DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 19–28

REPRESENTATION OF THE LIE GROUP

OF AUTOMORPHISMS OF AN ALMOST

r-PARACONTACT RIEMANNIAN MANIFOLD

OF P-SASAKIAN TYPE

A. BUCKI

Abstract. In this paper the examples of Lie groups and Lie algebras on an almost r-paracontact Riemannian manifold M_{n} of P-Sasakian type are presented and their maximum dimensions are determined. It is proven that the set of all automorphisms of M_{n} is a Lie group, and the example of an almost r-paracontact Riemannian manifold of P-Sasakian type with the group of automorphisms of the maximum dimension is constructed. Finally, some representation of the group of automorphisms of an almost r-paracontact Riemannian manifold of P-Sasakian type is presented.

1. In this section we recall the definitions of almost r-paracontact Riemannian manifolds [3] and some types of these manifolds [4] together with some of their properties.

Definition 1.1. Let M_{n} be an n-dimensional Riemannian manifold with a positive definite metric g. 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} (1.2) η^{α}(X) = g(X, ξ_{α}), α ∈ (r) (1.3)

g(ϕX, ϕY) = g(X, Y) − ∑_{α} η^{α}(X)η^{α}(Y) (1.4)

for vector fields X, Y on M_{n} with a^{α}b_{α} =^{def} ∑_{α} a^{α}b_{α}. Then Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)}

1991 Mathematics Subject Classification. 53C15, 57S20, 20G05. Key words and phrases. Almost r-paracontact structures of P-Sasakian type, group of auto-

morphisms, representation of Lie group.

This paper is in final form and no version of it will be submitted for publication elsewhere.

is said to be an almost r-paracontact Riemannian structure on M_{n}, and M_{n} is an almost r-paracontact Riemannian manifold.

Remark 1.1 . From (1.1) through (1.4) we also have

ϕ(ξ_{α}) = 0, α ∈ (r) (1.5) η^{α} ∘ ϕ = 0, α ∈ (r) (1.6)

Φ(X, Y) =^{def} g(ϕX, Y) = g(X, ϕY) (1.7)

From (1.7) we get

(1.8)  ∇_{Z}Φ(X, Y) = g((∇_{Z}ϕ)X, Y)

Definition 1.2. If M_{n} is an almost r-paracontact manifold with the structure Σ = (ϕ, ξ_{α}, η^{α}, g)_{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.

Definition 1.3. An almost r-paracontact Riemannian manifold M_{n} with a structure Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)} is said to be of P-Sasakian type if

Φ(X, Y) = (∇_{Y} η^{α})(X) for all α ∈ (r) (1.9)

\begin{aligned}(∇_{Z}Φ)(X, Y) = & − ∑_{α} η^{α}(X)g(ϕY, ϕZ)− \\ & − ∑_{α} η^{α}(Y)g(ϕX, ϕZ) \end{aligned} (1.10)

for all vector fields X, Y, Z on M_{n}.

Remark 1.2. The conditions (1.9) and (1.10) are equivalent respectively to:

ϕX = ∇_{X}ξ_{α} for all α ∈ (r) (1.11)

\begin{aligned}(∇_{Y} ϕ)(X) & = − ∑_{β} η^{β}(X)[Y − η^{α}(Y)ξ_{α}] \\ & − [g(X, Y) − ∑_{α} η^{α}(X)η^{α}(Y)] ∑_{β} ξ_{β} \end{aligned} (1.12)

Corollary 1.1. On a manifold M_{n} an almost r-paracontact Riemannian structure Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)} of P-Sasakian type is normal and all η^{α} are closed.

Remark 1.3 [5]. The Ricci tensor Ric satisfies

(1.13)  Ric(X, ξ_{α}) = −(n − r) ∑_{β} η^{β}(X), α ∈ (r)

Remark 1.4. On an almost r-paracontact Riemannian manifold M_{n} with the structure Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)} the tensor ϕ has constant eigenvalues 1, −1, and 0. Let p, q, and r be their multiplicities, respectively, with p+q+r = n = dim M_{n}. M_{n} is said to be of type (p, q). If T_{x}M_{n} is the tangent space to M_{n} at a point x ∈ M_{n}, then M_{n} has an orthonormal frame {e_{1}, …, e_{p}, e_{p+1}, …, e_{p+λ}, e_{p+q+1} = ξ_{1}, …, e_{n} = ξ_{r}} with ϕ(e_{a}) = e_{a}, ϕ(e_{p+λ}) = −e_{p+λ}, ϕ(e_{p+q+α}) = 0, a = 1, …, p, λ = 1, …, q, α = 1, …, r. This frame is called an adapted frame.

Remark 1.5. The indices run always through the following ranges A, B, C, ⋯ ∈ (n), a, b, c, ⋯ ∈ (p), λ, μ, ν, ⋯ ∈ (q), α, β, γ, ⋯ ∈ (r), i, j, k, ⋯ ∈ (p + q), with λ^{∗} = p + λ and α^{∗} = p + q + α.

2. In this section the Lie algebra of infinitesimal automorphisms of an almost rparacontact Riemannian manifold of P-Sasakian type is defined and its maximunm dimension is found.

Definition 2.1. A vector field X on (M_{n}, Σ) is an infinitesimal automorphism of M_{n} if the following conditions are satisfied

(2.1)  L_{X}ϕ = 0, L_{X}η_{α} = 0, L_{X}ξ_{α} = 0, L_{X}g = 0, α ∈ (r)

where L_{X} is the Lie derivative with respect to a vector field X.

Remark 2.1. An infinitesimal automorphism of M_{n} is an infinitesimal isometry. Proposition 2.1. The set L of all infinitesimal automorphisms of M_{n} is a Lie

algebra.

Now we shall study the maximum dimension of L. The equations (2.1), in local coordinates, may be written as follows

L_{X}ϕ_{B}^{A} = ∇_{C}ϕ_{B}^{A}X^{C} − ∇_{C}X^{A}ϕ_{B}^{C} + ∇_{B}X^{C}ϕ_{C}^{A} = 0 (2.2)

L_{X}ξ_{α}^{A} = ∇_{B}ξ_{α}^{A}X^{B} − ∇_{B}X^{A}ξ_{α}^{B} = 0 (2.3)

L_{X}η_{A}^{α} = ∇_{C}η_{B}^{α}X^{C} + ∇_{B}X^{C}η_{C}^{α} = 0 (2.4)

L_{X}g_{AB} = ∇_{A}X^{C}g_{CB} + ∇_{B}X^{C}g_{CA} = 0 (2.5)

The maximum dimension of L is equal to the number of independent algebraic solutions X^{A} and ∇_{B}X^{C} of (2.2) – (2.5) at a point P of M_{n}. Let {e_{a}, e_{λ^{∗}}, e_{α^{∗}}} be an adapted frame at a point P of M_{n}. Then the structure tensors of Σ are given by

(2.6)

g_{AB} = δ_{AB}

ϕ_{B}^{A} = \{ \begin{matrix}δ_{b}^{a} & if A = a, B = b \\ −δ_{μ}^{λ} & if A = λ^{∗}, B = μ^{∗} \\ 0 & otherwise \end{matrix}

ξ_{α}^{A} = δ_{α^{∗}}^{A}, η_{B}^{α} = δ_{B}^{α^{∗}}, α ∈ (r)

Consider the solutions of (2.2) – (2.5) at P such that X(P) = 0. Then the formulas (2.3) – (2.5) are reduced to

∇_{α^{∗}}X^{A}(P) = 0 (2.7)

∇_{B}X^{α^{∗}}(P) = 0 (2.8)

∇_{A}X^{B}(P) + ∇_{B}X^{A}(P) = 0 (2.9)

respectively. From (2.2), if A = a, B = λ^{∗}, and if A = μ^{∗}, B = b, we get

(2.10)  ∇_{λ^{∗}}X^{a}(P) = 0,  ∇_{b}X^{μ^{∗}}(P) = 0

Hence, we conclude that the matrix [∇_{A}X^{B}(P)] has the form

(2.11)  [ \begin{matrix}A_{p} & 0 & 0 \\ 0 & B_{q} & 0 \\ 0 & 0 & 0_{r} \end{matrix} ]

where A_{p}, B_{q} are skew symmetric matrices of orders p and q respectively, and 0_{r} is the zero matrix of order r. Therefore, the number of independent matrices of the form (2.11) at a point P is \frac{1}{2}p(p − 1) + \frac{1}{2}q(q − 1). On the other hand, the number of non-vanishing vector fields at P is at most n. Thus we have

Theorem 2.1. \frac{1}{2}p(p − 1) + \frac{1}{2}q(q − 1) + n, with p + q + r = n = dim M_{n} is the maximum dimension of the Lie algebra L of all infinitesimal automorphisms of an almost r-paracontact Riemannian manifold M_{n} of P-Sasakian type.

3. In this section the maximum dimension of the group of isometries of an almost r-paracontact Riemannian manifold of P-Sasakian type is determined.

Suppose that an almost r-paracontact Riemannian manifold M_{n} of P-Sasakian type with the structure Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)} is compact. From (1.11), for any

ξ_{α} we have div ξ_{α} = trace{X → ∇_{X}ξ_{α}} = ∑_{i=1}^{n} g(∇_{e_{i}}ξ_{α}, e_{i}) = ∑_{i=1}^{n} g(ϕe_{i}, e_{i}). Hence,

(3.1)  div ξ_{α} = p − q, α ∈ (r)

By Green’s theorem [10] we get 0 = ∫_{M_{n}} div ξ_{α} dM_{n} = p − q. We obtain

Proposition 3.1. If an almost r-paracontact Riemannian manifold M_{n} of P-Sasakian type is compact, then p = q = \frac{1}{2}(n − r) and n − r is even.

Now, from Theorem 2.1 we get

Theorem 3.1. \frac{1}{4}(n − r)(n − r − 2) + n is the maximum dimension of the Lie algebra L of all infinitesimal automorphisms of an almost r-paracontact, compact, Riemannian manifold M_{n} of P-Sasakian type.

Now, suppose that X is an infinitesimal isometry, i.e., X satisfies (2.5). Denote by

(3.2)  D^{α} = η_{A}^{α}X^{A}, α ∈ (r)

Making use of (1.7), (1.9), and (1.10) we get

(3.3)  \begin{aligned}g^{BA}∇_{B}∇_{A}D^{α} & = −(n − r) ∑_{β} D^{β} + 2ϕ_{A}^{B}∇_{B}X^{A} \\ & + g^{CB}η_{A}^{α}∇_{C}∇_{B}X^{A}, \end{aligned} α ∈ (r)

From (2.5) we get

(3.4)  ∇_{A}X^{A} = 0,  ξ_{α}^{A}∇_{A}X^{C}g_{CB} + ξ_{α}^{A}∇_{B}X^{C}g_{CA} = 0

Differentiating covariantly (3.4)_{2} and making use of (1.11) and the fact that Φ is symmetric we get

(3.5)  2ϕ_{B}^{A}∇_{A}X^{B} + ξ_{α}^{A}∇_{B}∇_{A}X^{B} + g^{BC}η_{A}^{α}∇_{B}∇_{C}X^{A} = 0, α ∈ (r)

and from the Ricci identity, (3.4)_{1} and (1.13) we get

(3.6)  ξ_{α}^{A}∇_{B}∇_{A}X^{B} = −(n − r) ∑_{β} D^{β}, α ∈ (r)

Hence, from (3.3) we have g^{BA}∇_{A}∇_{B}D^{α} = 0, α ∈ (r). Thus, by Green’s theorem [10] we have

(3.7)  D^{α} = η_{A}^{α}X^{A} = constant, α ∈ (r)

Differentiating covariantly (3.7) and making use of (1.9) we get

(3.8)  L_{X}η^{α} = 0, α ∈ (r)

From (3.8) and (2.5) we have

(3.9)  L_{X}ξ_{α} = 0, α ∈ (r)

For any vector field Y we have the identity [10]

(3.10)  L_{X}∇_{B}Y^{A} = ∇_{B}L_{X}Y^{A} + (L_{X}Γ_{BC}^{A})Y^{C}

Making use (1.11) we get for Y^{A} = ξ_{α}^{A}

(3.11)  L_{X}ϕ_{B}^{A} = ∇_{B}(L_{X}ξ_{α}^{A}) + (L_{X}Γ_{BC}^{A})ξ_{α}^{C}, α ∈ (r)

From (2.5) we have L_{X}Γ_{BC}^{A} = 0, so from (3.9) and (3.11) we get

(3.12)  L_{X}ϕ = 0

Thus we obtain

Theorem 3.2. On a compact almost r-paracontact Riemannian manifold of P-Sasakian type every infinitesimal isometry is an infintesimal automorphism.

We have

Proposition 3.2 [6]. Let M be a complete Riemannian manifold, then the Lie algebra I of all infinitesimal isometries of M is isomorphic to the Lie algebra i(M) of the group of isometries I(M) of M.

Combining Thoorems 3.1, 3.2 and Proposition 3.2 we obtain

Theorem 3.3. \frac{1}{4}(n − r)(n − r − 2) + n is the maximum dimension of the group of isometries of a complete, compact, almost r-paracontact Riemannian manifold M_{n} of P-Sasakian type.

Corollary 3.1. The maximum dimension of the group of isometries of a complete, compact, P-Sasakian manifold M_{n} is \frac{1}{4}(n^{2} + 3).

4. In this section it is shown that the automorphism group of an almost rparacontact Riemannian manifold of P-Sasakian type is a Lie group and the example of the manifold with the maximum dimensional automorphism group is presented.

Definition 4.1. Let M_{n} with Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)} be an almost r-paracontact Riemannian manifold of P-Sasakian type. A diffeomorphism f of M_{n} onto itself is an automorphism of M_{n} if it preserves all structure tensors, i.e.,

(4.1)  ϕ ∘ f_{∗} = f_{∗} ∘ ϕ, f_{∗}(ξ_{α}) = ξ_{α}, f^{∗}η^{α} = η^{α}, f^{∗}g = g, α ∈ (r)

where f_{∗} is the differential of f and f^{∗} is the dual map of f_{∗}.

Proposition 4.1. The set of all automorphisms of an almost r-paracontact Riemannian manifold M_{n} of P-Sasakian type constitutes a group which is called the automorphism group of M_{n} and is denoted by A(M_{n}).

Theorem 4.1. The automorphism group A(M_{n}) of an almost r-paracontact Riemannian manifold M_{n} of P-Sasakian type is a Lie group.

Proof. According to (4.1) the group A(M_{n}) is a subgroup of the isometry group I(M_{n}) of the manifold M_{n} with the same metric tensor g. The group A(M_{n}) is closed in I(M_{n}), so it is a Lie group.

Remark 4.1. For any given infinitesimal automorphism X of an almost rparacontact Riemannian manifold M_{n} of P-Sasakian type exp(tX) (−∞ < t < ∞) is an automorphism of M_{n}. Also, any automorphism f of M_{n} can be generated by an infinitesimal automorphism X.

Taking this into consideration we get

Proposition 4.2. The Lie algebra of all infinitesiaml automorphisms of an almost r-paracontact Riemannian manifold M_{n} of P-Sasakian type is isomorphic to the Lie algebra of the group A(M_{n}).

Combining Proposition 4.2 and Theorem 2.1 we obtain

Theorem 4.2. \frac{1}{2}p(p + 1) + \frac{1}{2}q(q + 1) + r, with p + q + r = n, is the maximum dimension of the automorphism group A(M_{n}) of an almost r-paracontact Riemannian manifold M_{n} of P-Sasakian type.

Now we have

Theorem 4.3. R^{n} with the structure Σ = (ϕ, ξ_{α}, η^{α}, g)_{α∈(r)} given, in local coordinates (x^{1}, …, x^{n}), by

(4.2 )  ϕ_{B}^{A} = \{ \begin{matrix}δ_{b}^{a} & if A = a, B = b \\ −δ_{μ}^{λ} & if A = λ^{∗}, B = μ^{∗} \\ 0 & otherwise \end{matrix}

ξ_{α}^{A} = δ_{α^{∗}}^{A}, α ∈ (r) (4.3)

η_{B}^{α} = δ_{B}^{α^{∗}} α ∈ (r) (4.4)

(4.5) g_{AB} = \{ \begin{matrix}g_{ab} = e^{2(x^{p+q+1}+,...,+x^{n})}δ_{ab} \\ g_{λ^{∗}μ^{∗}} = e^{−2(x^{p+q+1}+,...,+x^{n})}δ_{λμ} \\ g_{α^{∗}β^{∗}} = δ_{αβ} \\ 0 all others \end{matrix}

is the almost r-paracontact Riemannian manifold of P-Sasakian type whose the automorphism group A(R^{n}) has the maximum dimension \frac{1}{2}p(p+1)+ \frac{1}{2}q(q +1)+r.

Proof. In [4] it is shown that (R^{n}, Σ) is an almost r-paracontact Riemannian manifold of P-Sasakian type. Now, to find the dimension of A(R^{n}) it suffices to determine the dimension of the Lie algebra L of all infinitesimal automorphisms of R^{n}. To this end, we have to solve the system (2.2) through (2.5) for a vector field V. Making use of (4.5) we calculate

(4.6)  Γ_{BC}^{A} = \{ \begin{matrix}−e^{2(x^{p+q+1}+,...,+x^{n})} & if B = C = α^{∗}, A = β^{∗} \\ e^{−2(x^{p+q+1}+,...,+x^{n})} & if B = C = λ^{∗}, A = β^{∗} \\ δ_{αβ} & if A = a, B = b, C = β^{∗} \\ −δ_{λμ} & if A = λ^{∗}, B = μ^{∗}, C = β^{∗} \\ 0 & otherwise \end{matrix}

From (2.3), (2.4), and (2.2) we get ∂_{B}ξ_{β}^{A}V^{B}−∂_{B}V^{A}ξ_{β}^{B} = 0, ∂_{C}η_{B}^{β}V^{C}+∂_{B}V^{C}η_{C}^{β} = 0, and ∂_{C}ϕ_{B}^{A}V^{C} − ∂_{C}V^{A}ϕ_{B}^{C} + ∂_{B}V^{C}ϕ_{C}^{A} = 0, respectively, and after having used (4.3), (4.4), and (4.2) we obtain

(4.7)  ∂_{β^{∗}}V^{A} = 0, ∂_{B}V^{β^{∗}} = 0, ∂_{λ^{∗}}V^{a} = 0, ∂_{b}V^{μ^{∗}} = 0

Putting A = a, B = b in (2.5) and also putting A = λ^{∗}, B = μ^{∗} in (2.5), and making use of (4.5) and (4.6) give us

2δ_{ab} ∑_{β^{∗}} V^{β^{∗}} + ∂_{a}V^{b} + ∂_{b}V^{a} = 0 (4.8)

−2δ_{λμ} ∑_{β^{∗}} V^{β^{∗}} + ∂_{λ^{∗}}V^{μ^{∗}} + ∂_{μ^{∗}}V^{λ^{∗}} = 0 (4.9)

If we put any other set of values for A and B in (2.5) we get no new relation. Putting a = b and a ≠ b in (4.8), and also λ = μ and λ ≠ μ in (4.9) yield

∂_{a}V^{a} + ∑_{β^{∗}} V^{β^{∗}} = 0, ∂_{a}V^{b} + ∂_{b}V^{a} = 0 (4.10)

∂_{λ^{∗}}V^{λ^{∗}} − ∑_{β^{∗}} V^{β^{∗}} = 0, ∂_{λ^{∗}}V^{μ^{∗}} + ∂_{μ^{∗}}V^{λ^{∗}} = 0 (4.11)

From (4.7)_{1} we obtain V^{β^{∗}} = c^{β^{∗}}, where c^{β^{∗}} is a constant for any β. Let

(4.12)  ∑_{β^{∗}} V^{β^{∗}} = ∑_{β^{∗}} c^{β^{∗}} = c

Because of (4.7)_{1} and (4.7)_{3}, V^{a} depends only on x^{1}, …, x^{p}, so from (4.10)_{1} we get

(4.13)  V^{a} = −cx^{a} + h^{a}(x^{1}, …, x̄^{a}, …, x^{p})

where h^{a} are differentiable functions and x̄^{a} means that a-th variable is omitted. Substituting (4.13) into (4.10)_{2} gives

(4.14)  ∂_{a}h^{b}(x^{1}, …, x̄^{b}, …, x^{p}) + ∂_{b}h^{a}(x^{1}, …, x̄^{a}, …, x^{p}) = 0

From this we get ∂_{b}(∂_{b}h^{a}) = 0, hence ∂_{b}h^{a} = c_{b}^{a} and

(4.15)  h^{a}(x^{1}, …, x^{p}) = c_{b}^{a}x^{b} + c^{a}

where c_{b}^{a} and c^{a} are constants satisfying (because of (4.14))

(4.16)  c_{b}^{a} + c_{a}^{b} = 0

Thus, from (4.13) we have

(4.17)  V^{a} = −cx^{a} + c_{b}^{a}x^{b} + c^{a}

Because of (4.7)_{1} and (4.7)_{4} V^{λ^{∗}} depends only on x^{p+1}, …, x^{p+q}, so from (4.11)_{1} we get

(4.18)  V^{λ^{∗}} = cx^{λ^{∗}} + h^{λ^{∗}}(x^{p+1}, …, x̄^{p+λ}, …, x^{p+q})

Substituting (4.18) into (4.11)_{2} we obtain

(4.19)  ∂_{λ^{∗}}h^{μ^{∗}} + ∂_{μ^{∗}}h^{λ^{∗}} = 0

Hence ∂_{λ^{∗}}(∂_{λ^{∗}}h^{μ^{∗}}) = 0, and

(4.20)  h^{λ^{∗}}(x^{p+1}, …, x^{p+q}) = c_{μ^{∗}}^{λ^{∗}}x^{μ^{∗}} + c^{λ^{∗}}

where c_{μ^{∗}}^{λ^{∗}} and c^{λ^{∗}} are constants satisfying (because of (4.19))

(4.21)  c_{μ^{∗}}^{λ^{∗}} + c_{λ^{∗}}^{μ^{∗}} = 0

Thus, from (4.18) we have

(4.22)  V^{λ^{∗}} = cx^{λ^{∗}} + c_{μ^{∗}}^{λ^{∗}}x^{μ^{∗}} + c^{λ^{∗}}

Now, making use of (4.17), (4.22), (4.12), (4.16), and (4.21), any infinitesimal automorphism V on our manifold is of the form

V = (−cx^{a} + c_{b}^{a}x^{b} + c^{a})\frac{∂}{∂x^{a}} + (cx^{λ^{∗}} + c_{μ^{∗}}^{λ^{∗}}x^{μ^{∗}} + c^{λ^{∗}})\frac{∂}{∂x^{λ^{∗}}}

+c^{α^{∗}} \frac{∂}{∂x^{α^{∗}}} = c^{a} \frac{∂}{∂x^{a}} + c^{λ^{∗}} \frac{∂}{∂x^{λ^{∗}}} + ∑_{λ<μ} c_{λ^{∗}}^{μ^{∗}}(x^{λ^{∗}} \frac{∂}{∂x^{μ^{∗}}} − x^{μ^{∗}} \frac{∂}{∂x^{λ^{∗}}})

+ ∑_{a<b} c_{a}^{b}(x^{a} \frac{∂}{∂x^{b}} − x^{b} \frac{∂}{∂x^{a}}) + c^{α^{∗}}(\frac{∂}{∂x^{α^{∗}}} − x^{a} \frac{∂}{∂x^{a}} + x^{λ^{∗}} \frac{∂}{∂x^{λ^{∗}}})

Hence

(4.23)  V = c^{a}E_{a} + c^{λ^{∗}}E_{λ^{∗}} + c^{α^{∗}}E_{α^{∗}} + ∑_{a<b} c_{a}^{b}E_{b}^{a} + ∑_{λ^{∗}<μ^{∗}} c_{λ^{∗}}^{μ^{∗}}E_{μ^{∗}}^{λ^{∗}}

where

E_{a} = \frac{∂}{∂x^{a}} , E_{λ^{∗}} = \frac{∂}{∂x^{λ^{∗}}} , E_{α^{∗}} = \frac{∂}{∂x^{α^{∗}}} − x^{a} \frac{∂}{∂x^{a}} + x^{λ^{∗}} \frac{∂}{∂x^{λ^{∗}}} ,

E_{b}^{a} = x^{a} \frac{∂}{∂x^{b}} − x^{b} \frac{∂}{∂x^{a}} (a < b), E_{μ^{∗}}^{λ^{∗}} = x^{λ^{∗}} \frac{∂}{∂x^{μ^{∗}}} − x^{μ^{∗}} \frac{∂}{∂x^{λ^{∗}}} (λ < μ)

The set of vector fields {E_{a}, E_{λ^{∗}}, E_{α^{∗}}, E_{b}^{a}, E_{μ^{∗}}^{λ^{∗}}, a < b, λ^{∗} < μ^{∗}} is a basis for the Lie algebra L of all infinitesimal automorphisms of the almost r-paracontact Riemannian manifold of P-Sasakian type R^{n} with the structure Σ given by (4.2) – (4.5). This set consists of p+q+r+ \frac{1}{2}(p^{2} −p)+ \frac{1}{2}(q^{2} −q) = \frac{1}{2}p(p+1)+ \frac{1}{2}q(q+1)+r vectors and this completes the proof.

5. In this section some representation of the Lie group of automorphisms of an almost r-paracontact Riemannian manifold of P-Sasakian type is presented.

Let Σ = (ϕ, ξ_{α}, η^{α}) be an almost r-paracontact structure of P-Sasakian type on a Riemannian manifold M_{n} and let A(M_{n}) be the group of automorphisms of M_{n}. Let V = R^{n} ⊕ R_{q}^{p} be a vector space, where R_{q}^{p} is a vector space generated by the set of n × n-matrices {e_{1}^{2}, e_{1}^{3}, e_{2}^{3}, e_{1}^{4}, …, e_{1}^{p}, …, e_{p−1}^{p}, e_{p}^{p+1}, e_{p}^{p+2}, e_{p+1}^{p+2}, …, e_{n−r−1}^{n−r}} with e_{l}^{k} = [a_{j}^{i}], where a_{j}^{i} = 1 if i = k, j = l and a_{j}^{i} = 0 otherwise. Let AutV be the group of automorphisms of V. Then δ : A(M_{n}) → AutV, f ↦ δ(f) = id_{V} is a representation of the Lie group A(M_{n}) of authomorphisms of M_{n}.

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Department of Mathematics Oklahoma School of Science and Mathematics Oklahoma City, OK 73104 USA E-mail: abucki@ossm.edu