Lobachevskii Journal of Mathematics Vol.8, 2001, 19–32

\begin{matrix}c \\ ◯ \end{matrix}A.Bucki

A.Bucki SOME PROPERTIES OF ALMOST R-PARACONTACT MANIFOLDS

(submitted by B.N.Shapukov)

Abstract. For an almost r-paracontact manifold M with a structure Σ, almost r-paracontact connections, D-connections, and pairs of connections compatible with Σ have been defined and discussed in [2]. In this paper, which is a continuation of [2], some properties of almost r-paracontact manifolds have been studied by means of the curvature and torsion tensor fields of these connections.

1. Almost r-paracontact manifolds [1]. In this section we recall the definition of an almost r-paracontact manifold and present some of its properties.

Definition 1.1. Let M_{n} be an n-dimensional differentiable manifold. If on M_{n} there exist: a tensor field ϕ of type (1,1), r vector fields ξ_{1}, ξ_{2}, …, ξ_{r} (r < n), r 1-forms η^{1}, η^{2}, …, η^{r} such that

η^{α}(ξ_{β}) = δ_{β}^{α},  α, β ∈ (r) = 1, 2, ..., r  (1.1)
ϕ^{2} = Id − η^{α} ⊗ ξ_{α}, where a^{α}b_{α} =^{def} ∑_{α} a^{α}b_{α}  (1.2)
η^{α} ∘ ϕ = 0,  α ∈ (r)  (1.3)

then Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} is said to be an almost r-paracontact structure on M_{n}, and M_{n} is an almost r-paracontact manifold.

From (1.1), (1.2), and (1.3) we also have

ϕ(ξ_{α}) = 0,  α ∈ (r)  (1.4)

There exists a positive Riemannian metric g on M_{n} such that

η^{α}(X) = g(X, ξ_{α}),  α ∈ (r)  (1.5)
g(ϕX, ϕY) = g(X, Y) − ∑_{α} η^{α}(X)η^{α}(Y)  (1.6)

Then, Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)} is called a metric almost r-paracontact structure on M_{n}, and g is said to be compatible Riemannian metric.

From (1.1) through (1.6) we get

g(ϕX, Y) = g(X, ϕY)  (1.7)

Proposition 1.1. An almost r-paracontact manifold M_{n} admits the following complementary distributions

D_{1} = {X; ϕX = X}  (1.8)
D_{2} = {X; ϕX = −X}  (1.9)
D_{3} = {X; ϕX = 0}  (1.8)

with dim D_{1} = p, dim D_{2} = q, dim D_{3} = r, and p + q + r = n.

Definition 1.2. If M_{n} is an almost r-paracontact manifold with the structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)}, then Σ is said to be normal if an almost product structure F defined on M_{n} × R^{r} by F (X, f^{α} \frac{d}{dt^{α}}) = (ϕX + f^{α}ξ_{α}, η^{α}(X)\frac{d}{dt^{α}}) is integrable, i.e., its Nijenhuis tensor field N_{F} vanishes.

Theorem 1.1. An almost r-paracontact structure Σ on M_{n} is normal if and only if N(X, Y) = N_{ϕ}(X, Y) − 2dη^{a}(X, Y)ξ_{α} = 0, where N_{ϕ} is the Nijenhuis tensor field for ϕ.

2. Projection operators on almost r-paracontact manifolds [2]. In this section various projection operators on an almost r-paracontact manifold are defined.

Definition 2.1. A multilinear operator Φ on an appropriate space is said to be a projection operator if Φ^{2} = Φ.

Definition 2.2. A set of operators {Φ_{i}} is said to be the set of complementary projection operators if ∑_{i} Φ_{i} = Id, Φ_{i}^{2} = Φ_{i}, Φ_{i}Φ_{j} = 0, i ≠ j.

Proposition 2.1 [3]. If Φ is a projection operator and Ψ = Id − Φ, then Φ and Ψ are complementary projection operators and all solutions of the equation Φx = y are of the form x = y + Ψw, where w is an arbitrary element.

Remark 2.1. If operators Φ and Ψ are tensor fields of type (2,2) and S, ϕ, X are tensor fields of type (1,2), (1,1), and (0,1) respectively, then the operations ΦΨ, ΦS, Φϕ, ΦX are expressed locally as follows: Φ_{kl}^{ij}Ψ_{in}^{ml}, Φ_{kl}^{ij}S_{mi}^{l}, Φ_{kl}^{ij}ϕ_{i}^{l}, Φ_{kl}^{ij}X^{k}.

Let Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} be an almost r-paracontact structure on M_{n}. We define the following operators on M_{n}.

ϕ_{1} = \frac{1}{2}(Id + ϕ − η^{α} ⊗ ξ_{α}) = \frac{1}{2}(ϕ^{2} + ϕ)  (2.1)
ϕ_{2} = \frac{1}{2}(Id − ϕ − η^{α} ⊗ ξ_{α}) = \frac{1}{2}(ϕ^{2} − ϕ)  (2.2)
ϕ_{3} = η^{α} ⊗ ξ_{α} = Id − ϕ^{2}  (2.3)
ϕ_{4} = ϕ^{2} = Id − ϕ_{3}  (2.4)

with the properties

\begin{matrix}ϕ = ϕ_{1} − ϕ_{2}, ϕϕ_{1} = ϕ_{1}ϕ = ϕ_{1}, ϕϕ_{2} = ϕ_{2}ϕ = −ϕ_{2}, \\ ϕϕ_{3} = ϕ_{3}ϕ = 0, ϕϕ_{4} = ϕ_{4}ϕ = ϕ \end{matrix}  (2.5)

Proposition 2.2. The operators ϕ_{1}, ϕ_{2}, ϕ_{3} are complementary projection operators on an almost r-paracontact manifold M_{n}.

Proposition 2.3. The operators ϕ_{3} and ϕ_{4} are complementary projection operators on an almost r-paracontact manifold M_{n}.

The distributions (1.8), (1.9), and (1.10) can be expressed as follows

D_{1} = {X; ϕ_{1}X = X}  (2.6)
D_{2} = {X; ϕ_{2}X = X}  (2.7)
D_{3} = {X; ϕ_{3}X = X}  (2.8)

Proposition 2.4. The distributions D_{i}, i = 1, 2, 3, are generated by the projection operators ϕ_{i}, respectively.

Let A, B, and C be tensor fields of type (2,2) defined as

A = \frac{1}{2}(Id ⊗ Id − ϕ ⊗ ϕ)  (2.9)
B = \frac{1}{2}(Id ⊗ Id + ϕ ⊗ ϕ)  (2.10)
C = \frac{1}{2}(ϕ_{3} ⊗ Id + Id ⊗ ϕ_{3} − ϕ_{3} ⊗ ϕ_{3})  (2.11)

The operators A, B, and C possess the following properties

\begin{matrix}A + B = Id ⊗ Id, AA = A − \frac{1}{2}C, BB = B − \frac{1}{2}C, \\ AB = BA = AC = CA = BC = CB = CC = \frac{1}{2}C \end{matrix}  (2.12)

Proposition 2.5. The operators

F = A + C = Id ⊗ Id − ϕ_{1} ⊗ ϕ_{1} − ϕ_{2} ⊗ ϕ_{2}  (2.13)
H = B − C = ϕ_{1} ⊗ ϕ_{1} + ϕ_{2} ⊗ ϕ_{2}  (2.14)

are complementary projection operators on an almost r-paracontact manifold M_{n}.

Proposition 2.6. The operators

P = F − ϕ_{3} ⊗ ϕ_{3} = Id ⊗ Id − ϕ_{1} ⊗ ϕ_{1} − ϕ_{2} ⊗ ϕ_{2} − ϕ_{3} ⊗ ϕ_{3} (2.15)
Q = H + ϕ_{3} ⊗ ϕ_{3} = ϕ_{1} ⊗ ϕ_{1} + ϕ_{2} ⊗ ϕ_{2} + ϕ_{3} ⊗ ϕ_{3}  (2.16)

are complementary projection operators on an almost r-paracontact manifold M_{n}.

3. Almost r-paracontact connections, D-connections, and pairs of compatible connections [2]. In this section the definitions and properties of an almost r-paracontact connection, a D-connection, and a pair of connections compatible with an almost r-paracontact structure are presented.

Definition 3.1. For an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} a linear connection Γ, given by its covariant derivative ∇, is said to be an almost r-paracontact if

∇_{X}ϕ = 0  (3.1)
∇_{X}η^{α} = 0,  α ∈ (r)  (3.2)

for any vector field X.

From (3.1) and (3.2) it follows

∇_{X}ξ_{α} = 0,  α ∈ (r)  (3.3)

Definition 3.2. For an almost r-paracontact Riemannian manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)} an almost r-paracontact connection \overline{Γ}, given by its covariant derivative \overline{∇}, is said to be a metric almost r-paracontact if

∇_{X}g = 0.  (3.4)

We have

Theorem 3.1. The general family of the almost r-paracontact connections \overline{Γ} on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} is given by

\overline{∇}_{X} = ∇_{X} + \frac{1}{2}(ϕ_{1}∇_{X}ϕ_{1} + ϕ_{2}∇_{X}ϕ_{2} + ϕ_{3}∇_{X}ϕ_{3} + 2∇_{X}η^{α} ⊗ ξ_{α}) + HW_{X} (3.5)

where H is defined by (2.14), W is an arbitrary tensor field of type (1,2) with W_{X}(Y) = W (X, Y), and ∇ is the covariant derivative of an arbitrary linear connection Γ on M_{n}.

Corollary 3.1. If the initial connection Γ is an almost r-paracontact connection on M_{n}, then the general family of almost r-paracontact connections \overline{Γ} on M_{n} is given by

\overline{∇}_{X} = ∇_{X} + HW_{X}  (3.6)

where W is an arbitrary tensor field.

Proposition 3.1. Let M_{n} be an almost r-paracontact Riemannian manifold with a structure Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)}. An almost r-paracontact connection \overline{Γ} given by (3.5) on M_{n} is metric if and only if there exists a tensor field W of type (1,2) with W (X, Y) = W_{X}(Y) satisfying

g(H_{X}W_{Z}, Y) + g(X, H_{Y} W_{Z}) = 0.  (3.7)

Theorem 3.2. The linear connection \overline{Γ} given by

\overline{∇}_{X} = ∇_{X} + \frac{1}{2}(ϕ_{1}∇_{X}ϕ_{1} + ϕ_{2}∇_{X}ϕ_{2} + ϕ_{3}∇_{X}ϕ_{3} + 2∇_{X}η^{α} ⊗ ξ_{α}) (3.8)

is a metric almost r-paracontact connection on an almost r-paracontact Riemannian manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)}, where ∇ is the Riemannian connection, i.e., \overline{∇}ϕ = 0, \overline{∇}η^{α} = 0, \overline{∇}ξ_{α} = 0, \overline{∇}g = 0.

Definition 3.3. Let D be a distribution on a manifold M_{n}. A linear connection Γ given by its covariant derivative ∇ on M_{n} is said to be a Dconnection, or D is said to be parallel with respect to Γ, if for any vector field Y and a vector field X from D the vector field ∇_{Y} X belongs to D.

Theorem 3.3. The distribution D_{i}, i = 1, 2, 3, given by (1.8)–(1.10) or (2.6)–(2.8), is parallel with respect to a linear connection Γ given by its covariant derivative ∇ on an almost r-paracontact manifold M_{n}, or Γ is a D_{i}-connection on M_{n}, if and only if

∇ϕ_{i} ∘ ϕ_{i} = 0,  i = 1, 2, 3.  (3.9)

If Γ is an almost r-paracontact connection, then from (2.1), (2.2), and (2.3) we get ∇ϕ_{i} = 0, and on account of Theorem 3.3 we obtain

Theorem 3.4. If a linear connection Γ on an almost r-paracontact manifold M_{n} is an almost r-paracontact connection, then the distributions D_{i}, i = 1, 2, 3 given by (1.8), (1.9) and (1.10) are parallel with respect to this connection.

Definition 3.4. A linear connection Γ on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} is said to be a D_{Σ}-connection if it is a D_{i}-connection, i = 1, 2, 3.

Theorem 3.5. Every almost r-paracontact connection Γ on an almost r-paracontact manifold M_{n} with a structure Σ is a D_{Σ}-connection.

Theorem 3.6. A linear connection Γ on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} is a D_{Σ}-connection if and only if

∇ϕ_{i} = 0,  i = i, 2, 3.  (3.10)

We also have

Theorem 3.7. The general family of the D_{Σ}-connections Γ̃ on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} is given by

˜∇_{X} = ∇_{X} + ϕ_{1}∇_{X}ϕ_{1} + ϕ_{2}∇_{X}ϕ_{2} + ϕ_{3}∇_{X}ϕ_{3} + QV_{X}  (3.11)

where Q is defined by (2.16), V is an arbitrary tensor field of type (1,2) with V_{X}(Y) = V (X, Y), and ∇ is the covariant derivative of an arbitrary linear connection Γ on M_{n}.

Corollary 3.2. If the initial connection Γ is a D_{Σ}-connection on M_{n}, then the general family of D_{Σ}-connections Γ̃ on M_{n} is given by

˜∇_{X} = ∇_{X} + QV_{X}  (3.12)

where V is an arbitrary tensor field.

Let Γ^{1} and Γ^{2} be two linear connections, given by their covariant derivatives

∇^{1} and ∇^{2}, on an almost r-paracontact manifold M_{n}.

Definition 3.5 [4]. The following derivatives

∇^{ij}_{Z}f = ∇^{ji}_{Z}f = Zf  (3.13)
∇^{ij}_{Z}Y = ∇^{i}_{Z}Y  (3.14)
∇^{ij}_{Z}ω = ∇^{i}_{Z}ω  (3.15)
(∇^{ij}_{Z}ψ)(A, B) = Zψ(A, B) − ψ(∇^{i}_{Z}A, B) − ψ(A, ∇^{j}_{Z}B)  (3.16)

are called mixed covariant derivatives for a function f, a vector field Y, a 1-form ω, and a tensor field ψ of type (1,1), (0,2), or (2,0), where i, j = 1, 2; i ≠ j.

Proposition 3.2. ∇^{m}_{Z} = \frac{1}{2}(∇^{1}_{Z} + ∇^{2}_{Z}) is a covariant differentiation op-

erator of a certain connection Γ^{m} on M_{n}.

Definition 3.6. The connection Γ^{m} on M_{n} is called a mean connection of

Γ^{1} and Γ^{2} if its mean covariant derivative is

∇^{m}_{Z} = \frac{1}{2}(∇^{1}_{Z} + ∇^{2}_{Z}).  (3.17)

Proposition 3.3. d_{Z} = ∇^{2}_{Z} − ∇^{1}_{Z} is a tensor field of type (1,1) on M_{n} for any vector field Z.

Definition 3.7. The tensor field d of type (1,2) defined by

d_{Z} = ∇^{2}_{Z} − ∇^{1}_{Z}  (3.18)

with d_{Z}(X) = d(Z, X) on M_{n} is called a deformation tensor field of connec-

tions Γ^{1} and Γ^{2}.

Definition 3.8. A pair of linear connections (Γ^{1}, Γ^{2}) on an almost rparacontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} is said to be compatible with Σ if

∇^{12}ϕ = 0,  ∇^{1}η^{α} = 0,  ∇^{2}ξ_{α} = 0,  α ∈ (r).  (3.19)

Theorem 3.8. A pair of linear connections (Γ^{1}, Γ^{2}) on an almost rparacontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} is compatible with Σ if and only if all structure tensor fields are parallel with respect to

both mixed covariant derivatives ∇^{12} and ∇^{21}.

Remark 3.1. From Theorem 3.11 we obtain the symmetry of compatibility,

i.e., a pair (Γ^{1}, Γ^{2}) is compatible with Σ if and only if (Γ^{2}, Γ^{1}) is compatible with Σ.

Theorem 3.9. A pair of linear connections (Γ^{1}, Γ^{2}) on an almost rparacontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} is compatible with Σ if and only if

(i) the mean connection Γ^{m} given by (3.17) is an almost r-paracontact connection on M_{n}

(ii) Bd_{Z} = 0, where B is given by (2.10).

Remark 3.2. The condition (ii) of Theorem 3.9 implies

Hd_{Z} = 0  (3.20)

where H is given by (2.14).

Hence we get

Theorem 3.10. If a pair of linear connections (Γ^{1}, Γ^{2}) on an almost rparacontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} is compatible with Σ then

∇^{1}_{Z} = ∇_{Z} + S_{Z} − \frac{1}{2}F W_{Z}  (3.21)
∇^{2}_{Z} = ∇_{Z} + S_{Z} + \frac{1}{2}F W_{Z}  (3.22)

where ∇ is an arbitrary linear connection on M_{n}, S_{Z} is given by (3.21), F is defined by (2.15), and W is an arbitrary tensor field of type (1,2) with W_{Z}(X) = W (Z, X).

4. Torsion tensors of almost r-paracontact connections and Dconnections. In this section torsion tensors of almost r-paracontact connections and D-connections are considered. Also, special pairs of connections compatible with a structure are discussed.

Let Γ, given by its covariant derivative ∇, be a linear connection with the torsion tensor T on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)}. Then, the tensor N from Theorem 1.1 can be expressed in the following form

\begin{matrix}N(X, Y) = ϕT (X, ϕY) + ϕT (ϕX, Y) − T (X, Y) − T (ϕX, ϕY) \\ +(∇_{X}ϕ)ϕY − (∇_{Y} ϕ)ϕX + (∇_{ϕX}ϕ)Y − (∇_{ϕY} ϕ)X \\ −η^{α}(X)∇_{Y} ξ_{α} + η^{α}(Y)∇_{X}ξ_{α} \end{matrix}  (4.1)

Now, suppose that Γ is an almost r-paracontact connection. Then, (4.1) becomes

N(X, Y) = ϕT (X, ϕY) + ϕT (ϕX, Y) − T (X, Y) − T (ϕX, ϕY)  (4.2)

Since X(η^{α}(Y)) = η^{α}(∇_{X}Y), we have: 2dη^{α}(X, Y) = η^{α}(∇_{X}Y − ∇_{Y} X − [X, Y]), or

2dη^{α} = η^{α} ∘ T,  α ∈ (r)  (4.3)

From (2.4) and (2.14) we have H = \frac{1}{2}(ϕ_{4} ⊗ ϕ_{4} + ϕ ⊗ ϕ), or

ϕ ⊗ ϕ = 2H − ϕ_{4} ⊗ ϕ_{4}  (4.4)

Now, we express the torsion tensor field T of the almost r-paracontact connection Γ in terms of the tensor field N and the operator H. For the tensor field T_{X} defined as T_{X}(Y) = T (X, Y) we have from (4.4) and Remark 2.1

ϕT_{Y} ϕ = 2HT_{Y} − ϕ_{4}T_{Y} ϕ_{4}  (4.5)

and for any vector field X we get

ϕT_{Y} ϕX = 2H_{X}T_{Y} − ϕ_{4}T_{Y} ϕ_{4}X  (4.6)

From (4.5) we have ϕT_{ϕY} ϕ^{2} = 2HT_{ϕY} ϕ − ϕ_{4}T_{ϕY} ϕ_{4}ϕ, and using (2.4) and (2.5) we obtain

ϕT_{ϕY} ϕ_{4} = 2HT_{ϕY} ϕ − ϕ_{4}T_{ϕY} ϕ  (4.7)

From (4.7) we get

H_{Y} T_{ϕX}ϕ − H_{X}T_{ϕY} ϕ − \frac{1}{2}ϕ_{4}T_{ϕX}ϕY + \frac{1}{2}ϕ_{4}T_{ϕY} ϕX = \frac{1}{2}ϕT_{ϕX}ϕ_{4}Y − \frac{1}{2}ϕT_{ϕY} ϕ_{4}X

Hence

ϕ_{4}T_{ϕX}ϕY = \frac{1}{2}ϕT_{ϕ_{4}Y} ϕX − \frac{1}{2}ϕT_{ϕ_{4}X}ϕY + H_{Y} T_{ϕX}ϕ − H_{X}T_{ϕY} ϕ  (4.8)

and using (4.6) we obtain

\begin{matrix}ϕ_{4}T_{ϕX}ϕY = \frac{1}{2}ϕ_{4}T_{ϕ_{4}X}ϕ_{4}Y − \frac{1}{2}ϕ_{4}T_{ϕ_{4}Y} ϕ_{4}X \\ −H_{Y} (T_{ϕ_{4}X} − T_{ϕX}ϕ) + H_{X}(T_{ϕ_{4}Y} − T_{ϕY} ϕ) \end{matrix}  (4.9)

Using Proposition 2.3 the tensor field N from (4.2) may be written in the form

N(X, Y) = −T_{X}Y − ϕ_{3}T_{ϕX}ϕY − ϕ_{4}T_{ϕX}ϕY + ϕT_{X}ϕY − ϕT_{Y} ϕX

Using (4.6) and (4.9) we obtain

N(X, Y) = −T_{X}Y − ϕ_{3}T_{ϕX}ϕY − ϕ_{4}T_{ϕ_{4}X}ϕ_{4}Y − ϕ_{4}T_{X}ϕ_{4}Y − ϕ_{4}T_{ϕ_{4}X}Y +H_{Y} (T_{2X+ϕ_{4}X} − T_{ϕX}ϕ) − H_{X}(T_{2Y +ϕ_{4}Y} − T_{ϕY} ϕ)

or

\begin{matrix}N(X, Y) = −ϕ_{3}T_{X}Y − ϕ_{3}T_{ϕX}ϕY − ϕ_{4}T_{X+ϕ_{4}X}(Y + ϕ_{4}Y) \\ +H_{Y} (T_{2X+ϕ_{4}X} − T_{ϕX}ϕ) − H_{X}(T_{2Y +ϕ_{4}Y} − T_{ϕY} ϕ) \end{matrix}  (4.10)

Using (2.5) and Proposition 2.3 we get

ϕ_{3}N(X, Y) = −ϕ_{3}T_{X}Y − ϕ_{3}T_{ϕX}ϕY  (4.11)

From (4.10) and (4.11) we obtain

ϕ_{4}T_{X+ϕ_{4}X}(Y + ϕ_{4}Y) = −ϕ_{4}N(X, Y)
+H_{Y} (T_{2X+ϕ_{4}X} − T_{ϕX}ϕ) − H_{X}(T_{2Y +ϕ_{4}Y} − T_{ϕY} ϕ)

or

\begin{matrix}T_{X+ϕ_{4}X}(Y + ϕ_{4}Y) = −ϕ_{4}N(X, Y) + ϕ_{3}T_{X+ϕ_{4}X}(Y + ϕ_{4}Y) \\ +H_{Y} (T_{2X+ϕ_{4}X} − T_{ϕX}ϕ) − H_{X}(T_{2Y +ϕ_{4}Y} − T_{ϕY} ϕ) \end{matrix}  (4.12)

From (2.3) and (4.3) we get

ϕ_{3} ∘ T = 2dη^{α} ⊗ ξ_{α}  (4.13)

From (4.6) we have

2H_{\frac{1}{2} X+ \frac{1}{2} ϕ_{3}X}T_{Y} = H_{X}T_{Y}  (4.14)

Substituting X by \frac{1}{2}X + \frac{1}{2}ϕ_{3}X and Y by \frac{1}{2}Y + \frac{1}{2}ϕ_{3}Y into (4.12) and making use of (4.13) and (4.14) we obtain

\begin{matrix}T (X, Y) = −\frac{1}{4}N(X + ϕ_{3}X, Y + ϕ_{3}Y) + 2dη^{α}(X, Y)ξ_{α}+ \\ +H_{Y} (T_{X− \frac{1}{4} ϕ_{4}X} − T_{\frac{1}{4} ϕX}ϕ) − H_{X}(T_{Y − \frac{1}{4} ϕ_{4}Y} − T_{\frac{1}{4} ϕY} ϕ) \end{matrix}  (4.15)

Now, consider the connection \overline{Γ} given by

\overline{∇}_{X} = ∇_{X} − H(T_{X− \frac{1}{4} ϕ_{4}X} − T_{\frac{1}{4} ϕX}ϕ)  (4.16)

In virtue of Corollary 3.1 this connection is also an almost r-paracontact connection and its torsion tensor field \overline{T} is

\overline{T} (X, Y) = 2dη^{α}(X, Y)ξ_{α} − \frac{1}{4}N(X + ϕ_{3}X, Y + ϕ_{3}Y)  (4.17)

Hence we obtain

Theorem 4.1. On an almost r-paracontact manifold M_{n} there exists an almost r-paracontact connection whose torsion tensor is given by (4.17).

We also have

Theorem 4.2. A tensor field T of type (1,2) with T (X, Y) = −T (Y, X) is the torsion tensor of an almost r-paracontact connection Γ on an almost r-paracontact manifold M_{n} if and only if it satisfies (4.2) and (4.3).

Proof. If T satisfies (4.2) and (4.3) it also satisfies (4.15) since this relation was obtained by using only (4.2) and (4.3). There exists an almost r-paracontact connection \overline{Γ} with its torsion tensor \overline{T} given by (4.17). So, if we consider the connection ∇_{X} = \overline{∇}_{X} + H(T_{X− \frac{1}{4} ϕ_{4}X} − T_{\frac{1}{4} ϕX}ϕ), which is an

almost r-paracontact connection, then its torsion tensor is exactly T. □

We also have

Theorem 4.3. On an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} there exists a symmetric almost r-paracontact connection Γ if and only if the following conditions are satisfied

(i) all 1-forms η^{α} are closed (ii) Σ is normal.

Proof. Suppose that there exists a symmetric almost r-paracontact connection on M_{n}. Then, on account of (4.2) and (4.3) the conditions (i) and (ii) are satisfied. Conversely, if (i) and (ii) are satisfied, then according to Theorem 4.1 there exists a symmetric almost r-paracontact connection on M_{n}. □

Using Theorem 3.5 we obtain from Theorem 4.3

Theorem 4.4. On an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} there exists a symmetric D_{Σ}-connection Γ if the following conditions are satisfied

(i) all 1-forms η^{α} are closed (ii) Σ is normal.

Now, let (Γ^{1}, Γ^{2}) be a pair of linear connections on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} such that

∇^{1}_{X} = ∇_{X},  and  ∇^{2}_{X} = ∇_{X} − T_{X}  (4.18)

where ∇ is a covariant derivative with respect to an arbitrary linear connection Γ and T is its torsion tensor field. Then, we have

Proposition 4.1. The mean connection Γ^{m} and the deformation tensor

field d of (Γ^{1}, Γ^{2}) defined by (4.18) are given by

∇^{m}_{X}Y = \frac{1}{2}(∇_{X}Y + ∇_{Y} X + [X, Y])  (4.19)
d_{X}(Y) = −T (X, Y).  (4.20)

Proposition 4.2. The mean connection Γ^{m} given by (4.19) is torsionless.

From Theorem 3.9 we obtain

Theorem 4.5. A pair of linear connections (Γ^{1}, Γ^{2}) defined by (4.18) on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} is compatible with Σ if and only if

(i) the mean connection Γ^{m} given by (4.19) is an almost r-paracontact connection on M_{n}

(ii) BT_{X} = 0, where B is given by (2.10).

Using Remark 3.2 and Proposition 2.1 we get

Corollary 4.1. If a pair of linear connections (Γ^{1}, Γ^{2}) defined by (4.18) on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} is compatible with Σ, then

T (X, Y) = F_{Y} W_{X}  (4.21)

where F is defined by (2.13), and W is an arbitrary tensor field of type (1,2) with W_{Z}(X) = W (Z, X).

Making use of Proposition 4.2 and Theorem 4.3 we get

Theorem 4.6. If on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} all 1-forms η^{α} are closed then Σ is normal if and

only if the pair of linear connections (Γ^{1}, Γ^{2}) defined by (4.18) is compatible with Σ.

5. Curvature tensors of almost r-paracontact connections and D-connections. In this section curvature tensors of almost r-paracontact connections and D-connections are considered. Also, mixed curvature and deformation operators for pairs of connections compatible with a structure are discussed.

Let Γ be an almost r-paracontact connection on an almost r-paracontact manifold M_{n}. Then we obtain

Theorem 5.1. The curvature tensor R_{XY} of an almost r-paracontact connection Γ on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} has the following properties

(i) R_{XY} ξ_{a} = 0, α ∈ (r)

(ii) η^{α} ∘ R_{XY} , α ∈ (r)

(iii) ϕ ∘ R_{XY} = R_{XY} ∘ ϕ

(iv) F R_{XY} = 0

(v) HR_{XY} = R_{XY} where F and H are given by (2.13) and (2.14).

Proof. The operator R_{XY} is given by

R_{XY} = ∇_{X}∇_{Y} − ∇_{Y} ∇_{X} − ∇_{[X,Y]}  (5.1)

The property (i) follows directly from (5.1). Since Γ is an almost r-paracon-tact connection we have η^{α}(∇_{X}Y) = Xη^{α}(Y) and ∇_{X}(ϕY) = ϕ∇_{X}Y. So, η^{α}R_{XY} Z = XY η^{α}(Z) − Y Xη^{α}(Z) − [X, Y]η^{α}(Z) = 0, and R_{XY} (ϕZ) = ϕR_{XY} Z. From (iii) and (ii) we get R_{XY} = ϕR_{XY} ϕ. From this, because F ϕ = 0 we obtain (iv) and (v). □

After lengthy calculations we also obtain

Theorem 5.2. For the metric almost r-paracontact connection \overline{Γ} given by (3.8) on an almost r-paracontact Riemannian manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)} its curvature tensor field \overline{R}_{XY} is of the form

\overline{R}_{XY} = HR_{XY} + S_{XY}  (5.2)

where

\begin{matrix}S_{XY} = \frac{1}{4}∇_{[X}ϕ∇_{Y]}ϕ + \frac{1}{2}∇_{[Y} η^{α} ⊗ ∇_{X]}ξ_{α} + \frac{1}{4}η^{α}∇_{[Y} ϕ ⊗ ∇_{X]}ϕξ_{α} \\ −\frac{1}{4}η^{α} ⊗ ∇_{[X}ϕ∇_{Y]}ϕξ_{α} − \frac{1}{4}η^{α}∇_{[X}ϕ∇_{Y]}ϕ ⊗ ξ_{α} \\ +\frac{1}{2}η^{α}(∇_{[Y} ξ_{β})∇_{X]}η^{β} ⊗ ξ_{α} + \frac{1}{2}η^{α}(∇_{[Y} ξ_{β})η^{β} ⊗ ∇_{X]}ξ_{α} \\ +\frac{1}{4}∇_{[X}η^{α}(∇_{Y]}ξ_{β})η^{β} ⊗ ξ_{α} − \frac{1}{4}η^{γ}(∇_{[Y} ξ_{β})η^{α}(∇_{X]}ξ_{γ})η^{β} ⊗ ξ_{α}, \end{matrix} (5.3)

H is given by (2.14), A_{[X}B_{Y]} denotes A_{X}B_{Y} − A_{Y} B_{X}, and ∇ is the Levi-Civita connection on M_{n}.

Making use of Proposition 2.1 we obtain from Theorem 5.2

Theorem 5.3. If the metric almost r-paracontact connection \overline{Γ} given by (3.8) on an almost r-paracontact Riemannian manifold M_{n} is flat, then the curvature tensor field R_{XY} of the Riemannian connection Γ generated by g is of the form

R_{XY} = −S_{XY} + F Z_{XY}  (5.4)

where S_{XY} is given by (5.3), F is given by (2.13), and Z is an arbitrary tensor field of type (1,3).

Now, let Γ be a D_{Σ}-connection on an almost r-paracontact manifold M_{n}. Then, we have

Theorem 5.4. The curvature tensor field R_{XY} of a D_{Σ}-connection Γ on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} has the following properties

(i) ϕ_{i} ∘ R_{XY} = R_{XY} ∘ ϕ_{i}, i = 1, 2, 3 (ii) P R_{XY} = 0 (iii) QR_{XY} = R_{XY}

where P and Q are given by (2.15) and (2.16) and ϕ_{i}’s are given by (2.1), (2.2), and (2.3).

Now, let (Γ^{1}, Γ^{2}) be a pair of linear connections on an almost r-paracontact manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)}.

Definition 5.1 [6]. The mixed curvature operator R^{m} for a pair of linear

connections (Γ^{1}, Γ^{2}) on M_{n} is defined as

R_{XY}^{m} = \frac{1}{2}([∇^{1}_{X}, ∇^{2}_{Y}] + [∇^{2}_{X}, ∇^{1}_{Y}] − ∇^{1}_{[X,Y]} − ∇^{2}_{[X,Y]}).  (5.5)

Definition 5.2 [6]. The deformation curvature operator R^{d} for a pair of

linear connections (Γ^{1}, Γ^{2}) on M_{n} is defined as

R_{XY}^{d} = \frac{1}{4}[d_{X}, d_{Y}].  (5.6)

We obtain

Theorem 5.5. For a pair of linear connections (Γ^{1}, Γ^{2}) compatible with an almost r-paracontact structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} on an almost r-paracon-tact manifold M_{n} the following identities take place

2R_{XY}^{m} + 4R_{XY}^{d} = R^{1}_{XY} + R^{2}_{XY}  (5.7)
R_{XY}^{m} + R_{XY}^{d} = R^{m}_{XY}  (5.8)

where R^{1}, R^{2}, and R^{m} are curvature tensor fields for Γ^{1}, Γ^{2}, and Γ^{m}. Also, we get Corollary 5.1. The tensor fields of mixed and deformation curvatures

R^{m} and R^{d} for a pair of linear connections (Γ^{1}, Γ^{2}) compatible with an almost r-paracontact structure Σ = (ϕ, ξ_{α}, η^{α})_{α∈(r)} on an almost r-paracontact manifold M_{n} have the following properties

R_{XY}^{m} ξ_{α} + R_{XY}^{d} ξ_{α} = o  (5.9)
η^{α}R_{XY}^{m} + η^{α}R_{XY}^{d} = 0  (5.10)
ϕR_{XY}^{m} − R_{XY}^{m} ϕ = −(ϕR_{XY}^{d} − R_{XY}^{d} ϕ)  (5.11)
ϕR_{XY}^{m} ϕ − R_{XY}^{m} = −(ϕR_{XY}^{d} ϕ − R_{XY}^{d})  (5.12)
F R_{XY}^{m} + F R_{XY}^{d} = 0  (5.13)
HR_{XY}^{m} − R_{XY}^{m} = HR_{XY}^{d} − R_{XY}^{d} .  (5.14)

REFERENCES

[1]. A. Bucki, A. Miernowski, Almost r-paracontact structures, Ann. Univ. Mariae Curie-

Skłodowska, Sect.A 39 (1985), 13–26. [2]. A. Bucki, Special almost r-paracontact connections, Acta Univ. Mathaei Belii Nat.

Sci. Ser. Math. 4 (1966), 3–15.

[3]. M. Obata, Hermitian manifolds with quaternion structures, Tohoku Math. J. 10

(1958), 11–18. [4]. A. P. Norden, Spaces with Affine Connections (in Russian), Nauka, Moscow, 1970. [5]. K. Yano, Differential Geometry on Complex and Almost Complex Space, Pergamon

Press, Oxford, 1965. [6]. I. Vaisman, Sur quelques formules du calcul de Ricci global, Comment. Math. Helv.

41 (1966/7), 74–87.

Department of Mathematics, Oklahoma School of Science and Mathematics, Oklahoma City, OK 73104, USA

E-mail address: abucki@ossm.edu

Received November, 27, 2000