C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 68, N umb er 1, Pages 559–571 (2019) D O I: 10.31801/cfsuasm as.438369
ISSN 1303–5991 E-ISSN 2618-6470
http://com munications.science.ankara.edu.tr/index.php?series= A 1
ON ALMOST -PARA-KENMOTSU MANIFOLDS SATISFYING
CERTAIN CONDITIONS
I. KÜPELI ERKEN
Abstract. In this paper, we study some remarkable properties of almost -para-Kenmotsu manifolds. We consider projectively ‡at, conformally ‡at and concircularly ‡at almost -para-Kenmotsu manifolds (with the -parallel ten-sor …eld h). Finally, we present an example to verify our results.
1. Introduction
The study of almost paracontact geometry was introduced by Kaneyuki and Williams in [7] and then it was continued by many other authors. A systematic study of almost paracontact metric manifolds was carried out in paper of Zamkovoy, [17]. However such manifolds were studied earlier, [1], [3], [4], [11]. These authors called such structures almost para-coHermitian. The curvature identities for dif-ferent classes of almost paracontact metric manifolds were obtained e.g. in [5], [6], [13], [17].
Considering the recent stage of the developments in the theory, there is an im-pression that the geometers are focused on problems in almost paracontact metric geometry which are created ad hoc.
Almost (para)contact metric structure is given by a pair ( ; ), where is a 1-form, is a 2-form and ^ nis a volume element. It is well known that then there
exists a unique vector …eld , called the characteristic (Reeb) vector …eld, such that i = 1, i = 0. The Riemannian or pseudo-Riemannian geometry appears if we try to introduce a compatible structure which is a metric or pseudo-metric g and an a¢ nor ((1; 1)-tensor …eld), such that
(X; Y ) = g( X; Y ); 2= (Id ): (1.1)
Received by the editors: December 20, 2017; Accepted: February 23, 2018. 2010 Mathematics Subject Classi…cation. 53B30, 53C25, 53D10.
Key words and phrases. Almost -para-Kenmotsu manifold, projectively ‡at, conformally ‡at, concircularly ‡at.
c 2 0 1 8 A n ka ra U n ive rsity C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a t h e m a t ic s a n d S ta t is t ic s
We have almost paracontact metric structure for = +1 and almost contact metric for = 1. Then, the triple ( ; ; ) is called almost paracontact structure or almost contact structure, resp.
Combining the assumption concerning the forms and , we obtain many di¤er-ent types of almost (para)contact manifolds, e.g. (para)contact if is contact form and d = , almost (para)cosymplectic if d = 0, d = 0, almost (para)Kenmotsu if d = 0, d = 2 ^ .
Classi…cations are obtained for contact metric, almost cosymplectic, almost -Kenmotsu and almost -cosymplectic manifolds, e.g [9], [10], for paracontact case [8].
The projective curvature tensor is an important tensor from the di¤erential geo-metric point of view. Let M be a (2n + 1)-dimensional semi-Riemannian manifold with metric g. The Ricci operator Q of (M; g) is de…ned by g(QX; Y ) = S(X; Y ), where S denotes the Ricci tensor of type (0; 2) on M . If there exists a one-to-one correspondence between each coordinate neighborhood of M and a domain in Euclidian space such that any geodesic of the semi-Riemannian manifold cor-responds to a straight line in the Euclidean space, then M is said to be locally projectively ‡at. For n 1, M is locally projectively ‡at if and only if the well known projective curvature tensor P vanishes. Here P is de…ned by [12]
P (X; Y )Z = R(X; Y )Z 1
2n[S(Y; Z)X S(X; Z)Y ] (1.2)
for all X; Y; Z 2 T (M), where R is the curvature tensor and S is the Ricci tensor. In fact M is projectively ‡at if and only if it is of constant curvature [16]. Thus the projective curvature tensor is the measure of the failure of a semi-Riemannian manifold to be of constant curvature.
In semi-Riemannian geometry, one of the basic interest is curvature properties and to what extend these determine the manifold itself. One of the important curvature properties is conformal ‡atness. The conformal (Weyl) curvature tensor is a measure of the curvature of spacetime and di¤ers from the semi-Riemannian curvature tensor. It is the traceless component of the Riemannian tensor which has the same symmetries as the Riemannian tensor. The most important of its special property that it is invariant under conformal changes to the metric. Namely, if g = kg for some positive scalar functions k, then the Weyl tensor satis…es the equation W = W . In other words, it is called conformal tensor. Weyl constructed a generalized curvature tensor of type (1; 3) on a semi-Riemannian manifold which vanishes whenever the metric is (locally) conformally equivalent to a ‡at metric; for this reason he called it the conformal curvature tensor of the metric. The Weyl conformal curvature tensor is de…ned by
C(X; Y )Z = R(X; Y )Z 1
2n 1[S(Y; Z)X S(X; Z)Y
+g(Y; Z)QX g(X; Z)QY ] + r
for all X; Y; Z 2 T (M), where R is the curvature tensor, S is the Ricci tensor and r = tr(S) is scalar curvature [15].
A necessary condition for a semi-Riemannian manifold to be conformally ‡at is that the Weyl curvature tensor vanish. The Weyl tensor vanish identically for 2 dimensional case. In dimensions 4, it is generally nonzero. If the Weyl tensor vanishes in dimensions 4, then the metric is locally conformally ‡at. So there exists a local coordinate system in which the metric is proportional to a constant tensor. For the dimensions greater than 3, this condition is su¢ cient as well. But in dimension 3 the vanishing of the equation c = 0, that is,
c(X; Y ) = (rXQ)Y (rYQ)X
1
2(2n 1)[(rXr)Y (rYr)X];
is a necessary and su¢ cient condition for the semi-Riemannian manifold being conformally ‡at, where c is the divergence operator of C, for all vector …elds X and Y on M . It should be noted that if the manifold is conformally ‡at and of dimension greater than 3, then C = 0 implies c = 0 [15].
The concircular curvature tensor C of a (2n + 1)-dimensional manifold is de…ned by
C(X; Y )Z = R(X; Y )Z r
2n(2n + 1)[g(Y; Z)X g(X; Z)Y ] (1.4) for all X; Y; Z 2 T (M), where R is the curvature tensor and r = tr(S) is scalar curvature [14], [16]. For n 1, M is coincular ‡at if and only if the well known coincular curvature tensor C vanishes.
The paper is organized in the following way.
Section 2 is preliminary section, we remember the class of almost paracontact metric manifolds which de…ned by
d = 0; d = 2 ^ ; (1.5)
where is a function. These manifolds are called almost -paracosymplectic [8]. They contain properly almost paracosymplectic, = 0, and almost -para-Kenmotsu, = const: 6= 0 manifolds. In this section, we remember basic properties of such manifolds.
Section 3 and 4 are devoted to properties of almost -para-cosymplectic and almost -para-Kenmotsu manifolds. Section 5 devoted to almost -para-Kenmotsu manifolds with the -parallel tensor …eld h.
In Section 6, 7 and 8 we study, respectively, projectively ‡at, conformally ‡at and concircularly ‡at almost -para-Kenmotsu manifolds (with the -parallel tensor …eld h). Finally, we present an example to verify our results.
2. Preliminaries
Let M be a (2n + 1)-dimensional di¤erentiable manifold and is a (1; 1) tensor …eld, is a vector …eld and is a one-form on M: Then ( ; ; ) is called an almost paracontact structure on M if
(i) 2= Id ; ( ) = 1,
(ii) the tensor …eld induces an almost paracomplex structure on the distrib-ution D = ker ; that is the eigendistribdistrib-utions D ; corresponding to the eigenvalues 1, have equal dimensions, dim D+= dim D = n.
The manifold M is said to be an almost paracontact manifold if it is endowed with an almost paracontact structure [17].
Let M be an almost paracontact manifold. M will be called an almost paracon-tact metric manifold if it is additionally endowed with a pseudo-Riemannian metric g of a signature (n + 1; n), i.e.
g( X; Y ) = g(X; Y ) + (X) (Y ): (2.1)
For such manifold, we have
(X) = g(X; ); ( ) = 0; = 0: (2.2)
Moreover, we can de…ne a skew-symmetric tensor …eld (a 2-form) by
(X; Y ) = g( X; Y ); (2.3)
usually called fundamental form. For an almost -paracosymplectic manifold, there exists an orthogonal basis fX1; : : : ; Xn; Y1; : : : ; Yn; g such that g(Xi; Xj) = ij,
g(Yi; Yj) = ij and Yi = Xi, for any i; j 2 f1; : : : ; ng. Such basis is called a
-basis.
An almost paracontact metric manifold is called Einstein if its Ricci tensor S satis…es the condition
S(X; Y ) = ag(X; Y ):
On an almost paracontact manifold, one de…nes the (1; 2)-tensor …eld N(1) by
N(1)(X; Y ) = [ ; ] (X; Y ) 2d (X; Y ) ; where [ ; ] is the Nijenhuis torsion of
[ ; ] (X; Y ) = 2[X; Y ] + [ X; Y ] [ X; Y ] [X; Y ] :
If N(1) vanishes identically, then the almost paracontact manifold (structure) is
said to be normal [17]. The normality condition says that the almost paracomplex structure J de…ned on M R J (X; d dt) = ( X + ; (X) d dt); is integrable.
3. Almost -Paracosymplectic manifolds
An almost paracontact metric manifold M2n+1, with a structure ( ; ; ; g) is
said to be an almost -paracosymplectic manifold [8], if
d = 0; d = 2 ^ ; (3.1)
For a particular choices of the function we have the following subclasses, almost -para-Kenmotsu manifolds, = const: 6= 0,
almost paracosymplectic manifolds, = 0.
If additionally normality condition is ful…lled, then manifolds are called -para-Kenmotsu or paracosymplectic, resp.
De…nition 1. [8] For an almost -paracosymplectic manifold, de…ne the (1; 1)-tensor …eld A by
AX = rX : (3.2)
Proposition 1. [8] For an almost -paracosymplectic manifold M2n+1, we have
i) L = 0; ii) g(AX; Y ) = g(X; AY ); iii) A = 0; iv) L = 2 ; v) (L g)(X; Y ) = 2g(AX; Y );
vi) (AX) = 0; vii) d = f if n > 2 (3.3)
where L indicates the operator of the Lie di¤erentiation, X, Y are arbitrary vector …elds on M2n+1 and f = i d .
Proposition 2. [8] For an almost -paracosymplectic manifold, we have
A + A = 2 ; (3.4)
r = 0: (3.5)
Let de…ne h = 12L : In the following proposition we establish some properties of the tensor …eld h:
Proposition 3. [8]For an almost -paracosymplectic manifold, we have the follow-ing relations
g(hX; Y ) = g(X; hY ); (3.6)
h + h = 0; (3.7)
h = 0; (3.8)
r = 2+ h = A: (3.9)
Corollary 1. [8]All the above Propositions imply the following formulas for the traces
tr(A ) = tr( A) = 0; tr(h ) = tr( h) = 0;
tr(A) = 2 n; tr(h) = 0: (3.10)
Proposition 4. [8]For an almost -paracosymplectic manifold, we have
Theorem 1. [8] Let (M2n+1; ; ; ; g) be an almost -paracosymplectic manifold.
Then, for any X; Y 2 (M2n+1);
R(X; Y ) = d (X)(Y (Y ) ) d (Y )(X (X) ) (3.12)
+ (X)( Y + hY ) (Y )( X + hX)
+(rX h)Y (rY h)X:
4. Almost -para-Kenmotsu manifolds
In this section, we give curvature properties of an almost -para-Kenmotsu man-ifold.
Theorem 2. [8]Let (M2n+1; ; ; ; g) be an almost -para-Kenmotsu manifold.
Then, for any X; Y 2 (M2n+1);
R(X; Y ) = (X)( Y + hY ) (Y )( X + hX)+(rX h)Y (rY h)X: (4.1)
Theorem 3. [8]Let (M2n+1; ; ; ; g) be an almost -para-Kenmotsu manifold.
Then, for any X 2 (M2n+1) we have
R(X; ) = lX = 2 2X 2 hX + h2X (r h)X; (4.2) (r h)X = 2 X 2 hX + h2X R(X; ) ; (4.3) 1 2(R( ; X) + R( ; X) ) = 2 2X h2X; (4.4) S(X; ) = 2n 2 (X) + g(div( h); X); (4.5) S( ; ) = 2n 2+ trh2: (4.6)
5. Almost -para-Kenmotsu manifolds with the -parallel tensor field h
For any vector …eld X on M2n+1, we can take X = XT + (X) ; XT is tan-gentially part of X and (X) the normal part of X. We say that any symmetric (1; 1)-type tensor …eld B on a semi-Riemannian manifold (M; g) is said to be a
-parallel tensor if it satis…es the equation
g((rXTB)YT; ZT) = 0;
for all tangent vectors XT, YT, ZT orthogonal to [2].
Proposition 5. Let (M2n+1; ; ; ; g) be an almost -para-Kenmotsu manifold. If the tensor …eld h is -parallel, then we have
(rX h)Y = (X)[ lY 2 2Y 2 hY + h2Y ]
(Y )[ hX h2X] g(Y; hX h2X) ; (5.1)
Proof. If we suppose that h is -parallel, we have 0 = g((rXT h)YT; ZT) = 0 0 = g((rX (X) h)Y (Y ) ; Z (Z) ) 0 = g((rX h)Y; Z) (X)g((r h)Y; Z) (Y )g((rX h) ; Z) (Z)g((rX h)Y; ) + (X) (Y )g((r h) ; Z) + (Y ) (Z)g((rX h) ; ) + (Z) (X)g((r h)Y; ) (X) (Y ) (Z)g((r h) ; ); for all vector …elds X; Y on M . After some calculations, we get
0 = g((rX h)Y; Z) (Z)g((rX h)Y; ) (X)g((r h)Y; Z) (Y )g((rX h) ; Z):
By (3.9), we obtain
(rX h)Y = (X)(r h)Y (Y )( hX h2X) g(Y; hX h2X) :
Using the fact that (r h)Y = (r h)Y and (4.3), we have the requested equation. Proposition 6. An almost -para-Kenmotsu manifold with the -parallel tensor …eld h satis…es the following relation
R(X; Y ) = (Y )lX (X)lY; (5.2)
where l = R(:; ) is the Jacobi operator with respect to the characteristic vector …eld .
Proof. With the help of the equations (4.1) and (5.1), we get (5.2).
Theorem 4. Let (M2n+1; ; ; ; g) be an almost -para-Kenmotsu manifold. If the tensor …eld h is -parallel, then is the eigenvector of Ricci operator on M2n+1. Proof. First, if we take the inner product of (5.2) with W , we have
g(R(X; Y ) ; W ) = (Y )g(lX; W ) (X)g(lY; W );
and then replacing X; W by ei in the last equation and taking summation over
i (Let fe1; e2; :::; e2n; g be an -basis of the tangent space at any point of the
manifold), we …nd 2n+1X i=1 "ig(R(ei; Y ) ; ei) = 2n+1X i=1 "i[ (Y )g(lei; ei) (ei)g(lY; ei)]
for all vector …elds X; Y on M . From the last equation, one can easily get
S(Y; ) = (Y )tr(l) (5.3)
On the other hand, (5.3) can be written as Q = tr(l) : So, this ends the proof.
Corollary 2. Let (M3; ; ; ; g) be an almost -para-Kenmotsu manifold. If the
tensor …eld h is -parallel then an almost -paracosymplectic ( ; ; )-manifold always exist on every open and dense subset of M:
6. Projectively flat almost -para-Kenmotsu manifolds (with the -parallel tensor field h)
Theorem 5. A projectively ‡at almost -para-Kenmotsu manifold (M2n+1; ; ; ; g)
has a scalar curvature
r = 2ntr( (r h)) + S( ; )(2n + 1): (6.1)
Proof. Let us suppose that almost -para-Kenmotsu manifold is projectively ‡at. If we take the inner product of (1.2) with W , we get
g(R(X; Y )Z; W ) = 1
2n[S(Y; Z)g(X; W ) S(X; Z)g(Y; W )]: Replacing W; X by in the last equation and using (4.2), (4.5), we get
S(Y; Z) = 2n 2g(Y; Z) + 2 g( Y; hZ) + g(hZ; hY )
+g((r h)Z; Y ) +2n1 (Y )g(div( h); Z) : (6.2) Considering the -basis and putting Y = Z = ei in (6.2), we obtain
2n+1X i=1 "iS(ei; ei) = 2n+1X i=1 "i2n 2g(ei; ei) + 2 g( ei; hei) + g(hei; hei) +g((r h)ei; ei) + 1 2n (ei)g(div( h); ei) r = 2n( 2(2n + 1) 2 tr( h) + tr(h2) tr( (r h)) + 1 2n(g(div( h); )):
Now, using (3.10), (4.5) and (4.6), we get the requested equation.
Theorem 6. A projectively ‡at -para-Kenmotsu manifold (M2n+1; ; ; ; g) is an Einstein manifold.
Proof. If we take h = 0 in the proof of Theorem 5, we obtain S(Y; Z) = 2n 2g(Y; Z). This means manifold is Einstein.
Theorem 7. Let (M2n+1; ; ; ; g) be an projectively ‡at almost -para-Kenmotsu
Proof. Let us suppose that (M2n+1; ; ; ; g) is projectively ‡at almost
-para-Kenmotsu manifold with the -parallel tensor …eld h. If we take the inner product of (1.2) with W , we get
g(R(X; Y )Z; W ) = 1
2n[S(Y; Z)g(X; W ) S(X; Z)g(Y; W )]:
By setting W = X = in the last equation and using Proposition 6 and Theorem 4, we obtain
g(R(Y; ) ; Z) = 1
2n[S(Y; Z) (Y )S( ; Z)] g(lY; Z) = 1
2n[S(Y; Z) (Y ) (Z)tr(l)]: If we set Y = Z = ei in the last equation, we end the proof.
7. Conformally flat almost -para-Kenmotsu manifolds
Theorem 8. A conformally ‡at almost -para-Kenmotsu manifold (M2n+1; ; ; ; g)
satis…es the following
0 = tr ( (r h)) : (7.1)
Proof. Let us suppose that almost -para-Kenmotsu manifold is conformally ‡at. If we take the inner product of (1.3) with W , we get
g(R(X; Y )Z; W ) = 1
2n 1[g(Y; Z)g(QX; W ) g(X; Z)g(QY; W ) +S(Y; Z)g(X; W ) S(X; Z)g(Y; W )]
r
2n(2n 1)[g(Y; Z)g(X; W ) g(X; Z)g(Y; W )]: By setting W = X = in the last equation and using (4.2), (4.5) and (4.6), we obtain
S(Y; Z) = (2n 1)( 2g(Y; Z) + 2 (Y ) (Z) 2 g( hY; Z) + g(h2Y; Z) g( (r h)Y; Z)) g(Y; Z)( 2n 2+ tr(h2) + (Z)( 2n 2 (Y ) +g(div( h); Y ) + (Y )( 2n 2 (Z) + g(div( h); Z)
+ r
Considering the -basis and putting Y = Z = ei in (7.2), we get 2n+1X i=1 "iS(ei; ei) = r = 2n+1X i=1 "i n (2n 1)( 2g(ei; ei) + 2 (ei) (ei) 2 g( hei; ei) +g(h2ei; ei) g( (r h)ei; ei)) g(ei; ei)( 2n 2+ tr(h2) + (ei)( 2n 2 (ei) + (div( h); ei) + (ei)( 2n 2 (ei) + (div( h); ei) + r 2n(g(ei; ei) (ei) (ei)) o :
Then by (3.10), (4.5) and (4.6)), we obtain r = tr ( (r h)) :
8. Concircularly flat almost -para-Kenmotsu manifolds (with the -parallel tensor field h)
Theorem 9. A concircularly ‡at almost -para-Kenmotsu manifold (M2n+1; ; ; ; g)
has a scalar curvature
r = (2n + 1)(S( ; ) tr( (r h)): (8.1)
Proof. Let us suppose that almost -para-Kenmotsu manifold is concircularly ‡at. If we take the inner product of (1.4) with W , we get
g(R(X; Y )Z; W ) = r
2n(2n + 1)[g(Y; Z)g(X; W ) g(X; Z)g(Y; W )]: By setting W = X = in the last equation and using (4.2), we obtain
2g( Y; Z) 2 g( hY; Z) + g(h2Y; Z)
g( (r h)Y; Z) (8.2)
= r
2n(2n + 1)(g(Y; Z) (Y ) (Z)):
Considering the -basis and putting Y = Z = ei in (8.2), we get 2n+1X i=1 "i 2g(ei; ei) + 2 (ei) (ei) 2 g( hei; ei) + g(h2ei; ei) g( (r h)ei; ei) = 2n+1X i=1 "i r 2n(2n + 1)(g(ei; ei) (ei) (ei)) : Then by (3.10), (4.5) and (4.6), we obtain (8.1).
Theorem 10. A concircularly ‡at -para-Kenmotsu manifold (M2n+1; ; ; ; g)
has a scalar curvature
r = 2 2n(2n + 1): (8.3)
Proof. If we take h = 0 in the proof of Theorem 9, we obtain the requested equation.
Theorem 11. Let (M2n+1; ; ; ; g) be an concircularly ‡at almost -para-Kenmotsu manifold with the -parallel tensor …eld h. Then r = tr(l)(2n + 1):
Proof. Let us suppose that (M2n+1; ; ; ; g) is concircularly ‡at almost
-para-Kenmotsu manifold with the -parallel tensor …eld h. If we take the inner product of (1.4) with W , we get
g(R(X; Y )Z; W ) = r
2n(2n + 1)[g(Y; Z)g(X; W ) g(X; Z)g(Y; W )]:
By setting W = X = in the last equation and using Proposition 6 and Theorem 4, we obtain
g(R(Y; ) ; Z) = g(lY; Z) = r
2n(2n + 1)[g(Y; Z) (Y ) (Z)]: If we set Y = Z = ei in the last equation, we end the proof.
9. Example
Now, we will give an example of a 3-dimensional para-Kenmotsu manifold ( = 1).
Example 1. We consider the 3-dimensional manifold M = f(x; y; z) 2 R3; z 6= 0g and the vector …elds
X = @ @x; X = @ @y; = (x + 2y) @ @x + (2x + y) @ @y + @ @z:
The 1-form = dz de…nes an almost paracontact structure on M with characteristic vector …eld = (x + 2y) @
@x+ (2x + y) @ @y+
@
@z. Let g, be the pseudo-Riemannian
metric and the (1; 1)-tensor …eld given by
g = 0 @ 1 0 1 2(x + 2y) 0 1 12(2x + y) 1 2(x + 2y) 1 2(2x + y) 1 (2x + y)2+ (x + 2y)2 1 A ; = 0 @ 01 10 (2x + y)(x + 2y) 0 0 0 1 A ; with respect to the basis @x@ ;@y@ ;@z@ .
Using (3.9) we have
rXX = ; r XX = 0; r X = 2 X;
rX X = 0; r X X = ; r X = 2X;
rX = X; r X = X; r = 0:
for = 1. Hence the manifold is a para-Kenmotsu manifold. One can easily compute,
R(X; X) = 0; R( X; ) = X; R(X; ) = X;
R(X; X) X = X; R( X; )~'X = ; R(X; ) X = 0;
R(X; X)X = X; R( X; )X = 0; R(X; )X = :
(9.1) We have constant scalar curvature as follows,
r = S(X; X) S( X; X) + S( ; ) = 6:
So, we conclude that M is a three dimensional projectively ‡at and concircularly ‡at para-Kenmotsu manifold for = 1.
References
[1] Bejan, C.L., Almost parahermitian structures on the tangent bundle of an almost para-coHermitian manifold, In: The Proceedings of the Fifth National Seminar of Finsler and Lagrange Spaces (Bra sov, 1988), 105–109, Soc. Stiinte Mat. R. S. Romania, Bucharest, 1989. [2] Boeckx, E. and Cho, J. T., -parallel contact metric spaces, Di¤ er. Geom. Appl. 22, (2005),
275-285.
[3] Buchner, K. and Rosca, R., Variétes para-coKählerian á champ concirculaire horizontale, C. R. Acad. Sci. Paris 285 (1977), Ser. A, 723-726.
[4] Buchner, K. and Rosca, R., Co-isotropic submanifolds of a para-coKählerian manifold with concicular vector …eld, J. Geometry, 25 (1985), 164-177.
[5] Cappelletti-Montano, B., Küpeli Erken, I. and Murathan, C., Nullity conditions in paracon-tact geometry, Di¤ . Geom. Appl. 30 (2012), 665-693.
[6] Dacko, P., On almost para-cosymplectic manifolds, Tsukuba J. Math. 28 (2004), 193-213. [7] Kaneyuki, S. and Williams, F. L., Almost paracontact and parahodge structures on manifolds,
Nagoya Math. J 1985; 99: 173-187.
[8] Küpeli Erken, I., Dacko, P. and Murathan, C., Almost -paracosymplectic manifolds, J. Geom. Phys., 88 (2015) 30-51.
[9] Öztürk, H., Aktan, N. and Murathan, C., Almost -cosymplectic ( ; ; )-spaces, Submitted. Available in Arxiv:1007.0527 [math. DG].
[10] Öztürk, H., Murathan, C., Aktan, N. and Turgut Vanl¬, A., Almost -cosymplectic f-manifolds, An. ¸Stiin¸t. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) 60 no. 1, (2014), 211-226. [11] Rossca, R. and Vanhecke, L., Súr une variété presque paracokählérienne munie d’une
con-nexion self-orthogonale involutive, Ann. Sti. Univ. “Al. I. Cuza” Ia si 22 (1976), 49-58. [12] Soos, G., Über die geodätischen Abbildungen von Riemannaschen Räumen auf projektiv
symmetrische Riemannsche Räume, Acta. Math. Acad. Sci. Hungar. Tom 9 (1958), 359-361. [13] We÷yczko, J., On basic curvature identities for almost (para)contact metric manifolds.
Avail-able in Arxiv: 1209.4731 [math. DG].
[14] Yano, K., Concircular geometry I, Concircular transformations, Proc. Imp. Acad. Tokyo, 16 (1940), 195-200.
[15] Yano, K. and Kon, M., Structures on manifolds, Series in Pure Mathematics, 3. World Sci-enti…c Publishing Co., Singapore, (1984).
[16] Yano, K. and Bochner, S., Curvature and Betti numbers, Annals of mathematics studies, 32, Princeton University Press,1953.
[17] Zamkovoy, S., Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom. 36 (2009), 37-60.
Current address : Faculty of Engineering and Natural Sciences, Department of Mathematics, Bursa Technical University, Bursa, TURKEY
E-mail address : irem.erken@btu.edu.tr
