62
S. Goto and H. Hino
λ
∧ λ
= λ
⊗ λ
− λ
⊗ λ
,
for all one-forms λ
and λ
. When the other convention of the numerical factor 2 dλ(X, Y ) = X λ(Y ) − Y λ(X ) − λ([X, Y ]) is adapted, the condition (4.3) is
replaced with
g(X, φY ) = dλ(X, Y ),
∀X, Y ∈ S T M.
In this paper (4.3) is used. Para-Sasakian manifolds are para-contact metric manifolds
satisfying the so-called normality condition.
Coordinate expressions were given for a para-contact metric manifold ( and a
para-Sasakian manifold ) (M, φ, ξ, λ, g) in [20]. They are summarized here. Let
(x, y, z) be coordinates on M with x = {x
1
, . . . , x
m
} and y = {y 1 , . . . , y m } such
that λ = dz −
m
a=1 y a dx
a . Since various dimensional manifolds will be discussed
in this paper the Einstein convention is not used throughout. Introduce the pseudoRiemannian metric tensor field, referred to as the Mrugala metric tensor field [30],
G =
1
2
m
a=1
dx
a
⊗ dy a + dy a ⊗ dx
a
+ λ ⊗ λ,
(4.4)
which is shown to induce a para-contact metric manifold (M, φ, ξ, λ, G). In what
follows we consider the case where y a > 0 for all a ∈ {1, . . . , m}. Introduce the
co-frame { θ
0
, θ
1
− , θ
1
+ , . . . , θ
m
− , θ
m
+ } and frame {e 0 , e
−
1 , e
+
1 , . . . , e
−
m , e
+
m } with
θ
0
:= λ,
θ
a
± :=
1
2
√ y a
y a dx
a
± dy a
,
e 0 := ξ,
e
±
a :=
√ y a
1
y a
∂
∂ x a + y a
∂
∂z
±
∂
∂ y a
,
so that
θ
0
(e 0 ) = 1,
θ
a
+ (e
+
b ) = θ
a
− (e
−
b ) = δ
a
b ,
others vanish,
where δ
a
b is the Kronecker delta, giving unity for a = b and zero otherwise. One can
then show that
G = θ
0
⊗ θ
0
+
m
a=1
θ
a
+ ⊗ θ
a
+ − θ
a
− ⊗ θ
a
−
, ξ =
∂
∂z
,
φ = −
m
a=1
θ
a
− ⊗ e
+
a + θ
a
+ ⊗ e
−
a
.
Some of relations are explicitly verified below. First, one has
φ( e
+
a ) = − e
−
a , φ(e
−
a ) = − e
+
a , and φ( e 0 ) = 0.