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Lobachevskii Journal of Mathematics Vol.8, 2001, 19^32
©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 E, almost r-paracontact connections, D-connections, and pairs of connections compatible with E 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 Mn be an n-dimensional differentiable manifold. If on Mn there exist: a tensor field (p of type (1,1), r vector fields £i, £2>•••>&• (r < n), r 1-forms rj1, rj2,..., rjr such that
Tja(€<3)=S$, a, (3 E (r) = 1,2,..., r (1.1)
(j)2 = Id - rf* ® Za, where aaba d= J2 aaba (1.2)
a
rf o = 0, a E (r) (1.3)
then E = (<j>,Ca^Va)a£(r) is said to be an almost r-paracontact structure on Mn, and Mn is an almost r-paracontact manifold.
Prom (1.1), (1.2), and (1.3) we also have
0(£a) =0, aE (r) (1.4)
There exists a positive Riemannian metric g on Mn such that
Va(X) = g(X,£a), a E (r) (1.5)
g(<f>X,<f>Y) = g(X,Y) - Y,Va(x)va(Y) (1.6)
Then, E = (qi, £a, ?7a, g)a£(r) is called a metric almost r-paracontact structure
on A/,,., and g is said to be compatible Riemannian metric.
Prom (1.1) through (1.6) we get
g(4>X,Y)=g(X,4>Y) (1.7)
Proposition 1.1. An almost r-paracontact manifold Mn admits the following complementary distributions
Dx = {X; 0X = X} (1.8)
D2 = {X; 0X = (1.9)
D3 = {X; 0X = 0} (1.8)
with dim 1)\ = p, dimD2 = q, dim O.i = r, and p + q + r = n.
Definition 1.2. If Mn is an almost r-paracontact manifold with the structure E = (0, Va)ae(r)i then E is said to be normal if an almost product
structure F defined on Mn x Rr by F(X, fa^) = (0X + /a£a, r]a{X)^)
is integrable, i.e., its Nijenhuis tensor field Np vanishes.
Theorem 1.1. An almost r-paracontact structure E on Mn is normal if and only if N(X,Y) = N^,(X,Y) — 2drja(X,Y)£a = 0, where N<p is the Nijenhuis tensor field for 0.
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 <E> on an appropriate space is said to be a projection operator if <E>2 = <&.
Definition 2.2. A set of operators {<!>*} is said to be the set of complementary projection operators if V\ <1>; = Id. <j>? = <!>,. <!>,<!>; =0, i ^ j.
Proposition 2.1 [3]. If <E> is a projection operator and ^ = Id — then $ and are complementary projection operators and all solutions of the equation <frx = y are of the form x = y + ^w, where w is an arbitrary element.
Remark 2.1. If operators <E> and ^ are tensor fields of type (2,2) and S, 0, X are tensor fields of type (1,2), (1,1), and (0,1) respectively, then the operations $^1/, $5, <E>0, <E>X are expressed locally as follows: 1/^,
Let E = (0,£a, Va)ae(r) be an almost r-paracontact structure on Mn. We
define the following operators on Mn.
<f>i = \{Id+<t>®£a) = |(02 + 0) (2.1)
02 = \{Id- 0 ~ Va ® 6*) = |(02 - 0) (2.2)
03 = ® 6» = Id — 02 (2.3)
04 = 02 = Id 03 (2.4)
with the properties
0 = 01^02, 001=010 = 01, 002 = 020 = 02,
= 030 = 0, 004 = ^^ (2'5)
Proposition 2.2. The operators 0i, 02, 03 are complementary projection operators on an almost r-paracontact manifold Mn.
Proposition 2.3. JTie operators 03 and 04 are complementary projection operators on an almost r-paracontact manifold Mn.
The distributions (1.8), (1.9), and (1.10) can be expressed as follows
D1 = {X
D2 = {X D3 = {X
0iX = X} (2.6)
02X = X} (2.7)
03X = X} (2.8)
Proposition 2.4. 77&e distributions D,, i = 1,2,3, are generated by the projection operators 0*, respectively.
Let A, S, and (7 be tensor fields of type (2,2) defined as
A = ^(Id® Id — 0 ® 0) (2.9)
B = \{Id ® Id + 0 ® 0) (2.10)
c* = 1(03 ®/d +Id® 03 ^ 03 ® 03) (2-11)
The operators A, S, and (7 possess the following properties
(2.12)
A + B = Id®Id, AA = A^-C, BB = B^-C, AB = BA = AC = CA = BC = CB = CC = -C
2
Proposition 2.5. The operators
F = A + C = Id ® Id - 0i ® 0i - 02 ® 02 (2.13)
H = B — C = 01 ® 01 + 02 ® 02 (2-14)
are complementary projection operators on an almost r-paracontact manifold Mn.
Proposition 2.6. The operators
P = F - 03 ® 03 = Id ® Id - 01 ® 01 - 02 ® 02 - 03 ® 03 (2.15)
Q = i? + 03 ® 03 = 01 ® 01 + 02 ® 02 + 03 ® 03 (2.16)
are complementary projection operators on an almost r-paracontact manifold Mn.
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 Mn with a structure E = (0, £a, Va)ae(r) a linear connection T, given by its covariant derivative V, is said to be an almost r-paracontact if
Vx0 = 0 (3.1)
VxVa = 0, a G (r) (3.2)
for any vector field X.
Prom (3.1) and (3.2) it follows
=0, a e (r) (3.3)
Definition 3.2. For an almost r-paracontact Riemannian manifold Mn with a structure E = (0, £a, r]a,g)ae^ an almost r-paracontact connection T, given by its covariant derivative V, is said to be a metric almost r-paracontact if
Vxg = o. (3.4)
We have
Theorem 3.1. The general family of the almost r-paracontact connections r on an almost r-paracontact manifold Mn with a structure E = (<f),£a,Va)ae(r) is given by
VX = Vx + ^(4>lVx4>l + 02Vx02 + 03Vx03 + 2VxVa ®£a) + HWx (3.5)
where H is defined by (2.14), W is an arbitrary tensor field of type (1,2)
with Wx(Y) = W{X,Y), and V is the covariant derivative of an arbitrary
linear connection T on Mn.
Corollary 3.1. If the initial connectionT is an almost r-paracontact connection on Mn, then the general family of almost r-paracontact connections r on Mn is given by
Vx = V.\ + HWx (3-6)
where W is an arbitrary tensor field.
Proposition 3.1. Let Mn be an almost r-paracontact Riemannian manifold with a structure E = (4>,£a,Vai §)a£(r)- An almost r-paracontact connection r given by (3.5) on Mn is metric if and only if there exists a tensor
field W of type (1,2) with W(X, Y) = Wx(Y) satisfying
g{HxWz, Y) + g(X, HYWZ) = 0. (3.7)
Theorem 3.2. The linear connection Г given by Vx = Vx + |(0iVx0i + 02 Vx4>2 + Фз^хФз + 2Vx??a ® £«) (3.8)
is a metric almost r-paracontact connection on an almost r-paracontact Rie-mannian manifold Mn with a structure E = (0,£а>?7а>g)a£(r)> where V is the Riemannian connection, i.e., V0 = 0, Vrja = 0, V£a = 0, Vg = 0.
Definition 3.3. Let D be a distribution on a manifold Mn. A linear connection Г given by its covariant derivative V on Mn is said to be a D-connection, or D is said to be parallel with respect to Г, if for any vector field
Y and a vector field X from D the vector field Vyl belongs to D.
Theorem 3.3. The distribution Di, i = 1,2,3, given by (1.8)-(1.10) or (2.6)-(2.8), is parallel with respect to a linear connection Г given by its covariant derivative V on an almost r-paracontact manifold Mn, or Г is a Di-connection on Mn, if and only if
V0i О<^ = 0, г = 1,2,3. (3.9)
If Г is an almost r-paracontact connection, then from (2.1), (2.2), and (2.3) we get Vфi = 0, and on account of Theorem 3.3 we obtain
Theorem 3.4. If a linear connection Г on an almost r-paracontact manifold Mn is an almost r-paracontact connection, then the distributions Di, % = 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 Г on an almost r-paracontact manifold Mn with a structure E = (0, £a, r]a)ae(r) is said to be a Dy,-connection if it is a Dj-connection, г = 1, 2, 3.
Theorem 3.5. Every almost r-paracontact connection Г on an almost r-paracontact manifold Mn with a structure E is a D^-connection.
Theorem 3.6. A linear connection Г on an almost r-paracontact manifold Mn with a structure E = (ф,^а,г1а)а£^ is a D^,-connection if and only if
V0i = 0, г = г, 2,3. (3.10)
We also have
Theorem 3.7. The general family of the D-^-connections Г on an almost r-paracontact manifold Mn with a structure E = (0, £a, Va)a£(r) given by
Vx = Vx + 01V X 01 + 02 V X 02 + 03 V X 03 + QVx (3-H)
where Q is defined by (2.16), V is an arbitrary tensor field of type (1,2) with
Vx(Y) = V(X,Y), and V is the covariant derivative of an arbitrary linear
connection Г on Mn.
Corollary 3.2. If the initial connection Г is a D-^-connection on Mn, then the general family of D-^-connections Г on Mn is given by
Vx=Vx + QVx (3.12)
where V is an arbitrary tensor field.
1 2
Let T and r be two linear connections, given by their covariant derivatives 1 2
V and V, on an almost r-paracontact manifold Mn.
Definition 3.5 [4]. The following derivatives
ij ji
Vzf = Vzf = Zf (3.13)
ij i
VZY = VZY (3.14)
ij i
= Vz^' (3.15)
ij i j (VZ^)(A,B) = Z%l>(A, B) - 'ipiyZA, B) - ip(A,VZB) (3.16)
are called mixed covariant derivatives for a function /, a vector field Y, a
1-form w, and a tensor field ij) of type (1,1), (0,2), or (2,0), where i,j =
1,2;
rn 1 2
Proposition 3.2. Vz = z + Vz) is a covariant differentiation op-
m
erator of a certain connection T on Mn.
rn
Definition 3.6. The connection T on Mn is called a mean connection of 1 2
r and r if its mean covariant derivative is
rn 1 2
Vz = 2^ +Vz). (3-17)
2 1
Proposition 3.3. dz = Vz — Vz is a tensor field of type (1,1) on Mn for any vector field Z.
Definition 3.7. The tensor field d of type (1,2) defined by
dz = Vz — Vz (3.18)
with dz(X) = d(Z, X) on Mn is called a deformation tensor field of connec-1 2 tions r and T.
1 2
Definition 3.8. A pair of linear connections (r,T) on an almost r-paracontact manifold Mn with a structure E = (0, £a, r]a)ae(r) is said to be compatible with E if
12 1 2
V0 = 0, V?f = 0, V£a=0, ae(r). (3.19)
1 2
Theorem 3.8. A pair of linear connections (r,T) on an almost r-
paracontact manifold Mn with a structure E = (<j),^a,ria)a£^ is compatible
with E if and only if all structure tensor fields are parallel with respect to
12 21
both mixed covariant derivatives V and V.
Remark 3.1. From Theorem 3.11 we obtain the symmetry of compatibility, 12 2 1 i.e., a pair (Г,Г) is compatible with E if and only if (Г, Г) is compatible
with E.
1 2
Theorem 3.9. A pair of linear connections (Г, Г) on an almost r-paracontact manifold Mn with a structure E = (ф,^а,г1а)а£^ is compatible with E if and only if
m
(i) the mean connection Г given by (3.17) is an almost r-paracontact connection on Mn
(ii) Bdz = 0, where В is given by (2.10).
Remark 3.2. The condition (ii) of Theorem 3.9 implies
Hdz = 0 (3.20)
where H is given by (2.14).
Hence we get
1 2
Theorem 3.10. If a pair of linear connections (Г, Г) on an almost r-
paracontact manifold Mn with a structure E = (ф,Са^'Па)а£(г) compatible
with E then
Vz = Vz + Sz^ \FWZ (3.21)
2
V z = Vz + Sz + .j Л1/ (3.22)
where V is an arbitrary linear connection on Mn, Sz is given by (3.21), F is defined by (2.15), and W is an arbitrary tensor field of type (1,2) with WZ(X) = W(Z,X).
4. Torsion tensors of almost r-paracontact connections and D-connections. 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 Г, given by its covariant derivative V, be a linear connection with the torsion tensor T on an almost r-paracontact manifold Mn with a structure ^ = {Фч Cai Va)ae(r)- Then, the tensor N from Theorem 1.1 can be expressed in the following form
N(X, Y) = фТ(Х, фУ) + фТ(фХ, Y) - T(X, Y) - Т(фХ, фУ)
+(Чхф)ф¥ - (ЧГФ)ФХ + {ЧфХф)¥ - (VфУф)Х (4.1)
^Va(X)VYta + Va(Y)Vxta
Now, suppose that Г is an almost r-paracontact connection. Then, (4.1) becomes
N(X, Y) = фТ(Х, фУ) + фТ(фХ, Y) - T(X, Y) - Т(фХ, фУ) (4.2)
Since X(tia(Y)) = 77a(VxY), we have: 2dr1a{X,Y) = 77a(VxY - VYX -[X,Y}),ot
2drfx = rf oT, a e (r) (4.3)
From (2.4) and (2.14) we have H = |(04 ® 04 + 0 ® 0), or
0 ® 0 = 2i? - 04 ® 04 (4.4)
Now, we express the torsion tensor field T of the almost r-paracontact connection T in terms of the tensor field N and the operator H. For the tensor field Tx defined as Tx(Y) = T(X, Y) we have from (4.4) and Remark 2.1
(:)Ty(;) = 2HTy — <:)\Ty<:)\ (4-5)
and for any vector field X we get
01V0X = 2H \Ty - 04Ty04X (4.6)
From (4.5) we have 0T^y02 = 2HTtpY4> — 04^y040, and using (2.4) and (2.5) we obtain
$T(pY 04 = 2HT(f>Y<t> — 04 T<pY<l> (4.7)
From (4.7) we get
HyT^-HxT^- 104T^x^Y + 104T(f,Y<j>X = \<t>T<j)x<t>AY — \<t>T<j,Y<l>4X Hence
04T^0r = |0%y0X - |0%x0r + HyT^x4> - HxT^y4> (4.8) and using (4.6) we obtain
1 J T J. V 1
x4>Y — ^aT^x^aY — 2 04^4^04-^ g.
^Hy(T^,4x - 3^X0) + Hx(T^4y - T^,y4>)
Using Proposition 2.3 the tensor field iV from (4.2) may be written in the form
Ar(-V. V) = -T.vV - faT^xW - 04^x0^ + 0Tx0F - 01V0X Using (4.6) and (4.9) we obtain
Ar(-V. V) = -T.vV - 03T^xW - 0,TrlK!X0,V - 04TX04^ - 04T^XY +Hy(T2x+4>4x - T<px4>) - Hx(T2y+4>4y - T^y^)
or
N{X, Y) = -<f>3TxY - 03^x0^ - faTx+tMY + 04Y)
+Hy(T2x+4>4x - T^X(j>) - Hx(T2y+4>4y - T^Y(j>)
Using (2.5) and Proposition 2.3 we get
(j)3N(X, Y) = -<f>3TXY - 0;{r,;,X0Y
(4.11)
From (4.10) and (4.11) we obtain
faTx+^xiY + 04Y) = -<f>4N(X, Y)
+Hy(T2x+4>4x ~ T^x^) ~ Hx(T2y+4>4y - T^y^)
or
Tx+4>4x(Y + 04Y) = -<j>4N(X, Y) + <5b3Tx+4>4x{Y + <f>AY) +Hy(T2x+4>4x ~ T^x^) ~ Hx(T2y+4>4y - T^y^)
From (2.3) and (4.3) we get
4>3oT = 2d if' ®
(4.13)
From (4.6) we have
2 H
Tv = H\Ty
(4.14)
Substituting X by \X + ^(p3X and Y by \Y + \4>3Y into (4.12) and making use of (4.13) and (4.14) we obtain
In virtue of Corollary 3.1 this connection is also an almost r-paracontact connection and its torsion tensor field T is
Hence we obtain
Theorem 4.1. On an almost r-paracontact manifold Mn 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 T on an almost r-paracontact manifold Mn if and only if it satisfies (4.2) and (4.3).
T(X, Y) = -\N{X + <f>3X, Y + <t>3Y) + 2dr/a(X, r)£a+
Now, consider the connection T given by
(4.16)
T(X, Y) = 2dVa(X, Y)Ca - \N{X + 4>3X, Y + 4>3Y) (4.17)
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 V with its torsion tensor T given by (4.17). So, if we consider the connection Vx = Vx + H{T i — T\ ,v4>), which is an
X—4 <p4X 4 (px
almost r-paracontact connection, then its torsion tensor is exactly T. □ We also have
Theorem 4.3. On an almost r-paracontact manifold Mn with a structure ^ = (0> Cm Va)a£(r) there exists a symmetric almost r-paracontact connection r if and only if the following conditions are satisfied
(i) all 1-forms r]a are closed
(ii) E is normal.
Proof. Suppose that there exists a symmetric almost r-paracontact connection on Mn. 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
Mn. □
Using Theorem 3.5 we obtain from Theorem 4.3
Theorem 4.4. On an almost r-paracontact manifold Mn with a structure E = (0, £a, Va)a£(r) there exists a symmetric D^,-connection T if the following conditions are satisfied
(i) all 1-forms r]a are closed
(ii) E is normal.
1 2
Now, let (r, T) be a pair of linear connections on an almost r-paracontact manifold Mn with a structure E = (0, £a, r]a)a£(r) such that
V.v = V.v. and V.v = V.v — T\ (4-18)
where V is a covariant derivative with respect to an arbitrary linear connection T and T is its torsion tensor field. Then, we have
rn
Proposition 4.1. The mean connection T and the deformation tensor 1 2
field d of (r,r) defined by (4.18) are given by
rn
VXY = 1(VxY + VyX + [X,Y}) (4.19)
dx{Y) = ^T{X,Y). (4.20)
rn
Proposition 4.2. The mean connection T given by (4.19) is torsionless. From Theorem 3.9 we obtain
1 2
Theorem 4.5. A pair of linear connections (I\r) defined by (4.18) on an almost r-paracontact manifold Mn with a structure E = (<j>,Ca^Va)a£(r) is compatible with E if and only if
m
(i) the mean connection T given by (4.19) is an almost r-paracontact connection on Mn
(ii) BTx = 0, where B is given by (2.10).
Using Remark 3.2 and Proposition 2.1 we get
1 2
Corollary 4.1. If a pair of linear connections (I\r) defined by (4.18) on an almost r-paracontact manifold Mn with a structure E = (<j),^a,ria)a£^ is compatible with E, then
T(X, Y) = FyWx (4.21)
where F is defined by (2.13), and W is an arbitrary tensor field of type (1,2) with WZ(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 Mn with a structure E = (0, £a, Va)a£(r) 1-forms r]a are closed then E is normal if and
1 2
only if the pair of linear connections (I\ T) defined by (4.18) is compatible with E.
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 T be an almost r-paracontact connection on an almost r-paracontact manifold Mn. Then we obtain
Theorem 5.1. The curvature tensor Rxy of an almost r-paracontact connection r on an almost r-paracontact manifold Mn with a structure E = (0? £<*> Va)a£(r) has the following properties
(i) Rxr^a = 0, a e (r)
(ii) rf- o Rxy, a G (r)
(iii) 4> o Rxy = Rxy ° 4>
(iv) FRxy = 0
(v) HRxy = Rxy
where F and H are given by (2.13) and (2.14).
Proof The operator Rxy is given by
Rxy = VxVy — VyVj — V[x,y]
(5.1)
The property (i) follows directly from (5.1). Since T is an almost r-paracon-tact connection we have i]a(VxY) = Xr]a(Y) and Vx(4>Y) = (pVxY. So, r]aRXyZ = XYrja(Z) - YXrja(Z) - [X,Y}r]a(Z) = 0, and Rxr((j>Z) =
(f)RxyZ. From (iii) and (ii) we get Rxy = <f>RxY<f>- From this, because
Fcf) = 0 we obtain (iv) and (v). □
After lengthy calculations we also obtain
Theorem 5.2. For the metric almost r-paracontact connection T given by
(3.8) on an almost r-paracontact Riemannian manifold Mn with a structure ^ = (0) ionrf* i d)ae(r) its curvature tensor field Rxy is of the form
Rxy = HRxy + Sxy (5-2)
where
Sxy = lV[x4>VY}4> + |V[y?f ® VX}(,a + \vaV\Y<f> ® VX}4>£,a ® V[x<pVYj№a - [X<№Y}<t> ® ia.
+ |»7a(V[r£/3)VX}VP ® + |?7a(V[y£/3)?7/3 ® VX](a
+ lV[X?f (W]^)/®^ - iv'r(V[YZf})Va(Vx&)ril3®Za,
H is given by (2.14), A[XBy] denotes AxBy — AyBx, and V is the Levi-Civita connection on Mn.
Making use of Proposition 2.1 we obtain from Theorem 5.2
Theorem 5.3. If the metric almost r-paracontact connection T given by
(3.8) on an almost r-paracontact Riemannian manifold Mn is flat, then the curvature tensor field Rxy of the Riemannian connection T generated by g is of the form
Rxy = ~Sxy + FZxy (5-4)
where Sxy is given by (5.3), F is given by (2.13), and Z is an arbitrary tensor field of type (1,3).
Now, let T be a Ds-connection on an almost r-paracontact manifold Mn. Then, we have
Theorem 5.4. The curvature tensor field Rxy of a D-^-connection T on an almost r-paracontact manifold Mn with a structure E = (<j),^a,ria)a£^ has the following properties
(i) 4>i o Rxy = Rxy °4>i, * = 1,2,3
(ii) PRxy = 0
(iii) QRxy = Rxy
where P and Q are given by (2.15) and (2.16) and 4>i’s are given by (2.1), (2.2), and (2.3).
1 2
Now, let (r, T) be a pair of linear connections on an almost r-paracontact manifold Mn with a structure E = (0, £a, r]a)ae(ry
Definition 5.1 [6]. The mixed curvature operator Rm for a pair of linear 1 2
connections (r, T) on Mn is defined as
Rxy = |([Vx,Vy] + [Vx,Vy] - V[x,y] - V[x,y])- (5.5)
Definition 5.2 [6]. The deformation curvature operator Rd for a pair of 1 2
linear connections (r, T) on Mn is defined as
rxy = l[dx,dY}. (5.6)
We obtain
1 2
Theorem 5.5. For a pair of linear connections (I\r) compatible with an almost r-paracontact structure E = (<j),^a,ria)a£^ on an almost r-paracon-tact manifold Mn the following identities take place
1 2
2 R™y + ARdXY = Rxy + Rxy (5.7)
m
Rxy + Rxy = Rxy (5.8)
1 2 m 1 2 to
where R, R, and R are curvature tensor fields for T, T, and I\
Also, we get
Corollary 5.1. The tensor fields of mixed and deformation curvatures
1 2
Rm and Rd for a pair of linear connections (I\ T) compatible with an almost r-paracontact structure E = (0, £a, 77a)a£(r) on an almost r-paracontact manifold Mn have the following properties
RxYta + RdXY^a = O (5.9)
if Rxy + V aRdxY = 0 (5.10)
<t>RxY ~ Rxy^ = ^(4>Rxy ~ rxy4>) (5-11)
0Rxy0 — Rxy = ~(<f>Rxy0 — Rxy) (5.12)
FR%Y + FRdXY = 0 (5.13)
HRxy — R^xy = HRxY — Rxy- (5-14)
REFERENCES
[1]. A. Bucki, A. Miernowski, Almost r-paracontact structures, Ann. Univ. Mariae Uurie-Sklodowska, 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: [email protected]
Received November, 27, 2000
