The Curvature Theory of Generalized Connection in Osc^{2}M

Irena Čomić

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 Classification 53B15, 53B50, 53C80

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

1 Introduction

Let M be a C^{∞}, n-dimensional manifold and E = (Osc^{2}M, 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 of T (E) and T^{∗}(E) will be presented.

Some point u ∈ E in the local charts (U, φ) and (U′, φ′) has coordinates (x^{i}, y^{i}, z^{i}) and (x^{i′}, y^{i′}, z^{i′}) respectively, i = 1, 2, …, n. In U ∩ U′ the allowable coordinate transformations are given by the equations

(1.1)  \begin{aligned}x^{i′} & = & x^{i′}(x)  rank|\frac{∂x^{i′}}{∂x^{i}} | = n \\ y^{i′} & = & \frac{∂x^{i′}}{∂x^{j}} y^{j} \\ z^{i′} & = & \frac{1}{2} \frac{∂^{2}x^{i′}}{∂x^{k}∂x^{j}} y^{k}y^{j} + \frac{∂x^{i′}}{∂x^{j}} z^{j}. \end{aligned}

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

The adapted basis B of T (E) is

(1.2)  B = \{\frac{δ}{δx^{i}} , \frac{δ}{δy^{i}} , \frac{∂}{∂z^{i}} \},

where

Balkan Journal of Geometry and Its Applications, Vol.1, No.2, 1996, pp. 21-29 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

(1.3)  \frac{δ}{δx^{i}} = \frac{∂}{∂x^{i}} − (1)N_{i}^{j} \frac{∂}{∂y^{j}} − (2)N_{i}^{j} \frac{∂}{∂z^{j}}

(1.4)  \frac{δ}{δy^{i}} = \frac{∂}{∂y^{i}} − (1)N_{i}^{j} \frac{∂}{∂z^{j}}

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

(1.5)  \frac{δ}{δx^{i}} = \frac{∂x^{i′}}{∂x^{i}} \frac{δ}{δx^{i′}} , \frac{δ}{δy^{i}} = \frac{∂x^{i′}}{∂x^{i}} \frac{δ}{δy^{i′}} , \frac{∂}{∂z^{i}} = \frac{∂x^{i′}}{∂x^{i}} \frac{∂}{∂z^{i′}}

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

(1.6)  (1)N_{i′}^{j′} = (1)N_{i}^{j} \frac{∂x^{i}}{∂x^{i′}} \frac{∂x^{j′}}{∂x^{j}} − \frac{∂y^{j′}}{∂x^{i}} \frac{∂x^{i}}{∂x^{i′}} ,

(1.7)  (2)N_{i′}^{j′} = (2)N_{i}^{j} \frac{∂x^{i}}{∂x^{i′}} \frac{∂x^{j′}}{∂x^{j}} + (1)N_{i}^{j} \frac{∂y^{j′}}{∂x^{j}} \frac{∂x^{i}}{∂x^{i′}} − \frac{∂z^{j′}}{∂x^{i}} \frac{∂x^{i}}{∂x^{i′}} .

The basis of T^{∗}(E) dual to B is

(1.8)  B^{∗} = {dx^{i}, δy^{i}, δz^{i}},

where

(1.9)  δy^{i} = dy^{i} + (1)M_{j}^{i}dx^{j}

(1.10)  δz^{i} = dz^{i} + (1)M_{j}^{i}dy^{j} + (2)M_{j}^{i}dx^{j}.

Theorem 1.2. The elements of B^{∗} ((1.8)) with respect to (1.1) are transformed as

(1.11)  dx^{i′} = \frac{∂x^{i′}}{∂x^{i}} dx^{i}, δy^{i′} = \frac{∂x^{i′}}{∂x^{i}} δy^{i}, δz^{i′} = \frac{∂x^{i′}}{∂x^{i}} δz^{i}

if for the nonlinear connections (1)M and (2)M the following equations are valid

(1.12)  (1)M_{j}^{i} = (1)N_{j}^{i}

(1.13)  (2)M_{j}^{i} = (2)N_{j}^{i} + (1)N_{m}^{i}(1)N_{j}^{m}.

If we denote as T_{H}, T_{V_{1}}, T_{V_{2}} the subspaces of T (E) spanned by \{\frac{δ}{δx^{i}} \}, \{\frac{δ}{δy^{i}} \}, \{\frac{∂}{∂z^{i}} \} and as T_{H}^{∗}, T_{V_{1}}^{∗}, T_{V_{2}}^{∗} the subspaces of T^{∗}(E) spanned by \{dx^{i}\}, \{δy^{i}\}, {δz^{i}} respectively, then

T (E) = T_{H} ⊕ T_{V_{1}} ⊕ T_{V_{2}}, T^{∗}(E) = T_{H}^{∗} ⊕ T_{V_{1}}^{∗} ⊕ T_{V_{2}}^{∗}.

For the further examinations it is useful to introduce different kinds of indices. Indices i, j, h, k, l = \overline{1, n} will be used in T_{H} and T_{H}^{∗} a, b, c, d, e, f = \overline{n + 1, …, 2n} in T_{V_{1}} and T_{V_{1}}^{∗}, p, q, r, s, t = \overline{2n + 1, …, 3n} in T_{V_{2}} and T_{V_{2}}^{∗}. The Greek letters as indices will take values from 1 to 3n. Using these notations the adapted basis B and B^{∗} given by (1.2) and (1.8) have the form:

(1.14)  B = \{\frac{δ}{δx^{i}} , \frac{δ}{δy^{a}} , \frac{∂}{∂z^{p}} \} = \{δ_{i}, δ_{a}, ∂_{p}\} = \{δ_{α}\},

(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)  \begin{aligned}(a) \frac{δ}{δx^{i}} & = & \frac{∂}{∂x^{i}} = \frac{∂}{∂x^{i}} − (1)N_{i}^{b} \frac{∂}{∂y^{b}} − (2)N_{i}^{q} \frac{∂}{∂z^{q}} \\ (b) \frac{δ}{δy^{a}} & = & \frac{∂}{∂y^{a}} − (1)N_{a}^{q} \frac{∂}{∂z^{q}} \\ (c) δy^{a} & = & dy^{a} + (1)N_{i}^{a}dx^{i} \\ (d) δz^{q} & = & dz^{q} + (1)N_{a}^{q}dy^{a} + ((2)N_{j}^{q} + (1)N_{j}^{r}(1)N_{r}^{q})dx^{j} = \\ & = & dz^{q} + (1)N_{a}^{q}δy^{a} + (2)N_{j}^{q}dx^{j}. \end{aligned}

In (1.16)

(1)N_{i}^{j} = (1)N_{i}^{b} = (1)N_{a}^{q}, (2)N_{i}^{q} = (2)N_{i}^{j}

if i = a(modn) and j = b = q(modn).

Some vector field X ∈ T (E) and some 1-form w ∈ T^{∗}(E) expressed in the bases B and B^{∗} have the form:

(1.17)  \begin{aligned}X & = & X^{i} \frac{δ}{δx^{i}} + X^{a} \frac{δ}{δy^{a}} + X^{p} \frac{∂}{∂z^{p}} = X^{α}δ_{α} \\ w & = & w_{j}dx^{j} + w_{b}δy^{b} + w_{q}δz^{q} = w_{β}δ^{β}. \end{aligned}

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

\begin{aligned}X^{i′} & = & X^{i} \frac{∂x^{i′}}{∂x^{i′}} , X^{a′} = X^{a} \frac{∂y^{a′}}{∂y^{a}} , X^{p′} = X^{p} \frac{∂z^{p′}}{∂z^{p}} , \\ w_{j′} & = & w_{j} \frac{∂x^{j}}{∂x^{j′}} , w_{b′} = w_{b} \frac{∂y^{b}}{∂y^{b′}} , w_{q′} = w_{q} \frac{∂z^{q}}{∂z^{q′}} , \end{aligned}

because if i = a = p(modn) we have

\frac{∂x^{i′}}{∂x^{i}} = \frac{∂y^{a′}}{∂y^{a}} = \frac{∂z^{p′}}{∂z^{p}} .

2 The generalized connection on T (E)

Let ∇ : T (E) × T (E) → T (E) be a linear connection such that ∇ : (X, Y) → ∇_{X}Y ∈ T (E), ∀X, Y ∈ T (E). Definition 2.1. The generalized connection on T (E) is a linear connection ∇ determined by:

(2.1)  \begin{aligned}(a) ∇_{δ_{i}}δ_{β} & = & F_{β i}^{κ}δ_{κ}, \\ (b) ∇_{δ_{a}}δ_{β} & = & C_{β a}^{κ}δ_{κ} \\ (c) ∇_{∂_{p}}δ_{β} & = & L_{β p}^{κ}δ_{κ}, \end{aligned}

where β = j or β = b or β = q and

(2.2)  T_{...}^{..κ}δ_{κ} = T_{...}^{..k}δ_{k} + T_{...}^{..c}∂_{c} + T_{...}^{..r}∂_{r}.

We shall use the abbreviated form of (2.1):

(2.3)  ∇_{δ_{α}}δ_{β} = Γ_{β α}^{κ}δ_{κ}.

From (2.1) and (2.3) follows:

If α = i, then Γ = F; if α = a, then Γ = C; if α = p, then Γ = L.

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

(2.4)  \begin{matrix}∇_{δ_{i}}X = X_{|i}^{α}δ_{α} & , & X_{|i}^{α} = δ_{i}X^{α} + F_{β i}^{α} X^{β} \\ ∇_{∂_{a}}X = X^{α}|_{a}δ_{α} & , & X^{α}|_{a} = ∂_{a}X^{α} + C_{β a}^{α}X^{β} \\ ∇_{∂_{p}}X = X^{α}∥_{p}δ_{α} & , & X^{α}∥_{p} = ∂_{p}X^{α} + L_{β p}^{α}X^{β}, \end{matrix}

where

(2.5)  Γ_{β .}^{.}X^{β} = Γ_{j .}^{.}X^{j} + Γ_{b .}^{.}X^{b} + Γ_{q .}^{.}X^{q}, (Γ = F or Γ = C or Γ = L)

Theorem 2.1. If X and Y are vector fields in T E expressed in basis B, ∇ the generalized connection defined by (2.1), then

(2.6)  ∇_{Y} X = (X_{|β}^{α})Y^{β}δ_{α},

where

(2.7)  _{...|β}^{...}Y^{β} = _{...|j}^{...}Y^{j} + _{...}^{...}|_{b}Y^{b} + _{...}^{...}∥_{q}Y^{q}.

Theorem 2.2. All covariant derivatives X_{|i}^{α}, X^{α}|_{a}, X^{α}∥_{p} (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 following which have the form:

(2.8)  F_{j i}^{k} = F_{j′ i′}^{k′} \frac{∂x^{i′}}{∂x^{i}} \frac{∂x^{k}}{∂x^{k′}} \frac{∂x^{j′}}{∂x^{j}} + \frac{∂^{2}x^{k′}}{∂x^{i}∂x^{j}} \frac{∂x^{k}}{∂x^{k′}}

(2.9)  F_{b i}^{c} = F_{b′ i′}^{c′} \frac{∂x^{i′}}{∂x^{i}} \frac{∂y^{b′}}{∂y^{b}} \frac{∂y^{c}}{∂y^{c′}} + \frac{∂^{2}y^{c′}}{∂x^{i}∂y^{b}} \frac{∂z^{r}}{∂y^{c′}}

(2.10)  F_{q i}^{r} = F_{q′ i′}^{r′} \frac{∂x^{i′}}{∂x^{i}} \frac{∂z^{q′}}{∂z^{q}} \frac{∂z^{r}}{∂z^{r′}} + \frac{∂^{2}z^{r′}}{∂x^{i}∂z^{q}} \frac{∂z^{r}}{∂z^{r′}}

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

The Curvature Theory 25

(2.11)  T (X, Y) = ∇_{X}Y − ∇_{Y} X − [X, Y] = T_{α β}^{κ}Y^{α}X^{β}δ_{κ},

where

(2.12)  T_{α β}^{κ} = Γ_{α β}^{κ} − Γ_{β α}^{κ}

except the following components:

(2.13)  \begin{aligned}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} \end{aligned}

where

(2.14)  \begin{aligned}K_{i j}^{c} & = & δ_{j}(1)N_{i}^{c} − δ_{i}(1)N_{j}^{c}, \\ K_{i j}^{r} & = & δ_{j}(2)N_{i}^{r} − δ_{i}(2)N_{j}^{r} + (1)N_{c}^{r}K_{i j}^{c}, \\ K_{i b}^{c} & = & δ_{b}(1)N_{i}^{c}, \\ K_{i b}^{r} & = & δ_{b}(2)N_{i}^{r} − δ_{i}(1)N_{b}^{r} + (1)N_{c}^{r}K_{i b}^{c}, \\ K_{i q}^{r} & = & ∂_{q}(2)N_{i}^{r}, \\ K_{a b}^{r} & = & δ_{b}(1)N_{a}^{r} − δ_{a}(1)N_{b}^{r}. \end{aligned}

Proof. By direct calculation we obtain

(2.15)  [X, Y] = X^{α}(∂_{α}Y^{β})∂_{β} − Y^{α}(∂_{α}X^{β})∂_{β} + X^{α}Y^{β}(∂_{α}∂_{β} − ∂_{β}∂_{α}),
(2.16)  X^{α}Y^{β}(∂_{α}∂_{β} − ∂_{β}∂_{α}) = X^{α}Y^{β}K_{α β}^{κ}∂_{κ} =

= X^{i}Y^{j}(K_{i j}^{c}δ_{c} + K_{i j}^{r}∂_{r}) + (X^{i}Y^{b} − Y^{i}X^{b})(K_{i b}^{c}δ_{c} + K_{i b}^{r}∂_{r})+

+(X^{i}Y^{q} − Y^{i}X^{q})K_{i q}^{r}∂_{r} + X^{a}Y^{b}K_{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}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[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)  \begin{matrix}∇_{X}∇_{Y} Z = ∇_{X^{α}∂_{α}}∇_{Y^{β}∂_{β}}Z^{γ}∂_{γ} = \\ ∇_{X^{α}∂_{α}}[Y^{β}(∂_{β}Z^{γ})∂_{γ} + Y^{β}Z^{γ}Γ_{γ β}^{κ}∂_{κ}] = \\ X^{α}(∂_{α}Y^{β})(∂_{β}Z^{γ})∂_{γ} + X^{α}Y^{β}(∂_{α}∂_{β}Z^{γ})∂_{γ}+ \\ X^{α}Y^{β}(∂_{β}Z^{γ})Γ_{γ α}^{κ}∂_{κ} + X^{α}(∂_{α}Y^{β})Z^{γ}Γ_{γ β}^{κ}∂_{κ}+ \\ X^{α}Y^{β}(∂_{α}Z^{γ})Γ_{γ β}^{κ}∂_{κ} + X^{α}Y^{β}Z^{γ}(∂_{α}Γ_{γ β}^{κ})∂_{κ}+ \\ X^{α}Y^{β}Z^{γ}Γ_{γ β}^{θ}Γ_{θ α}^{κ}∂_{κ}. \end{matrix}

From (2.15) and (2.16) follows

(3.3)  \begin{aligned}∇_{[XY]}Z & = & X^{α}(∂_{α}Y^{β})(∂_{β}Z^{γ})∂_{γ} + X^{α}(∂_{α}Y^{β})Z^{γ}Γ_{γ β}^{κ}∂_{κ} \\ & & −Y^{α}(∂_{α}X^{β})(∂_{β}Z^{γ})∂_{γ} − Y^{α}(∂_{α}X^{β})Z^{γ}Γ_{γ β}^{κ}∂_{κ} \\ & & +X^{α}Y^{β}[(∂_{α}∂_{β} − ∂_{β}∂_{α})Z^{γ}]∂_{γ}+ \\ & & X^{α}Y^{β}Z^{γ}K_{α β}^{θ}Γ_{γ θ}^{κ}∂_{κ}. \end{aligned}

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

(3.4)  \begin{aligned}R(X, Y)Z & = & [K_{γ βα}^{κ}X^{α}Y^{β} − (K_{i j}^{c}C_{γ c}^{κ} + K_{i j}^{q}L_{γ q}^{κ})X^{i}Y^{j}− \\ & & (K_{i b}^{c}C_{γ c}^{κ} + K_{i b}^{r}L_{γ r}^{κ})(X^{i}Y^{b} − Y^{i}X^{b})− \\ & & K_{i q}^{r}L_{γ r}^{κ}(X^{i}Y^{q} − Y^{i}X^{q}) − K_{a b}^{r}L_{γ r}^{κ}X^{a}Y^{b}]Z^{γ}∂_{κ}, \end{aligned}

where

(3.5)  (3.5)K_{γ βα}^{κ} = (∂_{α}Γ_{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 T_{H}, T_{V_{1}}, T_{V_{2}} respectively), so on the T E we have 3^{4} = 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)  \begin{matrix}R_{γ ji}^{κ} = K_{γ ji}^{κ} − K_{i j}^{c}C_{γ c}^{κ} − K_{i j}^{q}L_{γ q}^{κ}, \\ R_{γ ib}^{κ} = K_{γ ib}^{κ} + K_{i b}^{c}C_{γ c}^{κ} + K_{i b}^{r}L_{γ r}^{κ} = −R_{γ bi}^{κ}, \\ R_{γ iq}^{κ} = K_{γ iq}^{κ} + K_{i q}^{p}L_{q p}^{κ} = −R_{γ qi}^{κ}, \\ R_{γ ba}^{κ} = K_{γ ba}^{κ} − K_{a b}^{r}L_{γ r}^{κ} = −R_{γ ab}^{κ}. \end{matrix}

As R_{γ βα}^{κ} = −R_{γ αβ}^{κ} we can write (3.4) in the form:

(3.8)  \begin{aligned}R(X, Y)Z & = & [\frac{1}{2}K_{γ βα}^{κ}(X^{α}Y^{β} − Y^{α}X^{β})− \\ & & \frac{1}{2}(K_{i j}^{c}C_{γ c}^{κ} + K_{i j}^{q}L_{γ q}^{κ})(X^{i}Y^{j} − Y^{i}X^{j})− \\ & & \frac{1}{2}(K_{i b}^{c}C_{γ c}^{κ} + K_{i b}^{r}L_{γ r}^{κ}(X^{i}Y^{b} − Y^{i}X^{b})+ \\ & & \frac{1}{2}(K_{i b}^{c}C_{γ c}^{κ} + K_{i b}^{r}L_{γ r}^{κ})(Y^{i}X^{b} − X^{i}Y^{b})− \\ & & \frac{1}{2}K_{i q}^{r}L_{γ r}^{κ}(X^{i}Y^{q} − Y^{i}X^{q})+ \\ & & \frac{1}{2}K_{i q}^{r}L_{γ r}^{κ}(Y^{i}X^{q} − X^{i}Y^{q})]− \\ & & \frac{1}{2}K_{a b}^{r}L_{γ r}^{κ}(X^{a}Y^{b} − Y^{a}X^{b})Z^{γ}δ_{κ} \end{aligned}

For (β, α) = (j, i) the sum of the first and the second line in (3.8) is equal to \frac{1}{2}R_{γ ji}^{κ}(X^{i}Y^{j} − Y^{i}X^{j}), for (β, α) = (b, i) the sum of the first and the third line in (3.8) is equal to \frac{1}{2}R_{γ bi}^{κ}(X^{i}Y^{b} − Y^{i}X^{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 = \frac{1}{2}R_{γ βα}^{κ}(X^{α}Y^{β} − Y^{α}X^{β})Z^{γ}δ_{κ},

where the components of R 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} = δ_{i}C_{γ b}^{κ} − F_{γ i}^{θ}C_{θ b}^{κ} − ∂_{b}F_{γ i}^{κ} + C_{γ b}^{θ}F_{θ i}^{κ} − K_{i b}^{c}C_{γ c}^{κ} − K_{i b}^{r}L_{γ r}^{κ} =
δ_{i}C_{γ b}^{κ} − F_{γ i}^{k}C_{k b}^{κ} − F_{γ i}^{c}C_{c b}^{κ} − F_{γ i}^{r}C_{r b}^{κ} − ∂_{b}F_{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}^{c}C_{γ c}^{κ} − K_{i b}^{r}L_{γ r}^{κ}.

4 Ricci identities for ∇

From (2.6) it follows

(4.1)  ∇_{X}∇_{Y} Z = [(Z_{|β}^{γ})Y^{β}]_{|α}X^{α}δ_{γ} = (Z_{|β|α}^{γ}Y^{β} + Z_{|β}^{γ}Y _{|α}^{β})X^{α}δ_{γ}.

From (2.6), (2.15) and (2.16) we obtain

(4.2)  ∇_{[X,Y]}Z = Z_{|β}^{γ}[X, Y]^{β}δ_{γ} = A + B,

where

(4.3)  A = Z_{|β}^{γ}[X^{α}(∂_{α}Y^{β}) − Y^{α}(∂_{α}X^{β})]δ_{γ} =
= Z_{|β}^{γ}[X^{α}Y _{|α}^{β} − Y^{α}X_{|α}^{β} − (Γ_{θ α}^{β} − Γ_{α θ}^{β})X^{α}Y^{θ}]δ_{γ}

(4.4)  \begin{aligned}B & = & X^{i}Y^{j}[Z^{γ}|_{c}K_{i j}^{c} + Z^{γ}∥_{q}K_{i j}^{q}]δ_{γ}+ \\ & & (X^{i}Y^{b} − Y^{i}X^{b})[Z^{γ}|_{c}K_{i b}^{c} + Z^{γ}∥_{r}K_{i b}^{r}]δ_{γ} \\ & & (X^{i}Y^{q} − Y^{i}X^{q})δ_{γ} + X^{a}Y^{a}Z^{γ}∥_{r}K_{a b}^{r}δ_{γ}. \end{aligned}

Taking into account (2.13) and (2.14) we obtain

(4.5)  A + B = [Z_{|β}^{γ}(X^{α}Y _{|α}^{β} − Y^{α}X_{|α}^{β}) − Z_{|κ}^{γ}T_{β α}^{κ}X^{α}Y^{β}]δ_{γ}.

From (4.1), (4.2) and (4.5) we obtain

(4.6)  \begin{aligned}R(X, Y)Z & = & (Z_{|β|α}^{γ} − Z_{|α|β} + Z_{|κ}^{γ}T_{β α}^{κ})X^{α}Y^{β}δ_{γ} = \\ & & \frac{1}{2}(Z_{|β|α}^{γ} − Z_{|α|β}^{γ} + Z_{|κ}^{γ}T_{β α}^{κ})(X^{α}Y^{β} − Y^{α}X^{β})δ_{γ}. \end{aligned}

From (4.6) and (3.9) it follows:

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

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

(4.7) contains 3^{3} 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_{|j|i}^{γ} − Z_{|i|j}^{γ} + Z_{|k}^{γ}T_{j i}^{k} + Z^{γ}|_{c}T_{j i}^{c} + Z^{γ}∥_{p}T_{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^{γ}∥_{p|i} − Z_{|i}^{γ}∥_{p} + Z_{|k}^{γ}T_{p i}^{k} + Z^{γ}|_{c}T_{p i}^{c} + Z^{γ}∥_{r}T_{p i}^{r} =
= R_{k pi}^{γ}Z^{k} + R_{c pi}^{γ}Z^{c} + R_{r pi}^{γ}Z^{r}, e.t.c.

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