2 The generalized connection on T (E)

in Osc ² M

Irena ˇ Comi´c

Abstract

In the paper are presented the explicit expressions for the components of the torsion and curvature of a generalized connection on the 2-osculator bundle of a real differentiable manifold; the corresponding Ricci identities for the generalized connection are also derived.

Mathematics Subject Classification53B15, 53B50, 53C80

Key words: osculator, bundle, generalized connection, curvature, torsion.

1 Introduction

LetM be aC^∞,n-dimensional manifold andE= (Osc²M, π, M) its 2-osculator bundle. The K-osculator bundle was studied among others in [15], [16], [17], where the adapted basis is determined. As we here need more types of indices, the adapted basis ofT(E) andT^∗(E) will be presented.

Some point u ∈ E in the local charts (U, ϕ) and (U⁰, ϕ⁰) has coordinates (xⁱ, yⁱ, zⁱ) and (xⁱ⁰, yⁱ⁰, zⁱ⁰) respectively, i= 1,2, . . . , n. InU∩U⁰ the allowable coordinate transformations are given by the equations

(1.1)

xⁱ⁰ = xⁱ⁰(x) rank|^∂x_∂xⁱi⁰|=n yⁱ⁰ = ^∂x_∂xⁱj⁰y^j

zⁱ⁰ = ¹_2∂x^∂2k^x∂x^i0jy^ky^j+^∂x_∂xⁱj⁰z^j.

It can be shown, that the transformations of type (1.1) form a group.

The adapted basisB ofT(E) is

(1.2) B={ δ

δxⁱ, δ δyⁱ, ∂

∂zⁱ}, where

Balkan Journal of Geometry and Its Applications, Vol.1, No.1, 1996, pp. 21-29 c

°Balkan Society of Geometers, Geometry Balkan Press

(1.3) δ δxⁱ = ∂

∂xⁱ −(1)N_i^j ∂

∂y^j −(2)N_i^j ∂

∂z^j

(1.4) δ

δyⁱ = ∂

∂yⁱ −(1)N_i^j ∂

∂z^j

Theorem 1.1.The element ofB ((1.2)) with respect to (1.1) are transformed as tensors, i.e.

(1.5) δ

δxⁱ =∂xⁱ⁰

∂xⁱ δ δxⁱ⁰, δ

δyⁱ = ∂xⁱ⁰

∂xⁱ δ δyⁱ⁰, ∂

∂zⁱ = ∂xⁱ⁰

∂xⁱ

∂

∂zⁱ⁰

if the nonlinear connections (1)N and (2)N are transformed in the following way:

(1.6) (1)N_i^j0⁰ =(1)N_i^j∂xⁱ

∂xⁱ⁰

∂x^j0

∂x^j −∂y^j0

∂xⁱ

∂xⁱ

∂xⁱ⁰,

(1.7) (2)N_i^j0⁰ =(2)N_i^j∂xⁱ

∂xⁱ⁰

∂x^j0

∂x^j +(1)N_i^j∂y^j0

∂x^j

∂xⁱ

∂xⁱ⁰ −∂z^j0

∂xⁱ

∂xⁱ

∂xⁱ⁰. The basis ofT^∗(E) dual toB is

(1.8) B^∗={dxⁱ, δyⁱ, δzⁱ}, where

(1.9) δyⁱ=dyⁱ+(1)Mⁱ_jdx^j

(1.10) δzⁱ=dzⁱ+(1)Mⁱ_jdy^j+(2)Mⁱ_jdx^j.

Theorem 1.2.The elements ofB^∗((1.8)) with respect to (1.1) are transformed as

(1.11) dxⁱ⁰ = ∂xⁱ⁰

∂xⁱdxⁱ, δyⁱ⁰ = ∂xⁱ⁰

∂xⁱδyⁱ, δzⁱ⁰ =∂xⁱ⁰

∂xⁱδzⁱ

if for the nonlinear connections(1)Mand(2)Mthe following equations are valid

(1.12) (1)Mⁱ_j=(1)N_jⁱ

(1.13) (2)Mⁱ_j =(2)N_jⁱ+(1)N_mⁱ(1)N_j^m.

If we denote asTH,TV1,TV2 the subspaces ofT(E) spanned by{_δx^δi},{_δy^δi}, {_∂z^∂i} and as T_H^∗, T_V^∗₁, T_V^∗₂ the subspaces of T^∗(E) spanned by {dxⁱ}, {δyⁱ}, {δzⁱ}respectively, then

T(E) =TH⊕TV1⊕TV2, T^∗(E) =T_H^∗ ⊕T_V^∗₁⊕T_V^∗₂.

For the further examinations it is useful to introduce different kinds of indices.

Indicesi, j, h, k, l= 1, nwill be used inTH andT_H^∗ a, b, c, d, e, f =n+ 1, . . . ,2n inTV1 andT_V^∗₁,p, q, r, s, t= 2n+ 1, . . . ,3ninTV2 andT_V^∗₂. The Greek letters as indices will take values from 1 to 3n. Using these notations the adapted basisB andB^∗ given by (1.2) and (1.8) have the form:

(1.14) B={ δ

δxⁱ, δ δy^a, ∂

∂z^p}={δi, δa, ∂p}={δα}, (1.15) B^∗ ={dx^j, δy^b, δz^p}={δ^β}, where from (1.3), (1.4), (1.9) and (1.10) we get

(1.16)

(a) _δx^δi = _∂x^∂i = _∂x^∂i−(1)N_i^b_∂y^∂b−(2)N_i^q_∂z^∂q

(b) _δy^δa = _∂y^∂a −(1)N_a^q_∂z^∂q

(c) δy^a = dy^a+(1)N_i^adxⁱ

(d) δz^q = dz^q+(1)N_a^qdy^a+ ((2)N_j^q+(1)N_j^r(1)N_r^q)dx^j =

= dz^q+(1)N_a^qδy^a+(2)N_j^qdx^j. In (1.16)

(1)N_i^j =(1)N_i^b=(1)N_a^q, (2)N_i^q =(2)N_i^j ifi=a(modn) andj=b=q(modn).

Some vector field X ∈ T(E) and some 1-form w ∈T^∗(E) expressed in the basesB andB^∗ have the form:

(1.17) X = X^{i δ}_δxi +X^{a δ}_δya +X^{p ∂}_∂zp =X^αδα

w = wjdx^j+wbδy^b+wqδz^q =wβδ^β.

With respect to (1.1) the coordinates of X and w transform in the following way:

Xⁱ⁰ = X^{i ∂x}_∂xⁱ_i⁰0, X^a0 =X^{a ∂y}_∂y^aa⁰, X^p0 =X^{p ∂z}_∂z^pp⁰, wj⁰ = wj∂x^j

∂x^j0, wb⁰ =wb∂y^b

∂y^b0, wq⁰ =wq∂z^q

∂z^q0, because ifi=a=p(modn) we have

∂xⁱ⁰

∂xⁱ =∂y^a0

∂y^a = ∂z^p0

∂z^p.

2 The generalized connection on T (E)

Let∇ :T(E)×T(E)→T(E) be a linear connection such that ∇ : (X, Y)→

∇XY ∈T(E),∀X, Y ∈T(E).

Definition 2.1.The generalized connection on T(E) is a linear connection∇ determined by:

(2.1)

(a) ∇δiδβ = F_{β i}^κδκ, (b) ∇δaδβ = C_{β a}^κ δκ

(c) ∇_∂pδ_β = L_{β p}^κ δ_κ,

whereβ=j orβ =b orβ=qand

(2.2) T_...^..κδκ=T_...^..kδk+T_...^..c∂c+T_...^..r∂r. We shall use the abbreviated form of (2.1):

(2.3) ∇δαδβ = Γ_{β α}^κ δκ. From (2.1) and (2.3) follows:

Ifα=i, then Γ =F; ifα=a, then Γ =C; ifα=p, then Γ =L.

Proposition 2.1.If X is the vector field ((1.17)) defined on E, then the fol- lowing equations are valid:

(2.4)

∇δiX =X^α_|iδα , X^α_|i=δiX^α+F_{β i}^αX^β

∇∂aX=X^α|aδα , X^α|a=∂aX^α+C_{β a}^αX^β

∇∂pX =X^αkpδα , X^αkp=∂pX^α+L_{β p}^α X^β, where

(2.5) Γ_{β .}^. X^β = Γ_{j .}^. X^j+ Γ_{b .}^.X^b+ Γ_{q .}^. X^q, (Γ =F or Γ =Cor Γ =L) Theorem 2.1.If X andY are vector fields inT E expressed in basisB,∇ the generalized connection defined by (2.1), then

(2.6) ∇YX = (X^α_|β)Y^βδα, where

(2.7) ^..._...|βY^β=^..._...|jY^j+^..._...|_bY^b+^..._...k_qY^q.

Theorem 2.2. All covariant derivatives X_|i^α, X^α|a, X^αkp (α = j, or α = b, or α=p) from (2.4) are transformed as tensors with respect to (1.1) if all connection coefficients from (2.1) are transformed as tensors, except the follow- ing which have the form:

(2.8) F_{j i}^k =F_j^k0⁰i⁰

∂xⁱ⁰

∂xⁱ

∂x^k

∂x^k0

∂x^j0

∂x^j + ∂²x^k0

∂xⁱ∂x^j

∂x^k

∂x^k0

(2.9) F_{b i}^c =F_b0^c0i⁰

∂xⁱ⁰

∂xⁱ

∂y^b0

∂y^b

∂y^c

∂y^c0 + ∂²y^c0

∂xⁱ∂y^b

∂z^r

∂y^c0

(2.10) F_{q i}^r =F_q^r0⁰i⁰

∂xⁱ⁰

∂xⁱ

∂z^q0

∂z^q

∂z^r

∂z^r0 + ∂²z^r0

∂xⁱ∂z^q

∂z^r

∂z^r0

Theorem 2.3. The torsion tensorT for the generalized connection ∇ has the form:

(2.11) T(X, Y) =∇XY − ∇YX−[X, Y] =T_{α β}^κ Y^αX^βδκ, where

(2.12) T_{α β}^κ = Γ_{α β}^κ −Γ_{β α}^κ except the following components:

(2.13)

T_{j i}^c = F_{j i}^c −F_{i j}^c +K_{j i}^c

T_{j b}^c = C_{j b}^c −F_{b j}^c +K_{j b}^c =−T_{b j}^c T_{j i}^r = F_{j i}^r −F_{i j}^r +K_{j i}^r

T_{j b}^r = C_{j b}^r −F_{b j}^r +K_{j b}^r =−T_{b j}^r T_{j q}^r = L_{j q}^r −F_{q j}^r +K_{j q}^r =−T_{q j}^r T_{b a}^r = C_{b a}^r −C_{a b}^r +K_{a b}^r

where

(2.14)

K_{i j}^c = δj(1)N_i^c−δi(1)N_j^c,

K_{i j}^r = δj(2)N_i^r−δi(2)N_j^r+(1)N_c^rK_{i j}^c, K_{i b}^c = δb(1)N_i^c,

K_{i b}^r = δb(2)N_i^r−δi(1)N_b^r+(1)N_c^rK_{i b}^c, K_{i q}^r = ∂q(2)N_i^r,

K_{a b}^r = δb(1)N_a^r−δa(1)N_b^r. Proof.By direct calculation we obtain

(2.15) [X, Y] =X^α(∂αY^β)∂β−Y^α(∂αX^β)∂β+X^αY^β(∂α∂β−∂β∂α),

(2.16) X^αY^β(∂α∂β−∂β∂α) =X^αY^βK_{α β}^κ ∂κ=

=XⁱY^j(K_{i j}^cδc+K_{i j}^r∂r) + (XⁱY^b−YⁱX^b)(K_{i b}^cδc+K_{i b}^r∂r)+

+(XⁱY^q−YⁱX^q)K_{i q}^r∂r+X^aY^bK_{a b}^r∂r.

Substituting (2.15) and (2.16) into (2.11) we obtain (2.12) and (2.13).

3 The curvature theory of ∇

The curvature tensor for the generalized connection∇is defined as usual (3.1) R(X, Y)Z =∇X∇YZ− ∇Y∇XZ− ∇_[X,Y]Z.

If we use the notations

X =X^α∂α, Y =Y^β∂β, Z=Z^γ∂γ, then we have

(3.2)

∇X∇YZ =∇X^α∂α∇_Y^β_∂βZ^γ∂γ =

∇X^α∂α[Y^β(∂βZ^γ)∂γ+Y^βZ^γΓ_{γ β}^κ ∂κ] = X^α(∂αY^β)(∂βZ^γ)∂γ+X^αY^β(∂α∂βZ^γ)∂γ+ X^αY^β(∂βZ^γ)Γ_{γ α}^κ ∂κ+X^α(∂αY^β)Z^γΓ_{γ β}^κ ∂κ+ X^αY^β(∂αZ^γ)Γ_{γ β}^κ ∂κ+X^αY^βZ^γ(∂αΓ_{γ β}^κ )∂κ+

X^αY^βZ^γΓ_{γ β}^θ Γ_{θ α}^κ ∂κ. From (2.15) and (2.16) follows

(3.3)

∇_[XY]Z = X^α(∂αY^β)(∂βZ^γ)∂γ+X^α(∂αY^β)Z^γΓ_{γ β}^κ ∂κ

−Y^α(∂αX^β)(∂βZ^γ)∂γ−Y^α(∂αX^β)Z^γΓ_{γ β}^κ ∂κ

+X^αY^β[(∂α∂β−∂β∂α)Z^γ]∂γ+ X^αY^βZ^γK_{α β}^θ Γ_{γ θ}^κ ∂κ.

Substituting (3.2) and (3.3) into (3.1) we obtain

(3.4)

R(X, Y)Z = [K_{γ βα}^κ X^αY^β−(K_{i j}^cC_{γ c}^κ +K_{i j}^qL_{γ q}^κ )XⁱY^j− (K_{i b}^cC_{γ c}^κ +K_{i b}^rL_{γ r}^κ )(XⁱY^b−YⁱX^b)−

K_{i q}^rL_{γ r}^κ (XⁱY^q−YⁱX^q)−K_{a b}^r L_{γ r}^κ X^aY^b]Z^γ∂κ, where

(3.5) (3.5)K_{γ βα}^κ = (∂αΓ_{γ β}^κ −Γ_{γ α}^θ Γ_{θ β}^κ )−(α, β).

As the indices α, β, γ, κ belong to one of the sets {i, j, k, l, . . .}, {a, b, c, d, . . .}, {p, q, r, s, t, . . .} (corresponding to TH,TV1,TV2 respectively), so on the T E we have 3⁴ = 81 types of curvature tensors. It is meaningless to introduce different letters as R, P, S for the curvature tensors as in Finsler geometry.

We shall denote

(3.6) R_{γ βα}^κ =K_{γ βα}^κ

for all (β, α) except when (β, α) = (j, i), (β, α) = (i, b), (β, α) = (i, q) and (β, α) = (b, a). In these cases we have

(3.7)

R_{γ ji}^κ =K_{γ ji}^κ −K_{i j}^cC_{γ c}^κ −K_{i j}^qL_{γ q}^κ ,

R_{γ ib}^κ =K_{γ ib}^κ +K_{i b}^cC_{γ c}^κ +K_{i b}^rL_{γ r}^κ =−R_{γ bi}^κ , R_{γ iq}^κ =K_{γ iq}^κ +K_{i q}^pL_{q p}^κ =−R_{γ qi}^κ ,

R_{γ ba}^κ =K_{γ ba}^κ −K_{a b}^rL_{γ r}^κ =−R_{γ ab}^κ . AsR_{γ βα}^κ =−R_{γ αβ}^κ we can write (3.4) in the form:

(3.8)

R(X, Y)Z = [¹₂K_{γ βα}^κ (X^αY^β−Y^αX^β)−

1

2(K_{i j}^cC_{γ c}^κ +K_{i j}^qL_{γ q}^κ )(XⁱY^j−YⁱX^j)−

1

2(K_{i b}^cC_{γ c}^κ +K_{i b}^rL_{γ r}^κ (XⁱY^b−YⁱX^b)+

1

2(K_{i b}^cC_{γ c}^κ +K_{i b}^rL_{γ r}^κ )(YⁱX^b−XⁱY^b)−

1

2K_{i q}^rL_{γ r}^κ (XⁱY^q−YⁱX^q)+

1

2K_{i q}^rL_{γ r}^κ (YⁱX^q−XⁱY^q)]−

1

2K_{a b}^rL_{γ r}^κ (X^aY^b−Y^aX^b)Z^γδκ

For (β, α) = (j, i) the sum of the first and the second line in (3.8) is equal to

1

2R_{γ ji}^κ (XⁱY^j−YⁱX^j), for (β, α) = (b, i) the sum of the first and the third line in (3.8) is equal to ¹₂R_{γ bi}^κ (XⁱY^b−YⁱX^b) etc.

From (3.4)–(3.8) follows

Theorem 3.1. The curvature tensor of the generalized connection ∇ has the form

(3.9) R(X, Y)Z =1

2R_{γ βα}^κ (X^αY^β−Y^αX^β)Z^γδκ, where the components ofR are determined by (3.6) and (3.7).

Formula (3.9) is short and elegant, but the explicit form of curvature tensor is much longer, for instance if (β, α) = (b, i) from (3.5) and (3.7) we have:

R_{γ bi}^κ = δiC_{γ b}^κ −F_{γ i}^θC_{θ b}^κ −∂bF_{γ i}^κ +C_{γ b}^θ F_{θ i}^κ −K_{i b}^cC_{γ c}^κ −K_{i b}^rL_{γ r}^κ = δ_iC_{γ b}^κ −F_{γ i}^kC_{k b}^κ −F_{γ i}^cC_{c b}^κ −F_{γ i}^rC_{r b}^κ −∂_bF_{γ i}^κ+

C_{γ b}^k F_{k i}^κ +C_{γ b}^c F_{c i}^κ +C_{γ b}^r F_{r i}^κ −K_{i b}^cC_{γ c}^κ −K_{i b}^rL_{γ r}^κ .

4 Ricci identities for ∇

From (2.6) it follows

(4.1) ∇X∇YZ= [(Z^γ_|β)Y^β]_|αX^αδγ = (Z^γ_|β|αY^β+Z^γ_|βY^β_|α)X^αδγ. From (2.6), (2.15) and (2.16) we obtain

(4.2) ∇[X,Y]Z =Z^γ_|β[X, Y]^βδγ =A+B, where

(4.3) A=Z^γ_|β[X^α(∂αY^β)−Y^α(∂αX^β)]δγ =

=Z^γ_|β[X^αY^β_|α−Y^αX^β_|α−(Γ_{θ α}^β −Γ_{α θ}^β )X^αY^θ]δγ

(4.4)

B = XⁱY^j[Z^γ|cK_{i j}^c +Z^γkqK_{i j}^q]δγ+

(XⁱY^b−YⁱX^b)[Z^γ|cK_{i b}^c +Z^γkrK_{i b}^r]δγ

(XⁱY^q−YⁱX^q)δγ+X^aY^aZ^γkrK_{a b}^rδγ. Taking into account (2.13) and (2.14) we obtain

(4.5) A+B= [Z^γ_|β(X^αY^β_|α−Y^αX^β_|α)−Z^γ_|κT_{β α}^κ X^αY^β]δγ. From (4.1), (4.2) and (4.5) we obtain

(4.6) R(X, Y)Z = (Z^γ_|β|α−Z|α|β+Z^γ_|κT_{β α}^κ )X^αY^βδγ =

1

2(Z^γ_|β|α−Z^γ_|α|β+Z^γ_|κT_{β α}^κ )(X^αY^β−Y^αX^β)δγ. From (4.6) and (3.9) it follows:

Theorem 4.3.The Ricci equations for the generalized connection ∇ have the form:

(4.7) Z^γ_|β|α−Z^γ_|α|β+Z^γ_|κT_{β α}^κ =R_{κ βα}^γ Z^κ.

(4.7) contains 3³ types of Ricci equations, because each Greek index may be the element from one of the sets:{i, j, h, k, l},{a, b, c, d, e},{p, q, r, s, t}.

For (β, α) = (j, i) (4.7) becomes

Z^γ_|j|i−Z^γ_|i|j+Z^γ_|kT_{j i}^k +Z^γ|cT_{j i}^c +Z^γkpT_{j i}^p =

=R_{k ji}^γ Z^k+R_{c ji}^γ Z^c+R_{p ji}^γ Z^p, for (β, α) = (p, i) (4.7) takes the form

Z^γkp|i−Z^γ_|ikp+Z^γ_|kT_{p i}^k +Z^γ|cT_{p i}^c +Z^γkrT_{p i}^r =

=R^γ_{k pi}Z^k+R_{c pi}^γ Z^c+R_{r pi}^γ Z^r, e.t.c.

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