbd , ◦
a a e a e a a e (f) S abcd = (γbc||d − γbd||c ) + (γbd γec − γbc γed ) − γbe Tdc .
Consequently, ◦
◦
α d α ǫ α ǫ , Rαµ − Cβ|µ ) − Cǫ γβµ + γβǫ γµα − γβd (g) Rβµ := Rαβµα = (γβµ|α ◦
◦
ǫ d (h) R := g βµ Rβµ = 12 (Ωαµ µ|α − Cα Ωαµ µ ) − C µ |µ + γ αµ ǫ γµα − γ αµ d Rαµ , ◦
◦
α ǫ α ǫ ǫ α d (i) P βc := − P αβαc = (Cβ||c − γβc|α ) + Cǫ γβc + γβǫ (Cαc − γαc ) + γβd Pαc , ◦
◦
d d d e d ǫ e d ) + Cd γbν − γbe γdν − γbǫ Cνd − γbd Pνe , (j) P bν := P dbνd = (Cb|ν − γbν||d ◦
◦
d d d e (k) S bc := S dbcd = (γbc||d − Cb||c ) − Cd γbc + γbe γcd , ◦
◦
c (l) S := g bc S bc = 12 (Ωad d||a − Ca Ωad d ) − C d ||d + γ ad c γda , α α a a where Ωαβµ := γβµ + γµβ , Ωabc := γbc + γcb .
Proof. We prove (a) and (c) only. The other formulae of the first part are proved in a similar manner. The second part is obtained directly by applying the suitable contractions. (a) We have ◦
◦
◦
◦
◦
◦
◦
◦
d Rαβµν = δµ Γαβν − δν Γαβµ + Γǫβν Γαǫµ − Γǫβµ Γαǫν + C αβd Rνµ ǫ α ǫ α α ) ) − (Γǫβµ − γβµ )(Γαǫµ − γǫµ ) + (Γǫβν − γβν ) − δν (Γαβµ − γβµ = δµ (Γαβν − γβν α α α d (Γαǫν − γǫν ) + (Cβd − γβd )Rνµ α α ǫ α ǫ α ǫ α ǫ = Rβµν + (δν γβµ + γβµ Γαǫν − γǫµ Γβν ) − (δµ γβν + γβν Γαǫµ − γǫν Γβµ ) α d ǫ α ǫ α − γβd Rνµ + (γβν γǫµ − γβµ γǫν ) α α ǫ α ǫ α α ǫ α d = (γβµ|ν − γβν|µ ) + (γβν γǫµ − γβµ γǫν ) − γβǫ Λνµ − γβd Rνµ
13
(c) We have ◦
◦
◦
◦
◦
P αβνc = ∂˙c Γαβν − C αβc o ν + C αβd P dνc |
=
∂˙c Γαβν
=
α Pβνc
−
α ∂˙c γβν
◦
α α α α d d − Cβc|ν + (Cβc|ν − C αβc o ν ) + (Cβd − γβd )(Pνc + γcν ) |
α α α α o α d α d α d − ∂˙c γβν + {Cβc|ν − (Cβc − γβc ) | ν } + Cβd γcν − γβd γcν − γβd Pνc
α ǫ α α ǫ α d α ǫ α α ǫ = − ∂˙c γβν + (Cβc γǫν − Cǫc γβν − Cβd γcν ) + (γβc|ν − γβc γǫν + γǫc γβν α d α d α d α d + γβd γcν ) + Cβd γcν − γβd γcν − γβd Pνc α α ǫ α α ǫ ǫ α ǫ α α d = γβc|ν − (∂˙c γβν + γβν Cǫc − γǫν Cβc ) + (γβν γǫc − γβc γǫν ) − γβd Pνc α α ǫ α ǫ α α ǫ α d = (γβc|ν − γβν||c ) + (γβν γǫc − γβc γǫν ) − γβǫ Cνc − γβd Pνc
e can be Theorem 3.5. The non-vanishing curvature tensors of the dual d-connection D expressed in the form: a eα = Λα + Sβ,µ,ν C α Rµν (a) R , βνµ βa µν|β
e a = Rd T a , (b) R µν db bνµ α ǫ (c) Peβµc = Λαµβ||c + Λαǫβ Cµc , a a a d (d) Pebµc = Tbc|µ + Tdb Pµc , a a (e) Sebcd = Tdc||b .
Consequently, eβν := R eα = − Cν|β + Sβ,ν,αC α Ra , (f) R βνα βa αν e := g βµ R eβµ = − C µ |µ , (g) R α ǫ (h) Peβc := −Peβαc = Cβ||c + Λαβǫ Cαc , a a d (i) Pebµ := Peβµa = Cb|µ + Tdb Pµa , a (j) Sebd := Sebda = − Cd||b ,
(k) Se := g bd Sebd = − C d ||d . Proof. (b) is a consequence of the commutation formula (d) of Lemma 3.1 applied to the dual d-connection, taking into account (2.1), (2.18) and (2.19). (b) could be also ea and C a = T a + obtained from Theorem 1.9 (b) and Theorem 3.2, noting that Γabµ = Γ bµ bd bd a e Cbd . We next prove (a) and (c) of the first part. The second part follows immediately by applying the suitable contractions.
14
(a) We have ˜α R βνµ = = = =
eα − δµ Γ eα + Γ eǫ Γ eα eǫ e α eα a δν Γ βµ βν βµ ǫν − Γβν Γǫµ + Cβa Rµν α a δν Γαµβ − δµ Γανβ + Γǫµβ Γανǫ − Γǫνβ Γαµǫ + Cβa Rµν α a {δν Γαµβ + Γǫµβ (Λανǫ + Γαǫν )} − {δµ Γανβ + Γǫνβ (Λαµǫ + Γαǫµ )} + Cβa Rµν α α a α α a (Rµνβ − Cµa Rβν + δβ Γαµν + Γǫµν Γαǫβ ) − (Rνµβ − Cνa Rβµ + δβ Γανµ α a Rµν + Γǫνµ Γαǫβ ) − (Γǫµβ Λαǫν + Γǫνβ Λαµǫ ) + Cβa α a = (δβ Λαµν + Γαǫβ Λǫµν − Γǫµβ Λαǫν − Γǫνβ Λαµǫ ) + Sβ,µ,ν Cβa Rµν α a = Λαµν|β + Sβ,µ,ν Cβa Rµν
(c) We have α eα − C eα e + C eα Pe d = ∂˙c Γα − C α e + C α P d Peβµc = ∂˙c Γ βµ βd µc µβ βd µc βc|µ βc|µ α α α α d − Cβc + Cβd Pµc ) + ∂˙c Λαµβ + (Cβc|µ = (∂˙c Γαβµ − Cβc|µ e|µ ) α α ǫ = Pβµc + (∂˙c Λαµβ + Λǫµβ Cǫc − Λαµǫ Cβc ) α α ǫ = Λµβ||c + Λǫβ Cµc
b Theorem 3.6. The non-vanishing curvature tensors of the symmetic d-connection D can be expressed in the form bα = 1 (Λα − Λα ) + 1 (Λǫ Λα − Λǫ Λα ) + 1 (Λǫ Λα ), (a) R νµ βǫ βν µǫ βµ νǫ βνµ βν|µ βµ|ν 2 4 2 ba = 1 R ea , (b) R bνµ bνµ 2 α α (c) Pbβµc = 21 Peβµc , a a , (d) Pbbµc = 12 Pebµc a a a a e a e a (e) Sbbcd = 21 (Tbc||d − Tbd||c ) + 41 (Tbce Tde − Tbd Tce ) + 21 (Tdc Teb ).
Consequently, bβν := R bα = (f) R βνα
1 2
eβν − 1 (Cα Λα + Λα Λǫ ), R νǫ αβ νβ 4
b := g βν R eβν = 1 R e − 1 Λαβ ǫ Λǫ , (g) R αβ 2 4 α (h) Pbβc := −Pbβαc = 12 Peβc , a (i) Pbbµ := Pbβµa =
1 2
Pebµ ,
a a a e (j) Sbbd := Sbbda = 12 Sebd − 41 (Ca Tdb + Tde Tab ), e . (k) Sb := g bd Sbbd = 21 Se − 41 T ab e Tab
Proof. Similar to the proof of Theorems 3.4 and 3.5. 15
4. Wanas tensors (W-tensors) The Wanas tensor, or simply the W -tensor, in the classical AP-geometry, is a tensor which measures the non-commutativity of covariant differentiations of the parallelization vector fields λ with respect to the dual connection: i
α Wβνµ := λβ ( λα e|νe|µ − λα e|µe|ν ) i
i
i
(4.1)
This tensor explicitely contains the curvature and torsion tensors. The W -tensor was first defined by M. Wanas [12] and has been used by F. Mikhail and M. Wanas [5] to construct a geometric theory unifying gravity and electromagnetism (GFT: generalized field theory). The scalar Lagrangian function of the GFT is obtained by double contractions of the tensor W αβ νµ . The symmetric part of the field equation obtained contains a second-order tensor representing the material distribution. This tensor is a pure geometric, not a phenomological, object. The skew part of the field equation gives rise to Maxwell-like equations. The use of the W -tensor has thus aided to construct a geometric theory via one single geometric entity, which Einstein was seeking for [1]. Various significant applications (e.g [12], [13], [15]) have supported such a theory. Recently, the authors of this paper investigated the most important properties of this tensor in the context of classical AP-geometry [17]. The W -tensor was also studied by the present authors in the context of generalized Lagrange spaces [16]. It should be noted that the W -tensor can be defined only in the context of APgeometry and its generalized versions (cf. e.g [16], [17]), since it is defined only in terms of the vector fields λ’s. Due to its importance in physical applications, we are going to investigate the properties of the W -tensor in the present section. The W -tensor (4.1) can be generalized in the context of the EAP-geometry as follows. Definition 4.1. Let (T M, λ) be an EAP-space. For a given d-connection D = (Γαµν , Γabµ , α a Cµc , Cbc ), the W -tensor is given by α a α a α a W = (Wβνµ , Wbνµ , Wβνc , Wbνc , Wβbc , Wbcd ), α (a) the hhh-tensor Wβνµ is defined by the formula α λα |ν|µ − λα |µ|ν = λǫ Wǫνµ , a (b) the hhv-tensor Wbνµ is defined by the formula a , λa |ν|µ − λa |µ|ν = λd Wdνµ α (c) the vhh-tensor Wβνc is defined by the formula α λα |ν||c − λα ||c|ν = λǫ Wǫνc , a (d) the vhv-tensor Wbνc is defined by the formula a , λa |ν||c − λa ||c|ν = λd Wdνc
16
α (e) the vvh-tensor Wβbc is defined by the formula α , λα ||b||c − λα ||c||b = λǫ Wǫbc a (f) the vvv-tensor Wbcd is defined by the formula a λa ||c||d − λa ||d||c = λe Wecd ,
where “ |” and “ ||” are the h- and v-covariant derivatives with respect to the given d-connection D. Theorem 2.1, together with (2.1), directly implies that the W -tensors of the canonical d-connection vanish identically. In view of Theorem 3.4, we obtain ◦
Theorem 4.2. The W -tensors corresponding to the natural metric d-connection D are given by: ◦
α α ǫ α ǫ α α ǫ (a) W αβνµ = (γβµ|ν − γβν|µ ) + (γβν γǫµ − γβµ γǫν ) − γβǫ Λνµ , ◦
a a d a d a a ǫ − γbν|µ ) + (γbν γdµ − γbµ γdν ) − γbǫ Λνµ , (b) W abνµ = (γbµ|ν ◦
ǫ α d α ǫ α α α ǫ α γβǫ ), γβd − γνc ) + (γcν − γβc γǫν − γβν||c ) + (γβν γǫc (c) W αβνc = (γβc|ν ◦
a a d a d a d a ǫ a (d) W abνc = (γbc|ν − γbν||c ) + (γbν γdc − γbc γdν ) + (γcν γbd − γνc γbǫ ), ◦
◦
◦
◦
(e) W αβbc = S αβcb , (f) W abcd = S abdc . Proof. We prove (c) only. The other formulae are derived in a similar manner. By definition, we have ◦ ◦ ◦ ◦ λǫ W αǫνc = λǫ P αǫνc − C ǫνc λα o| ǫ − P dνc λα ||o d . Consequently, by (2.1), (2.12) and (2.13), we obtain ◦
◦
ǫ ǫ d d W αβνc = P αβνc − ( λβ λα o| ǫ )(Cνc − γνc ) − ( λβ λα ||o d )(Pνc + γcν ) i
i
i
i
◦
ǫ α d α ǫ α d α = P αβνc + Cνc γβǫ + Pνc γβd − (γνc γβǫ − γcν γβd ) α α ǫ α ǫ α d α ǫ α = (γβc|ν − γβν||c ) + (γβν γǫc − γβc γǫν ) + (γcν γβd − γνc γβǫ )
Since λa e|µ = λα e||c = λa b|µ = λα b||c = 0, it follows, by definition, that fa = W fα = W ca = W c α = 0. W bνµ βbc bνµ βbc Proceeding as in Theorem 4.2, taking into account Theorem 3.5 and Theorem 3.6, we have the following 17
e Theorem 4.3. The non-vanishing W -tensors corresponding to the dual d-connection D are given by: f α = Λα + Λǫ Λα + Sν,µ,β C α Ra , (a) W νµ βǫ βνµ βa νµ νµ|β f α = Λα , (b) W βνc νβ||c fa = T a , (c) W bνc bc|ν fa = T a + T e T a . (d) W bdc dc be dc||b Theorem 4.4. The non-vanishing W -tensors corresponding to the symmetric db are given by: connection D cα = R bα , (a) W βνµ βµν cα = (b) W βνc ca = (c) W bνc
1 2 1 2
fα , W βνc fa , W bνc
c a = Sba . (d) W bcd bdc It is clear by the above theorem that the W -tensors corresponding to the symmetric d-connection give no new d-tensor fields. Remark 4.5. The W -tensors corresponding to a given d-connection can be also defined covariantly in the form ǫ λβ|µ|ν − λβ|ν|µ = λǫ Wβµν , with similar expressions for the other counterparts. These expressions give the same formulae (up to a sign) for the W -tensors obtained in Theorems 4.2, 4.3 and 4.4. Proposition 4.6. The following identities hold: ◦
c α = Sβ,µ,ν Ra C α (a) Sβ,µ,ν W αβνµ = Sβ,µ,ν W βνµ µβ νa f α = 2Sβ,µ,ν Ra C α (b) Sβ,µ,ν W βνµ µβ νa Proof. By Theorem 4.2, we have ◦
Sβ,µ,ν W αβνµ = = = =
α α α ǫ ǫ α ǫ α Sβ,µ,ν (γβµ|ν − γβν|µ ) + Sβ,µ,ν (γβǫ Λνµ ) + Sβ,µ,ν (γβν γǫµ − γβµ γǫν ) α ǫ α Sβ,µ,ν (Λβµ|ν + Λµν Λβǫ ) a α a α Sβ,µ,ν (Λαβµ|ν + Λǫβµ Λανǫ + Rβµ Cνa ) + Sβ,µ,ν Rµβ Cνa a α Sβ,µ,ν Rµβ Cνa ,
where in the last step we have used (3.3). The proof of the other part of (a) is achieved by applying the first Bianchi identity to the symmetric d-connection taking into account bα = C α . The proof of (b) is carried out Theorem 4.4 (a) together with the fact that C νa νa in a similar manner, again by using (3.3), taking into consideration Theorem 4.3 (a). 18
Summing up, the EAP-space has three distinct W -tensors (corresponding to the natural metric, dual and symmetric d-connections), each with six counterparts. Eight only out of the eighteen are independent, four coincide with the corresponding curvature tensors and four vanish identically.
5. Cartan-type case A drawback in the construction of the EAP-space is the fact that the nonlinear connection is assumed to exist a priori, independently of the vector fields λ’s defining the parallelization. It would be more natural and less arbitrary if the nonlinear connection were expressed in terms of these vector fields. In this case, all geometric objects of the EAP-space will be defined solely in terms of the building blocks of the space. Below, we impose an extra condition on the canonical d-connection, namely, being of Cartan type. The outcome of such a condition is the accomplishment of our requirment, besides many other interesting results. We first recall the definition of a Cartan type d-connection [7], [8]. α a Definition 5.1. A d-connection D = (Γαµν , Γabµ , Cµc , Cbc ) on T M is said to be of Cartan type if D hX C = 0; D vX C = vX; ∀X ∈ X(T M), where C = y a ∂˙a is the Loiuville vector field.
Locally, the above conditions are expressed in the form y a|µ = 0,
y a ||c = δca ,
(5.1)
or, equivalently, Nµa = y b Γabµ ,
a y b Cbc = 0.
(5.2)
Proposition 5.2. If a d-connection D is of Cartan type, then the following identities involving the torsions and curvatures hold: a a a a a Rµν = y bRbνµ , Pµc = y b Pbµc , Tbca = y dSdcb .
(5.3)
Proof. Follows by applying the commutation formulae (d), (e) and (f) of Lemma 3.1 to the vector field y a . Theorem 5.3. Let (T M, λ) be an EAP-space. Assume that the canonical d-connection D is of Cartan type. Then we have: (a) The expression y b (λa ∂µλb ) represents the coefficients of a nonlinear connection which i i coincides with the given nonlinear connection Nµa : Nµa = y b (λa ∂µ λb ).2 i
a a (b) Rµν = Pµc = Tbca = 0.
i
◦
a a a ea = C ba = C abc . Consequently, Cbc is symmetric, γbc = 0 and Cbc =C bc bc 2
A similar expression is found in “ArXiv: 0801.1132 [gr-qc]”, but in a completely different situation.
19
(c) λa e||b = λa b||b = λa
o b ||
= 0, gabe||c = gabb||c = gab ||o c = 0,
y a e||b = y a b||b = y a
o b ||
= δba .
(d) λa are positively homogeneous of degree 0 in y. Consequently, so are gab . (e) ∂˙b Nµa = Γabµ and Nµa is homogeneous. Consequently, Γabµ is positively homogeneous of degree 0 in y. ◦
a (f) γbµ = 0. Consequently, Γabµ = Γabµ . ◦
(g) P aµb = 0. Proof. We have a = y b ( λa ∂µ λb ). (a) Nµa = y b ( λa δµ λb ) = y b λa (∂µ − Nµc ∂˙c ) λb = y b ( λa ∂µ λb ) − Nµc y bCbc i
i
i
i
i
i
i
i
(b) is obtained from Proposition 5.2, taking into account Theorem 3.2. The vanishing a of γbc is readily obtained by Remark 2.8 and Tbca = 0. (c) is a direct consequence of (b). a a a (d) By the symmetry of Cbc , we have 0 = y b Cbc = y bCcb = λa (y b∂˙b λc ) so that, by (2.1), i i y b ∂˙b λc = 0. The result follows from Euler’s Theorem. a = ∂˙b Nµa − Γabµ = 0 so that y b∂˙b Nµa = y b Γabµ = Nµa . (e) follows from the fact that Pµb This could be also deduced from the expression obtained for Nµa in (a), taking into account that λa (hence λa ) are positively homogeneous of degree 0 in y.
(f) By (e), we have Γabµ = ∂˙b Nµa . Consequently, by definition of the natural metric d◦ a . connection (Theorem 1.13), 21 g ac (δν gbc − gdc ∂˙b Nνd − gbd ∂˙c Nνd ) = Γabµ − Γabµ = − γbµ 1 d d ˙ ˙ Multiplying by gae , we get 2 (δν gbe − gde ∂b Nν − gbd ∂e Nν ) = −γebµ . This implies that γebµ is symmetric in the indices e, b. By Proposition 2.6, γebµ is also skewa symmetric in the indices e, b. Consequently, γebµ vanishes so that γbµ = g ae γebµ = 0. ◦
a a (g) is a direct consequence of the relation P aµb = Pµb + γbµ , taking into account that a a Pµb = γbµ = 0. ◦
e D b and D are Corollary 5.4. If the canonical d-connection is of Cartan type, then D, also of Cartan type. In what follows, we assume that the canonical d-connection is of Cartan type. The next two results are immediate consequences of the fact that a a a a Rµν = Pµc = Tbca = γbc = γbµ = 0,
taking into consideration Proposition 4.6 and Theorems 3.4, 3.5, 3.6, 4.2, 4.3 and 4.4. Proposition 5.5. The following relations hold: ◦
◦
◦
(a) Rabµν = P abµc = S abcd = 0, 20
e α = Λα , (b) R βµν νµ|β ea = R ba = Pea = Pba = Sea = Sba = 0. (c) R bµν bµν bµc bµc bcd bcd ◦
◦
(d) W αβνµ = Rαβµν , ◦
◦
◦
fa = W cα = W fa = W c a = 0, (e) W abµν = W abµc = W abcd = W bµc bµc bcd bcd ◦
c α = Sβ,µ,ν W f α = 0. (f) Sβ,µ,ν W αβνµ = Sβ,µ,ν W βνµ βνµ ◦
c α and W f α satisfy the first Bianchi identity of the Riemannian Consequently, W αβνµ , W βνµ βνµ curvature tensor. Theorem 5.6. The independent non-vanishing W -tensors are given by: ◦
α α ǫ α ǫ α ǫ α (a) W αβνc = (γβc|ν − γβν||c ) + (γβν γǫc − γβc γǫν ) − γνc γβǫ
f α = Λα + Λǫ Λα , (b) W νµ βǫ βνµ νµ|β f α = Λα (c) W βµc µβ||c To sum up, the assumption that the canonical d-connection being of Cartan type implies that all the geometric objects defined in the EAP-space are expressed in terms of the vector fields λ’s only. The curvature of the nonlinear connection vanishes and the three other defined d-connections of the EAP-space, namely the dual, symmetric and the natural metric d-connections, are also of Cartan type. Moreover, there are only seven non-vanishing curvature tensors and only three independent non-vanishing W -tensors, some of which have simpler expressions than that obtained in the general case. The EAP-space becomes richer as new relations among its various geometric objects - which are not valid in the general case - emerge. Accordingly, the EAP-space in this case becomes more tangible, thus more suitable for physical applications. We end this section with the following table. Table 3: EAP-geometry under the Cartan type case Connection
Coefficients
Torsion
Curvature
Canonical
α a ˙ a (Γα µν , ∂b Nν , Cµc , Cbc )
α (Λα µ ν , 0, Cµc , 0, 0)
(0, 0, 0, 0, 0, 0)
Dual
α a ˙ a (Γα νµ , ∂b Nν , Cµc , Cbc )
α (−Λα µν , 0, Cµc , 0, 0)
eα , 0, Pe α , 0, 0, 0) (R βµν βµc
Symmetric
α a ˙ a (Γα (µν) , ∂b Nν , Cµc , Cbc )
α (0, 0, Cµc , 0, 0)
bα , 0, Pb α , 0, 0, 0) (R βµν βµc
Natural
α a ˙ a ( Γα µν , ∂b Nν , C µc , Cbc )
◦
◦
◦
(0, 0, C α µc , 0, 0)
21
◦
◦
◦
α α ( Rα βµν , 0, P βµc , 0, S βcd , 0)
6. Berwald-type case In this section we assume that the canonical d-connection D is of Berwald type [8], [10]. The consequences of this assumption are investigated. α a Definition 6.1. A d-connection D = (Γαµν , Γabµ , Cµc , Cbc ) on T M is said to be of Berwald type if α ∂˙b Nµa = Γabµ ; Cµc = 0. (6.1) α Proposition 6.2. If the canonical d-connection D is of Cartan type such that Cµc = 0, then it is of Berwald type. a Proof. Follows from the fact that 0 = Pµb = ∂˙b Nµa − Γabµ .
Theorem 6.3. Let (T M, λ) be an EAP-space. Assume that the canonical d-connection D is of Berwald type. Then we have: a (a) Pµb =0
(b) λµ are functions of the positional argument x only. Consequently, so are gµν . ◦
α (c) C αµc = 0. Consequently, γµc = 0. ◦
(d) The coefficients Γαµν and Γαµν are functions of the positional argument x only and are given respectively by Γαµν (x) = ( λi α ∂ν λi µ )(x),
◦
Γαµν (x) =
1 αǫ g (∂µ gνǫ + ∂ν gµǫ − ∂ǫ gµν )(x). 2
α (e) Λαµν and γµν are functions of the positional argument x only. ◦
◦
a (f) γbµ = P abµ = 0. Consequently, Γabµ = Γabµ . a Proof. The proof is straightforward except for the relation γbµ = 0, which can be proved in exactly the same manner as (f) of Theorem 5.3. ◦
e D b and D Corollary 6.4. If the canonical d-connection D is of Berwald type, then D, are also of Berwald type. In what follows, we assume that the canonical d-connection is of Berwald type. The next two results are immediate consequences of the fact that a α α a Pµc = Cµc = γµc = γbµ = 0,
taking into account Theorems 3.4, 3.5, 3.6, 4.2, 4.3 and 4.4. Proposition 6.5. The following relations hold: ◦
a d (a) Rabµν = γbd Rµν ,
e α = Λα , (b) R βνµ µν|β
◦
◦
P αβνc = S αβcd = 0, α α Peβµc = Pbβµc = 0,
a fa , Pebµc =W bµc
22
◦
◦
fα = W c α = 0. (c) W abνµ = W αβνc = W βνc βνc Theorem 6.6. The independent non-vanishing W -tensors are given by: ◦
α α ǫ α ǫ α ǫ α (a) W αβνµ = (γβµ|ν − γβν|µ ) + (γβν γǫµ − γβµ ) − γβǫ γǫν Λνµ , ◦
a (b) W abνc = γbc|ν ,
f α = Λα + Λǫ Λα , (c) W νµ βǫ βνµ νµ|β fa = T a + T e T a . (d) W bdc dc be dc||b To sum up, the assumption that the canonical d-connection being of Berwald type implies that most of the purely horizontal geometric objects of the EAP-space become functions of the positional argument x only and coincide with the corresponding geometric objects of the classical AP-space. Moreover, the three other defined d-connections turn out also to be of Berwald type. Finally, in this case, there are twelve non-vanishing curvature tensors and four independent W -tensors. We end this section with the following table (compare with Table 3). Table 4: EAP-geometry under the Berwald type case Connection
Coefficients
Torsion
Curvature
Canonical
a ˙ a (Γα µν , ∂b Nν , 0, Cbc )
a a (Λα µ ν , Rµν , 0, 0, Tbc )
(0, 0, 0, 0, 0, 0)
Dual
a ˙ a (Γα νµ , ∂b Nν , 0, Ccb )
a a (−Λα µν , Rµν , 0, 0, −Tbc )
eα , R ea , 0, Pe α , 0, Sea ) (R βµν bµν βµc bcd
Symmetric
a ˙ a (Γα (µν) , ∂b Nν , 0, C(bc) )
a (0, Rµν , 0, 0, 0)
bα , R ba , 0, Pbα , 0, Sba ) (R βµν bµν βµc bcd
Natural
a ˙ a ( Γα µν , ∂b Nν , 0, C bc )
a (0, Rµν , 0, 0, 0)
a a a ( Rα βµν , Rbµν , 0, P bµc , 0, S bcd )
◦
◦
◦
◦
◦
◦
7. Recovering the classical AP-space We now assume that the canonical d-connection D is both Cartan and Berwald type. In view of Proposition 6.2, this condition is equivalent to the (apparently weaker) α condition that D is of Cartan type and Cµc = 0. We show that in this case the classical AP-space emerges, in a natural way, as a special case from the EAP-space. We refer to this condition as the CB-condition. CB-condition: Nµa = y b Γabµ ,
a y b Cbc = 0;
As easily checked, we have Theorem 7.1. Assume that the CB-condition holds. Then 23
α Cµc = 0.
(1) The four defined d-connections of the EAP-space coincide up to the hh-coefficients. Moreover, these hh-coefficients are functions of the positional argument x only and are identical to the coefficients of the corresponding four defined connections in the classical AP-space. (2) The torsion of the canonical d-connection and the contortion of the EAP-space are functions of the positional argument x only and are given by T = (Λαµν , 0, 0, 0, 0);
α C = (γµν , 0, 0, 0)
(3) The three non-vanishing curvature tensors are functions of the positional argument x only and are given by ◦
α α ǫ α ǫ α α ǫ (a) Rαβµν = (γβµ|ν − γβν|µ ) + (γβν γǫµ − γβµ γǫν ) + γβǫ Λµν
e α = Λα , (b) R βµν νµ|β bα = 1 (Λα − Λα ) + 1 (Λǫ Λα − Λǫ Λα ) + 1 (Λǫ Λα ) (c) R µν βǫ βµν βµ νǫ βν µǫ βµ|ν βν|µ 2 4 2 (4) There is only one W -tensor which is a function of the positional argument x only and is given by f α = Λα + Λǫ Λα W βνµ νµ|β νµ βǫ All other W-tensors vanish identically, or coincide with the corresponding curvature tensors. Consequently, the fundamental geometric objects of the EAP-space coincide with the corresponding fundamental geometric objects of the classical AP-space [17]. e D b Corollary 7.2. If the canonical d-connection D satisfies the CB-condition, then D, ◦ and D also satisfy the CB-condition. We end this section with the following two tables which summarize the geometry of the EAP-space under the CB-condition. Table 5: Fundamental connections of EAP-space under the CB-condition Connection
Coefficients
hh-Coefficients
Canonical
a ˙ a (Γα µν , ∂b Nν , 0, Cbc )
α Γα µν (x) = ( λ ∂ν λµ )(x)
Dual
e α , ∂˙b N a , 0, C a ) (Γ µν ν bc
e α (x) = Γα (x) Γ µν νµ
Symmetric
b α , ∂˙b N a , 0, C a ) (Γ µν ν bc
b α (x) = Γα (x) Γ µν (µν)
Natural
a ˙ a ( Γα µν , ∂b Nν , 0, Cbc )
◦
i
◦
Γα µν (x) =
24
1 2
i
g αǫ (∂µ gνǫ + ∂ν gµǫ − ∂ǫ gµν )(x)
Table 6: EAP-geometry under the CB-condition Connection
Coefficients
Torsion
Curvature
Canonical
a ˙ a (Γα µν , ∂b Nν , 0, Cbc )
(Λα µ ν , 0, 0, 0, 0)
(0, 0, 0, 0, 0, 0)
Dual
a ˙ a (Γα νµ , ∂b Nν , 0, Cbc )
(−Λα µν , 0, 0, 0, 0)
eα , 0, 0, 0, 0, 0) (R βµν
Symmetric
a ˙ a (Γα (µν) , ∂b Nν , 0, Cbc )
(0, 0, 0, 0, 0)
bα , 0, 0, 0, 0, 0) (R βµν
Natural
a ˙ a ( Γα µν , ∂b Nν , 0, Cbc )
(0, 0, 0, 0, 0)
( Rα βµν , 0, 0, 0, 0, 0)
◦
◦
Some comments on the CB-condition: a (a) It should be noted that the non-vanishing of the purely vertical tensors λa , Cbc , and a gab and the hv-coefficients Γbµ of the canonical d-connection may represent extra degrees of freedom which do not exist in the classical AP-context. Moreover, these vertical geometric objects still depend on the directional argument y. However, they actually do not contribute to the EAP-geometry under the CB-condition. This is because the torsion and the contortion tensors in this case have only one non-vanishing counterpart each, namely the purely horizontal components Λαµν and α γµν respectively.
(b) One reading of Theorem 7.1 is roughly that the “projection” of the geometric objects of the EAP-space on the base manifold M can be identified with the classical AP-geometry. The distinction which appears between the two geometries is due to the fact that the geometric objects of the EAP-space live in the double tangent bundle T T M → T M and not in the tangent bundle T M → M. Consequently, it can be said, roughly speaking, that the classical AP-space is a copy of the EAPspace equipped with the CB-condition, viewed from the perspective of the base manifold M.
8. Concluding remarks In the present article, we have constructed and developed a parallelizable structure analogous to the AP-geometry on the tangent bundle T M of M. Four linear connections, depending on one a priori given nonlinear connection, are used to explore the properties of this geometry. Different curvature tensors charaterizing this structure, together with their contractions, are computed. The different W -tensors are also derived. Extra conditions are imposed on the canonical d-connection, the consequences of which are investigated. Finally, a special form of the canonical d-connection is introduced under which the EAP-geometry reduces to the classical AP-geometry. On the present work, we have the following comments: 25
(1) Existing theories of gravity suffer from some problems connected to recent observed astrophysical phenomena, especially those admitting anisotropic behavior of the system concerned (e.g. the flatness of the rotation curves of spiral galaxies). So, theories in which the gravitational potential depends on both position and direction may be needed. Such theories are to be constructed in spaces admitting this dependence; a potential candidate is the EAP-space. This is one of the aims motivating the present work. (2) One possible physical application of the EAP-geometry would be the construction of a generalized field theory on the tangent bundle T M of M. This could be achieved by a double contraction of the purely horizontal W -tensor W αβ µν and the purely vertical W -tensor W ab cd to obtain respectively the “horizontal” scalar Lagrangian H := |λ|H := |λ|g αβ Hαβ , where |λ| := det(λα ) and Hαβ := ΛνǫαΛǫνβ − Cα Cβ and the “vertical” scalar Lagrangian V := ||λ||V := ||λ||g ab Vab , where ||λ|| := det(λa ) and d e Vab := Tea Tdb − Ca Cb . The field equations are obtained by the use of the Euler-Lagrange equations ∂H ∂ ∂H ∂ ∂H − ǫ( ) − e( ) = 0, ∂λβ ∂x ∂λβ,ǫ ∂y ∂λβ; e
(horizontal form)
∂V ∂ ∂V ∂ ∂V − ǫ( ) − e( ) = 0. (vertical form) ∂λb ∂x ∂λb,ǫ ∂y ∂λb; e The resulting field equations could be compared with those derived by M. Wanas [12] and R. Miron [6]. (3) Among the advantages of the classical AP-geometry are the ease in calculations and the diverse and its thorough applications [14]. For these reasons, we have kept, in this work, as close as possible to the classical AP-case. However, the extra degrees of freedom in our EAP-geometry have created an abundance of geometric objects which have no counterpart in the classical AP-geometry. Since the physical meaning of most of the geometric objects of the classical AP-structure is clear, we hope to attribute physical meaning to the new geometric objects appearing in the present work. The physical interpretation of the geometric objects existing in the EAP-space and not in the AP-geometry may need deeper investigation. The study of the Cartan type case, due to its simplicity, may be our first step in tackling the general case. (4) In conclusion, we hope that physicits working in AP-geometry would divert their attention to the study of EAP-geometry and its consequences, due to its wealth, relative simplicity and its close resemblance (at least in form) to the classical AP-geometry. We believe that the extra degrees of freedom offered by the EAPgeometry may give us more insight into the infrastructure of physical phenomena studied in the context of classical AP-geometry and thus help us better understand the theory of general relativity and its connection to other physical theories. 26
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