CURVATURE THEORY OF GENERALIZED CONNECTION IN J

Vol. 36, No. 1, 2006, 141-151

CURVATURE THEORY OF GENERALIZED CONNECTION IN J

M

Irena ˇComi´c¹, Mihai Anastasiei²

Abstract. The introduction ofJk²M manifold and its geometrical pre- sentation is given in [2]. Here, the generalized connection is defined, the torsion and curvature tensors are determined, and the Ricci equations are established.

AMS Mathematics Subject Classification (2000): 53B40

Key words and phrases: J_k²M manifold, generalized connection, curvature tensors, Ricci equation

1. Manifold J

M

Let M be a smooth manifold of dimension n and Jo,p(R^k, M) the set of germs of smooth mappings f : R^k → M with f(o) = p ∈ M. We say that f, g ∈ Jo,p(R^k, M) are equivalent up to order q if there exists a chart (U, ϕ) aroundpsuch that

(1.1) d^h_o(ϕ◦f) =d^h_o(ϕ◦g), 1≤h≤q,

where d means Frechet differentiation. It can be seen that if (1.1) holds for a chart (U, ϕ), it holds for any other chart (V, ψ) around p.

We denote byj_o,p^q f the equivalence class off (the coset off) and setJ_o,p^q = {j_o,p^q f, f ∈Jo,p(R^k, M)}. Then we putJ_k^qM = S

p∈M

J_o,p^q and define π:J_k^qM → M byπ(J_o,p^q ) =p.

One can see thatJ_m^qM has a structure of smooth manifold.

We notice that fork= 1, this manifold is just the manifold Osc^qM studied by R. Miron [5], which reduces to the tangent manifold forq= 1. Fork=nand q= 1, we get the manifold of frames over M and for k∈ {2,3, . . . , n−1} and q= 1 it can be identified to T M⊗ · · · ⊗T M (k times), which is the manifold supporting thek-Lagrange geometry, see R. [7].

For these reasons we confine ourselves to the casesk= 2,3, . . . , n−1 and for the sake of simplicity we takeq= 2. The caseqgreater than 2 can be similarly treated.

We also notice thatJ_k²M is the manifold of 2-jets of the sections of the fibre bundle R^k×M →R^k but the theory of jets from the book by D.J. Saunders [8] cannot be applied since the typical fibreM of this bundle is too general.

1Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia, e-mail:

[email protected]

2Faculty of Mathematics, University ”Al. I. Cuza” la¸si, 6600, Ia¸si, Romania, e-mail: anas- [email protected]

Instead of that theory we follow the ideas and techniques from thek-Lagrange geometry and from the geometry of Osc^qM spaces as well, see [1], [3], [6].

Let us come back to (1.1) for q = 2. Letting ϕ◦g : R^k → Rⁿ as fⁱ = fⁱ(t¹, . . . , t^k),gⁱ =gⁱ(t¹, . . . , t^k) this condition becomes

(1.1)’ fⁱ(o) =gⁱ(o) =ϕ(p), ∂fⁱ

∂t^α(o) = ∂gⁱ

∂t^α(o), ∂²fⁱ

∂t^α∂t^β(o) = ∂²gⁱ

∂t^α∂t^β(o), forα, β= 1,2, . . . , k. Let us set∂i: _∂x^∂i,∂α:=_∂t^∂α.

Now, for another local chart (V, ψ) around p such that ψ◦ϕ⁻¹ : xⁱ⁰ = xⁱ⁰(x¹, . . . , xⁿ), rank

³∂x⁰

∂x^k

´

=n, takingψ◦f andψ◦g asfⁱ⁰ =fⁱ⁰(t¹, . . . , t^k) and gⁱ⁰ = gⁱ⁰(t¹, . . . , t^k), respectively, we get fⁱ⁰ = xⁱ⁰(f^j(t¹, . . . , t^k)), gⁱ⁰ = xⁱ⁰(g^j(t¹, . . . , t^k)) as well as

(1.2)

∂fⁱ⁰

∂t^α(o) = ^∂x_∂xⁱj⁰(ϕ(p))^∂f_∂tα^j(o)

∂²fⁱ⁰

∂t^α∂t^β = _∂x^∂2j^x∂x^i0k(ϕ(p))^∂f_∂tα^j(o)^∂f_∂tβ^k(o) +^∂x_∂xⁱj⁰ ∂²f^j

∂t^α∂t^β.

By (1.2) it follows the independence of (1.1) on the chosen local chart.

For f :R^k →M with f(o) =ϕ(p)(x¹, . . . , xⁿ) we set y^αi= ^∂f_∂tαⁱ(o), z^αβi =

∂²fⁱ

∂t^α∂t^β(o) and define a mapping φ : π⁻¹(U) → ϕ(U)×R^kn ×R^{k(k+1)2 n} by φ([f]p) = (xⁱ, y^αi, z^αβi).

The mappingφis invertible, its inverse associating to (xⁱ, y^αi, z^αβi) the coset of the mappingϕ⁻¹◦T, whereT is the Taylor polynomial of second order with respect tot.

2. Decomposition of T (E). Integrability conditions

Let E = J_k²M be an n+kn+ 2⁻¹k(k+ 1)n dimensional C^∞ manifold.

Some point u ∈ J_k²M in the local charts (U, ϕ) and (U⁰, ϕ⁰) has coordinates (xⁱ, y^αi, z^αβi) and (xⁱ⁰, y^αi0, z^αβi0) respectively. InU∩U⁰, the allowable coordi- nate transformations are given by the equation

xⁱ⁰ =xⁱ⁰(x¹, x², . . . , xⁿ), rank(∂xⁱ⁰

∂xⁱ) =n, (2.1)

y^αi0 = ∂xⁱ⁰

∂xⁱy^αi=y^αi0(xⁱ, y^αi), α≤β, rank(y^αi) =k, z^αβi0 = ∂²xⁱ⁰

∂x^j∂x^hy^αjy^βh+∂xⁱ⁰

∂xⁱz^αβi=z^αβi0(xⁱ, y^αi, z^αβi) i, j, h, k, l= 1,2, . . . , n, α, β, γ, δ, κ, ε= 1,2, . . . , k.

Proposition 2.1. Transformations of type (2.1) form a pseudo group.

Proof. The neutral element of the group is given by the transformationxⁱ⁰ =xⁱ, y^αi0 =y^αi,z^αβi0 =z^αβi. Fromrank(^∂x_∂xⁱi⁰) =nit follows that (2.1) has inverse

Curvature theory of generalized connection in J_kM 143 transformation of the same type. If the pointuin some local chart (U⁰⁰, ϕ⁰⁰) has coordinates (xⁱ⁰⁰, y^αi00, z^αβi00), then inU⁰∩U⁰⁰ the transformation law is given by equation obtained from (2.1) in such a way, that the latin indices obtain one more prime. After some calculation it follows that the coordinates of the point uinU ∩U⁰⁰ satisfy the equation of type (2.1) if everywhere the index i⁰ is substituted byi⁰⁰.

Let us introduce the notations:

(2.2) ∂i= ∂

∂xⁱ, ∂αi= ∂

∂y^αi, ∂αβi = ∂

∂z^αβi,(α≤β).

The natural basis ¯B ofT(J_k²M) =T(E) is ¯B ={∂i, ∂αi, ∂αβi}.

If a change of coordinates (2.1) is performed, the elements of ¯B are trans- formed as follows:

∂i= (∂ixⁱ⁰)∂i⁰+ (∂i∂jxⁱ⁰)y^αj∂αi⁰+ [(∂i∂j∂hxⁱ⁰)y^βjy^γh+ (∂i∂jxⁱ⁰)z^βγj]∂βγi⁰

∂αi= (∂ixⁱ⁰)∂αi⁰+ 2(∂i∂jxⁱ⁰)y^γj∂αγi⁰

(2.3)

∂αβi= (∂ixⁱ⁰)∂αβi⁰. The adapted basis ofT(E) isB ={δi, δαi, δαβi}, where

(a) δi=∂i−N_i^αj∂αj−N_i^αβj∂αβj,(α≤β), (2.4)

(b) δαi=∂αi−N_αi^βγj∂βγj,(β≤γ), (c) δαβi=∂αβi.

The summation is going over both types of indices.

From (2.4) it follows that∂αβi is transformed asd tensor field, i.e. ∂αβi = (∂ixⁱ⁰)∂αβi⁰.

Proposition 2.2. The elements of B are transformed asd-tensor field, i.e.

(2.5) δi= (∂ixⁱ⁰)δi⁰ δαi= (∂ixⁱ⁰)δαi⁰,

if the nonlinear connection coefficients obey the following transformation law:

(a) N_αi^βγj0 ⁰(∂ixⁱ⁰) =N_αi^βγj(∂jx^j0)−(∂αiz^βγj0), (2.6)

(b) N_i^αj0 ⁰(∂ixⁱ⁰) =N_i^αj(∂jx^j0)−(∂iy^αj0),

(c) N_i^αβj0 ⁰(∂ixⁱ⁰) =N_i^αβj(∂jx^j0) +N_i^γj(∂γjz^αβj0)−∂iz^αβj0. Proof. The proof follows from (2.2)–(2.5).

If we denote by TH, TV1 and TV2 the subspaces of T(E) (at the point u) spanned by{δi},{δαi},{δαβi}, then we have

T(E) =TH⊕TV1⊕TV2, where dimTH =n,dimTV1 =nk,dimTV2 = 2⁻¹k(k+ 1)n.

The dual basis of ¯B is ¯B^∗={dxⁱ, dy^αi, dz^αβi}. By a change of coordinates (2.1) the element of ¯B^∗ are transformed as follows:

dxⁱ⁰ = (∂ixⁱ⁰)dxⁱ, (2.7)

dy^αi0 = (∂i∂jxⁱ⁰)y^αjdxⁱ+ (∂ixⁱ⁰)dy^αi,

dz^αβi0 = [(∂_i∂_j∂_hxⁱ⁰)y^αjy^βh+ (∂_i∂_jxⁱ⁰)z^αβj]dxⁱ+ (∂j∂hxⁱ⁰)(y^βhdy^αj+y^αhdy^βj) + (∂ixⁱ⁰)dz^αβi. The adapted basis ofT^∗(E) isB^∗={dxⁱ, δy^αi, δz^αβi}, where

δy^αj=dy^αj+M_i^αjdxⁱ (2.8)

δz^αβj=dz^αβj+M_γi^αβjdy^γi+M_i^αβjdxⁱ. The functionsM are, for the time being, undetermined.

Proposition 2.3. The necessary and sufficient conditions that the bases B and B^∗ to be dual to each other (when B¯ and B¯^∗ are dual) are the following equations:

(a) M_i^αj=N_i^αj (b)M_γi^αβj =N_γi^αβj,(α≤β) (2.9)

(c) M_i^αβj=N_i^αβj+N_i^γhN_γh^αβj,(α≤β).

Proof. The proof follows from (2.4)–(2.9).

Remark. The basesBandB^∗ are more general than those used in [2], but they are not in accordance with the operatorJα, where

(2.10) Jαδi=δαi, Jβδαi=δβαi, Jγδαβi= 0.

If we take the basis ˜B={δi, δαi, δαβi}, whereδi andδαβi are determined by (2.4a) and (2.4c) and

δαi=∂αi−N_i^βj∂αβj

then (2.10) are satisfied. The coresponding dual basis ˜B^∗is determined by (2.8), but now (2.9) has different form (see (2.16) in [2]).

Proposition 2.4. The horizontal distributionTH is integrable iff the following relations are satisfied

(2.11) K_{i j}^βk = ¯K_{i j}^βk =δjN_i^βk−δiN_j^βk= 0,

(2.12) K_{i j}^γδk = ¯K_{i j}^γδk +K_{i j}^βhN_βh^γδk= 0,

(2.13) ( ¯K_{i j}^γδk =δjN_i^γδk−δiN_j^γδk).

Curvature theory of generalized connection in J_kM 145 Proof. A straightforward calculation gives

(2.14) [δi, δj] =K_{i j}^κkδκk+K_{i j}^κρkδκρk, and from (2.11)–(2.13) it follows the statement.

Proposition 2.5. The vertical distributionTV1 is integrable iff K_{αi βj}^δκk = δβjN_αi^δκk−δαiN_βj^δκk= 0.

The proof follows from

(2.15) [δαi, δβj] =K_{αi βj}^δκk δδκk. TV2 is integrable distribution because

(2.16) [δαβi, δγδj] = 0.

3. The generalized connection on T (E)

Definition 3.1. The generalized connectionD:T(E)×T(E)→T(E),(X, Y)

→DXY,X, Y ∈T(E)is the linear connection defined by:

Dδiδj=F_{j i}^kδk+F_{j i}^κkδκk+F_{j i}^κρkδκρk

(3.1)

Dδiδγj=F_{γj i}^k δk+F_{γj i}^κkδκk+F_{γj i}^κρkδκρk

Dδiδγδj=F_{γδj i}^k δk+F_{γδj i}^κkδκk+F_{γδj i}^κρkδκρk

Dδαiδj=F_{j αi}^k δk+F_{j αi}^κk δκk+F_{j αi}^κρkδκρk

Dδαiδγj=F_{γj αi}^k δk+F_{γj αi}^κk δκk+F_{γj αi}^κρk δκρk

Dδαiδγδj=F_{γδj αi}^k δk+F_{γδj αi}^κk δκk+F_{γδj αi}^κρk δκρk

Dδαβiδj=F_{j αβi}^k δk+F_{j αβi}^κk δκk+F_{j αβi}^κρk δκρk

Dδαβiδγj=F_{γj αβi}^k δk+F_{γj αβi}^κk δκk+F_{γj αβi}^κρk δκρk

Dδαβiδγδj=F_{γδj αβi}^k δk+F_{γδj αβi}^κk δκk+F_{γδj αβi}^κρk δκρk (κ≤ρ).

Different types of linear connection in higher order geometries are given in [4]–[7].

Definition 3.2. If on the right-hand side of (3.1) all terms vanish except for the underlined then the generalized connection reduces to the distinguished d- connection.

If on the right-hand side of (3.1) all terms vanish except for F_{j i}^k, F_{γj αi}^κk , F_{γδj αβi}^κρk , the generalized connection reduces to the strongly distinguished (s.d.) connection.

For the sake of brevity it is convenient to use new kind of indices, the latin capitals, which take values from 1 ton+nk+ 2⁻¹k(k+ 1)n. Using them, (3.1) can be written in the form

DδIδJ=F_{J I}^KδK= (3.2)

F_{J I}^k δk+F_{J I}^κkδκk+F_J^κρk_I δκρk. LetX andY be vector fields determined onT(E) by:

(3.3) X=X^IδI =Xⁱδi+X^αiδαi+X^αβiδαβi, (3.4) Y =Y^JδJ =Y^jδj+Y^γjδγj+Y^γδjδγδj

then (3.2) has the form

(3.5) DXY =D_XIδIY^JδJ=X^I(δIY^J+F_HI^J Y^H)δJ =X^IY_|I^JδI.

Using the explicit forms of (3.3) and (3.4) for (3.5) we get the following propo- sition.

Proposition 3.1. The generalized connectionDcan be expressed by covariant derivatives in the form:

DXY = (XⁱY^j_|i+X^αiY^j_|αi+X^αβiY^j_|αβi)δj+ (3.6)

(XⁱY^γj_|i +X^αiY^γj_|αi+X^αβiY^γj_|αβi)δγj+ (XⁱY^γδj_|i +X^αiY^γδj_|αi+X^αβiY^γδj_|αβi)δγδj.

In (3.1), all the connection coefficients F are arbitrary smooth function of x,y andz, but they should satisfy prescribed transformation law with respect to (2.1). Our intention is to find these laws of transformations. All covariant derivatives

(3.7) Y^J_|I =δIY^J+F_{H I}^J Y^H =δIY^J+F_{h I}^JY^h+F_{νh I}^J Y^νh+F_{µνh I}^J Y^µνh which appear in (3.6) ared-tensor fields.

In (3.7) I∈ {i, αi, αβi},J ∈ {j, γj, γδj}.

For thed-connection (3.7) takes the form:

Y_|I^j =δIY^j+F_{h I}^j Y^h Y_|I^γj =δIY^γj+F_{δd I}^γjY^δd Y_|I^αβj =δIY^αβj+F_{µνh I}^αβjY^µνh

I∈ {i, κk, γκk}.

The necessary and sufficient conditions that all covariant derivatives Y^J_|I appeared in (3.6) bed-tensor fields are given in the following proposition.

Curvature theory of generalized connection in J_kM 147 Proposition 3.2. All the connection coefficients F_{H I}^J that appear in (3.7) transform as d-tensor fields except for the case whenI=i. Then we have (3.8) (F_H^J0⁰i⁰)(∂ixⁱ⁰)(∂hx^h0) =F_{H i}^J (∂jx^j0)−(∂h∂ix^j0).

Proof. If we suppose thatY^J_|i (I=i in (3.7)) isd-tensor field, then Y^J_|i(∂jx^j0) =Y^J_|i⁰0(∂ixⁱ⁰),

i.e.

(3.9) (δiY^J+F_{H i}^J Y^H)(∂jx^j0) = (δi⁰Y^J0+F_H^J0⁰i⁰Y^H0)(∂ixⁱ⁰).

As (see (2.4a) and (2.5))

δiY^J(∂jx^j0) =δi(Y^J∂jx^j0)−Y^Jδi(∂jx^j0) = (∂ixⁱ⁰)δi⁰Y^J0−Y^J(∂i∂jx^j0)

the substitution of the above equation in (3.9) gives

F_{H i}^J Y^H∂jx^j0−Y^H(∂h∂ix^j0) =F_H^J0⁰i⁰Y^H(∂hx^h0)(∂ixⁱ⁰)

from which follows (3.8). The connection coefficients from (3.8) appeared in the first three lines of (3.1).

If we putI =αi in (3.7), and suppose thatY^J_|αi is d-tensor field, then we get

Y^J_|αi(∂jx^j0) =Y^J_|αi⁰ 0(∂ixⁱ⁰)

(3.10) (δαiY^J+F_{H αi}^J Y^H)(∂jx^j0) = (δαi⁰Y^J0 +F_H^J0⁰αi⁰Y^H0)(∂ixⁱ⁰).

As (see (2.4b) and (2.5))

(δαiY^J)(∂jx^j0) =δαi(Y^J∂jx^j0)−Y^Jδαi(∂jx^j0) = (∂ixⁱ⁰)(δαi⁰Y^J0), ((δαi∂jx^j0) = 0)

the substitution of the above equation in (3.10) results in (3.11) F_{H αi}^J (∂jx^j0) =F_H^J0⁰αi⁰(∂hx^h0)(∂ixⁱ⁰).

From the above equation follows that all connection coefficients that ap- peared in the middle three lines of (3.1) are transformed asd-tensor fields.

In a similar way one can prove

(3.12) F_{H αβi}^J (∂jx^j0) =F_H^J0⁰αβi⁰(∂hx^h0)(∂ixⁱ⁰),

i.e. all connection coefficients that appeared in the last three lines of (3.1) are tensor fields.

The torsion tensor of the generalized connection D is determined by T(X, Y) =DXY −DYX−[X, Y].

A straightforward calculation gives

T(X, Y) ={(F_{J I}^K −F_{I J}^K)δK−[δI, δJ]}Y^JX^I. If we introduce the notation

(3.13) [δI, δJ] =K_{I J}^KδK, we get

(3.14) T(X, Y) = (F_{J I}^K −F_{I J}^K −K_{I J}^K)Y^JX^Iδ_K=T_{J I}^KY^JX^Iδ_K. Now the components of K_{J I}^K should be determined. [δi, δj], [δαi, δβj] and [δ_αβi, δ_γδj] are determined by (2.14), (2.15) and (2.16). Further, we obtain

[δ_i, δ_γj] =K_{i γj}^κk δ_κk+K_{i γj}^κρk δ_κρk, (3.15)

K_{i γj}^κk =δγjN_i^κk

K_{i γj}^κρk = ¯K_{i γj}^κρk + ¯K_{i γj}^δk M_δh^κρk K¯_{i γj}^κρk =δγjN_i^κρk−δiN_γj^κρk [δ_i, δ_γδj] =K_{i γδj}^κk ∂_κk+K_{i γδk}^κρk ∂_κρk K_{i γδj}^κk =∂γδjN_i^κk

K_{i γδj}^κρk =∂γδjN_i^κρk [δαi, δγδj] =K_{αi γδj}^κρk ∂κρk

K_{αi γδj}^κρk =∂γδjN_αi^κδk

Theorem 3.1. The components of the torsion tensor

T(X, Y) =T_{J I}^KY^JX^IδK

of the generalized connection D are determined by T_{J I}^K =F_{J I}^K −F_{I J}^K

except for the case when K_{I J}^K 6= 0(see (3.13), (3.14)), and then they have the

Curvature theory of generalized connection in J_kM 149 form (see (2.14), (2.15) and (3.15)):

(a) T_{j i}^κk =F_{j i}^κk −F_{i j}^κk −K_{i j}^κk (3.16)

(b) T_{j i}^κρk =F_{j i}^κρk −F_{i j}^κρk −K_{i j}^κρk (c) T_{γj αi}^κρk =F_{γj αi}^κρk −F_{αi γj}^κρk −K_{αi γj}^κρk (d) T_{γj i}^κk =F_{γj i}^κk −F_{i γj}^κk −K_{i γj}^κk (e) T_{γj i}^κρk =F_{γj i}^κρk −F_{i γj}^κρk −K_{i γj}^κρk (f) T_{γδj i}^κk =F_{γδj i}^κk −F_{i γδj}^κk −K_{i γδj}^κk (g) T_{γδj i}^κρk =F_{γδj i}^κρk −F_{i γδk}^κρk −K_{i γδj}^κρk (h) T_{γδj αi}^κρk =F_{γδj αi}^κρk −F_{αi γδj}^κρk −K_{αi γδj}^κρk

AsT_{J I}^K are components of thed-tensor field, using (3.16), (3.8), (3.11) and (3.12) we can obtain the transformation laws ofK_{J I}^K.

Proposition 3.3. K_{i j}^κk,K_{i j}^κρk,K_{αi γj}^κρk ,K_{αi γδj}^κρk are d-tensor fields, K_{i γj}^κk , K_{i γj}^κρk , K_{i γδj}^κk , K_{i γδj}^κρk

are not d-tensor fields and they transform in the following way:

(3.17) K_{i γj}^κk =K_i0^κkγj^{0 0}(∂ixⁱ⁰)(∂jx^j0)(∂k⁰x^k) + (∂i∂jx^k0)(∂k⁰x^k) (similar for the next three K).

Proof. From (3.8) it follows, that F_{j i}^κk −F_{i j}^κk is ad-tensor, and from (3.16a) follows that K_{i j}^κk is the difference of twod-tensors, so itself is ad-tensor.

As in (3.16d),T_{γj i}^κk andF_{i γj}^κk ared-tensors (see (3.11)), so F_{γj i}^κk −K_{i γj}^κk is a d-tensor. Using this fact and (3.8) we obtain (3.17).

For thed-connection and s.d. connection in (3.15) all terms K_{A B}^C remain, because they are not functions of different connection coefficients, they only depend onN andM, which are involved in adapted bases. In some components ofT_{J I}^K some Γ_{J I}^K vanish (see definition 3.2).

4. The curvature theory of generalized connection

The curvature tensor

(4.1) R(X, Y)Z=D_XD_YZ−D_YD_XZ−D_[X,Y]Z

can be calculated in the usual way. ForX =X^AδA,Y =Y^BδB,Z=Z^CδC we get

(4.2) DYZ =DY^BδBZ^CδC =Y^B(δBZ^C)δC+Y^BZ^CF_{C B}^D δD,

DXDYZ=DX^AδA[Y^B(δBZ^C)δC+Y^BZ^CF_{C B}^D δD] = (4.3)

X^A(δAY^B)(δBZ^C)δC+X^AY^BδA(δBZ^C)δC+ X^AY^B(δ_BZ^C)F_{C A}^D δ_D+X^A(δ_AY^B)Z^CF_{C B}^D δ_D+ X^AY^B(δAZ^C)F_{C B}^D δD+X^AY^BZ^C(δAF_{C B}^D )δD+ X^AY^BZ^CF_{C B}^E F_{E A}^D δD.

Further, using the notation [δ_A, δ_B] =K_{A B}^D δ_D, we get D[X,Y]Z=D[X^AδA,Y^BδB]Z^CδC = (4.4)

X^A(δ_AY^B)[(δ_BZ^C)δ_C+Z^CΓ_{C B}^D δ_D]− Y^B(δBX^A)[(δAZ^C)δC+Z^CΓ_{C A}^D δD] +

X^AY^B{[δA, δB]Z^C}δC+X^AY^BZ^CK_{A B}^E F_{C E}^D δD. Finally we obtain

Theorem 4.1. The curvature tensor of the generalized connectionDonT(E) is given by

(4.5) R(X, Y)Z =R_{C BA}^D X^AY^BZ^CδD, where

(4.6) R_{C BA}^D =K_{C BA}^D +F_{C E}^D K_{B A}^E ,

(4.7) K_{C BA}^D =δAF_{C B}^D +F_{C B}^E F_{E A}^D −δBF_{C A}^D −F_{C A}^E F_{E B}^D .

The values of K_{A B}^E which are different from zero are determined by (3.16).

Since the latin capitals as indices are connected with TH(i, j, k, h, . . .),TV1(αi, βj, γk, . . .) orTV2(αβi, γδj, κρk, . . .) there are 3⁴types of curvature tensors.

From (4.2) it follows

DYZ =Y^BZ^C_|BδC,

DX(DYZ) =X^A(Y^BZ^C_|B)|AδC= (4.8)

X^A(Y^B_|AZ^C_|B+Y^BZ^C_|B|A)δC. hand,side from (4.2) we get

(4.9) D_[X,Y]Z =Z^C_|D[X, Y]^DδC=A+B, where

A = Z^C_|B(X^AδAY^B−Y^AδAX^B)δC= (4.10)

[X^AY^B_AZ^C_|B−Y^BX^A_|BZ^C_|A− (F_{B A}^D −F_{A B}^D )X^AY^BZ^C_|D]δC,

Curvature theory of generalized connection in J_kM 151 (4.11) B=K_{A B}^D X^AY^BZ^C_|DδC.

Substituting (4.10) and (4.11) into (4.9), then (4.9) and (4.4) into (4.1), we obtain

(4.12) R(X, Y)Z= (Z^C_|B|A−Z^C_|A|B+T_{B A}^D Z^C_|D)X^AY^BδC. From (4.12) and (4.5) it follows

Theorem 4.2. The Ricci equations for the generalized connectionD have the form:

Z^C_|B|A−Z^C_|A|B+T_{B A}^D Z^C_|D=R_{D BA}^C Z^D, where

A∈ {i, αi, αβi}, B∈ {j, γj, γδj}, C∈ {k, εk, ερk}, D∈ {h, νh, νµh}.

References

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Matematiˇcki Vesnik 49 (1997), 15-22.

[3] ˇComi´c, I., The curvature theory of strongly distinguished connection in the recurrentk-Hamilton space. Indian J. Pure Appl. Math. 23(3) (1992), 189-202.

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Received by the editors February 22, 2006