C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 66, N umb er 2, Pages 205–217 (2017) D O I: 10.1501/C om mua1_ 0000000812 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
PARA-CONTACT PRODUCT SEMI-RIEMANNIAN SUBMERSIONS
YILMAZ GÜNDÜZALP
Abstract. We introduce the concept of para-contact product semi-Riemannian submersions from an almost para-contact metric manifold onto a semi-Riemannian product manifold. We provide an example and show that the vertical and horizontal distributions of such submersions are invariant with respect to the almost para-contact structure of the total manifold. Moreover, we investigate various properties of the O’Neill’s tensors of such submersions, …nd the in-tegrability of the horizontal distribution. The paper is also focused on the transference of structures de…ned on the total manifold.
Introduction
The theory of Riemannian submersion was introduced by O’Neill and Gray in [20] and [13], respectively. Later, Riemannian submersions were considered between almost complex manifolds by Watson in [23] under the name of almost Hermitian submersion. He showed that if the total manifold is a Kähler manifold, the base manifold is also a Kähler manifold. Since then, Riemannian submersions have been used as an e¤ective tool to describe the structure of a Riemannian manifold equipped with a di¤erentiable structure. Presently, there is an extensive literature on the Riemannian submersions with di¤erent conditions imposed on the total space and on the …bres. For instances, Riemannian submersions between almost contact manifolds were studied by Chinea in [6] under the name of almost contact submersions. Riemannian submersions have been also considered for quaternionic Kähler manifolds [14] and para-quaternionic Kähler manifolds [5]. This kind of submersions have been studied with di¤erent names by many authors(see [1], [2], [3], [11], [12],[15], [18] and more). On the other hand, in[16] Kaneyuki and Williams de…ned the almost para-contact structure on pseudo-Riemannian manifold M of dimension (2m+1) and constructed the almost para-complex structure on M2m+1 R:
Received by the editors: January 23, 2016; Accepted: December. 30, 2016. 2010 Mathematics Subject Classi…cation. 53C15, 53C12, 53C40.
Key words and phrases. Almost para-contact metric manifold, semi-Riemannian product man-ifold, semi-Riemannian submersion, para-contact product semi-Riemannian submersion.
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Semi-Riemannian submersions were introduced by O’Neill in his book[21]. It is known that such submersions have their applications in Klauza-Klein theories, Yang-Mills equations, strings, supergravity. For applications of semi-Riemannian submersions, see:[9]. Since semi-Riemannian Product manifolds and para-contact manifolds are semi-Riemannian manifolds, one should consider semi-Riemannian submersions between such manifolds. In this paper, we de…ne para-contact prod-uct semi-Riemannian submersions between almost para-contact metric manifold and semi-Riemannian product manifold, and study the geometry of such submer-sions. We observe that para-contact product semi-Riemannian submersion has also rich geometric properties.
The paper is organized as follows. In Section 2 we collect basic de…nitions, some formulas and results for later use. In section 3 we introduce the notion of contact product semi-Riemannian submersions and give an example of para-contact product semi-Riemannian submersion. Moreover, we investigate properties of O’Neill’s tensors and show that such tensors have nice algebraic properties for para-contact product semi-Riemannian submersions. We …nd the integrability of the horizontal distribution.
Preliminaries
In this section we are going to recall main de…nitions and properties of al-most para-contact metric manifolds, Riemannian product manifolds and semi-Riemannian submersions.
2.1. Almost para-contact metric manifolds. Let M be a (2m + 1)- dimen-sional di¤erentiable manifold. Let ' be a (1; 1) tensor …eld, a vector …eld and a 1-form on M: Then ('; ; ) is called an almost para-contact structure on M if
(i) ( ) = 1; '2= Id ;
(ii) the tensor …eld ' induces an almost para-complex structure on the dis-tribution D = ker ; that is, the eigendisdis-tributions D+; D corresponding to the
eigenvalues 1, -1 of '; respectively, have equal dimension m:
M is said to be almost contact manifold if it is endowed with an almost para-contact structure([7], [16],[19],[26]).
Let M be an almost para-contact manifold. M is called an almost para-contact metric manifold if it is additionally endowed with a pseudo-Riemannian metric g of signature (m + 1; m) such that
For such a manifold, we additionally have (X) = g(X; ); ' = 0; ' = 0: Moreover, we can de…ne a skew-symmetric 2-form by (X; Y ) = g(X; 'Y ); which is called the fundamental 2-form corresponding to the structure. Note that ^ is, up to a constant factor, the Riemannian volume element of M:
On an almost para-contact manifold, one de…nes the (1; 2) tensor …eld N(1) by
N(1)(X; Y ) = ['; '](X; Y ) 2d (X; Y ) ; (2) where ['; '] is the Nijenhuis torsion of ' given by
['; '](X; Y ) = '2[X; Y ] + ['X; 'Y ] '['X; Y ] '[X; 'Y ]: (3) If N(1) vanishes identically, then the almost para-contact manifold (structure) is
said to be normal ([26]). The normality condition says that the almost para-complex structure J de…ned on M R by J (X; f d dt) = ('X + f ; (X) d dt) (4) is integrable.
We note that an almost para-contact metric manifold (M; g; '; ; ) is called (a) normal, if N' 2d = 0; where N' is the Nijenhuis tensor of ';
(b) para-contact, if = d ;
(c) K-para-contact, if M is para-contact and Killing;
(d) para–cosymplectic, if r = 0 which implies r = 0; where r is the Levi-Civita connection on M ;
(f) almost para–cosymplectic, if d = 0 and d = 0;
(g) weakly para–cosymplectic, if M is almost para–cosymplectic and [R(X; Y ); '] = R(X; Y )' 'R(X; Y ) = 0;
(h) para–Sasakian, if = d and M is normal;
(j) quasi-para–Sasakian, if d = 0 and M is normal([8],[24],[26]).
It is known that an almost para-contact manifold is a para-Sasakian manifold if and only if (rX')Y = g(X; Y ) + (Y )X; for X; Y 2 (T M):
Lemma 2.1 ([26]). Let (M; '; ; ; g) be an almost para-contact metric manifold. Then we have
2g((rX')Y; Z) = d (X; Y; Z) d (X; 'Y; 'Z) N(1)(Y; Z; 'X)
+N(2)(Y; Z) (X) 2d ('Z; X) (Y )
+2d ('Y; X) (Z); (5)
where is the fundamental 2-form and
where L is the Lie derivative.
Moreover if M is para-contact, then we have
2g((rX')Y; Z) = N(1)(Y; Z; 'X) 2d ('Z; X) (Y )
+ 2d ('Y; X) (Z): (7)
For an almost para-contact metric manifold, the following identities are well known:
(rX')Y = rX'Y '(rXY ); (8)
(rX )(Y; Z) = g(Y; (rX')Z); (9)
(rX )Y = g(Y; rX ): (10)
2.2. Semi-Riemannian product manifolds. Let (M1; g1) and (M2; g2) be two
m1and m2 dimensional semi-Riemannian manifolds with constant indexes q1> 0;
q2 > 0; respectively. Let : M1 M2 ! M1 and : M1 M2 ! M2 the
projections which are given by (x; y) = x and (x; y) = y for any (x; y) 2 M1 M2;
respectively. We denote the product manifold by M = (M1 M2; g); where
g(X; Y ) = g1( X; Y ) + g2( X; Y )
for any X; Y 2 (T M) and means tangent mapping. Then we have 2 = ;
2 = ; = = 0 and + = I; where I is identity tranformation.
Thus (M; g) is an (m1+ m2) dimensional semi-Riemannian manifold with
con-stant index (q1+ q2): The semi-Riemannian product manifold M = M1 M2 is
characterized by M1 and M2 are totally geodesic submanifolds of M [25].
Now, if we put F = ; then we can easily see that F2= I and
g(F X; Y ) = g(X; F Y ); (11)
for any X; Y 2 (T M):
Denote the Levi-Civita connection on M with respect to g by r: Then, M is called a locally semi-Riemannian product manifold if F is parallel with respect to r; i.e.,
rXF = 0; X 2 (T M)([17]; [22]): (12)
2.3. Semi-Riemannian submersions. Let (M; g) and (B; g0) be two connected semi-Riemannian manifolds of index s(0 s dimM ) and s0(0 s0 dimB) respectively, with s > s0: Roughly speaking, a semi-Riemannian submersion is a
smooth map : M ! B which is onto and satis…es the following conditions: (i) p: TpM ! T (p)B is onto for all p 2 M;
(ii) The …bres 1(p0); p0 2 B; are semi-Riemannian submanifolds of M;
(iii) preserves scalar products of vectors normal to …bres.
The vectors tangent to …bres are called vertical and those normal to …bres are called horizontal. We denote by V the vertical distribution, by H the horizontal distrib-ution and by v and h the vertical and horizontal projection. An horizontal vector …eld X on M is said to be basic if X is related to a vector …eld X0 on B: It is
basic.
We recall that the sections of V; respectively H; are called the vertical vector …elds, respectively horizontal vector …elds. A semi-Riemannian submersion : M ! B determines two (1; 2) tensor …eld T and A on M; by the formulas:
T (E; F ) = TEF = hrvEvF + vrvEhF (13)
and
A(E; F ) = AEF = vrhEhF + hrhEvF (14)
for any E; F 2 (T M); where v and h are the vertical and horizontal projections (see [4],[10]). From (13) and (14), one can obtain
rUX = TUX + h(rUX); (15)
rXU = v(rXU ) + AXU ; (16)
rXY = AXY + h(rXY ); (17)
for any X; Y 2 (H); U 2 (V): Moreover, if X is basic then h(rUX) = h(rXU ) =
AXU:
We note that for U; V 2 (V); TUV coincides with the second fundamental form
of the immersion of the …bre submanifolds and for X; Y 2 (H); AXY = 12v[X; Y ]
re‡ecting the complete integrability of the horizontal distribution H: It is known that A is alternating on the horizontal distribution: AXY = AYX; for X; Y 2
(H) and T is symmetric on the vertical distribution: TUV = TVU; for U; V 2 (V):
We now recall the following result which will be useful for later.
Lemma 2.2(see [10],[21]). If : M ! B is a semi-Riemannian submersion and X; Y basic vector …elds on M; related to X0 and Y0 on B; then we have the
following properties
(1) h[X; Y ] is a basic vector …eld and h[X; Y ] = [X0; Y0] ;
(2) h(rXY ) is a basic vector …eld related to (r0X0Y0); where r and r0 are
the Levi-Civita connection on M and B;
(3) [E; U ] 2 (V); for any U 2 (V) and for any basic vector …eld E: Para-contact product semi-Riemannian submersions
In this section, we de…ne the notion of a para-contact product semi-Riemannian submersion, give an example and study the geometry of such submersions. We now de…ne a ('; F ) para-holomorphic map which is similar to the notion of a ('; F ) holomorphic map between almost contact metric manifold and almost Hermitian manifold.
De…nition 3.1. Let (M2m+1; '; ; ) be an almost para-contact manifold and
(Bn; F ) an almost product manifold, respectively. The map : M ! B is
('; F ) para-holomorphic if ' = F :
By using the above de…nition, we are ready to give the following notion. De…nition 3.2. Let (M; '; ; ; g) be an almost para-contact metric manifold and (B; F; g0) be a semi-Riemannian product manifold. A semi-Riemannian submersion
: M ! B is a called para-contact product semi-Riemannian submersion if it is ('; F )-para-holomorphic, as well.
We give an example of a para-contact product semi-Riemannian submersion. Example 3.1.Consider the following submersion de…ned by
: R52 ! R21 (x1; x2; y1; y2; z) ! ( x1+ x2 p 2 ; y1+ y2 p 2 ): Then, the kernel of is
V = Ker = SpanfV1= @ @ x1 + @ @ x2 ; V2= @ @ y1 + @ @ y2 ; = @ @zg and the horizontal distribution is spanned by
H = (Ker )?= SpanfX = @ @ x1 + @ @ x2 ; Y = @ @ y1 + @ @ y2g: Hence, we have g(X; X) = g0( X; X) = 2; g(Y; Y ) = g0( Y; Y ) = 2
Thus, is a semi-Riemannnian submersion. Moreover, we can easily obtain that satis…es
'X = F X; 'Y = F Y:
Thus, is a para-contact product semi-Riemannian submersion. By using De…nition 3.1, we have the following result.
Proposition 3.1.Let : M ! B be a para-contact product semi-Riemannian sub-mersion from an almost para-contact metric manifold M onto a semi-Riemannian product manifold B, and let X be a basic vector …eld on M; related to X0 on
The following result can be proved in a standard way.
Proposition 3.2.Let : M ! B be a para-contact product semi-Riemannian sub-mersion from an almost para-contact metric manifold M onto a semi-Riemannian product manifold B: If X; Y are basic vector …elds on M; related to X0; Y0 on
B, then, we have
(i) h(rX')Y is the basic vector …eld related to (r0X0F )Y0;
(ii) h[X; Y ] is the basic vector …eld related to [X0; Y0].
Next proposition shows that a para-contact product semi-Riemannian submer-sion puts some restrictions on the distributions V and H:
Proposition 3.3.Let : M ! B be a para-contact product semi-Riemannian sub-mersion from an almost para-contact metric manifold M onto a semi-Riemannian product manifold B. Then, the horizontal and vertical distributions are ' invari-ant.
Proof. Consider a vertical vector …eld U ; it is known that ('U ) = F ( U ): Since U is vertical and is a semi-Riemannian submersion, we have U = 0 from which ('U ) = 0 follows and implies that 'U is vertical, being in the kernel of
:
As concerns the horizontal distribution, let X be a horizontal vector …eld. We have g('X; U ) = g(X; 'U ) = 0 because 'U is vertical and X is horizontal. From g('X; U ) = 0 we deduce that 'X is orthogonal to U and then 'X is horizontal. Proposition 3.4.Let : M ! B be a para-contact product semi-Riemannian sub-mersion from an almost para-contact metric manifold M onto a semi-Riemannian product manifold B. Then, we have
(i) 0= holds on the horizontal distribution, only;
(ii) is a vertical vector …eld;
(iii) (X) = 0; for all horizontal vector …elds X;
Proof. We prove only statement (i), the other assertions can be obtained in a similar way. If X and Y are basic vector …elds on M; related to X0; Y0 on B; then using the de…nition of a para-contact product semi-Riemannian submersion, we have
0(X; Y ) = 0( X; Y ) = g0( X; F Y ) = g0( X; 'Y )
= g0(X; 'Y ) = g(X; 'Y ) = (X; Y ) which gives the proof of assertion(i).
Theorem 3.1.Let : M ! B be a para-contact product semi-Riemannian sub-mersion. If the total space M is an almost para-contact metric manifold with (rX')Y = 0; for X; Y 2 (H); then the base space B is a locally semi-Riemannian
product manifold.
Proof. Let X; Y and Z be basic vector …elds on M; related to X0; Y0 and Z0
on B. Since (rX')Y = 0 for X; Y 2 (H): we get
0 = g(Z; rX'Y 'rXY )
for Z 2 (H). Using (17) we obtain
0 = g(Z; hrX'Y ) g(Z; h'rXY )
Then, by using ' = F ; we get
0 = g0(Z0; r0X0F Y ) g0(Z0; F r0X0Y0):
Hence 0 = g0(Z0; (r0
X0F )Y0) which shows that B is a locally semi-Riemannian
product manifold.
Theorem 3.2.Let : M ! B be a para-contact product semi-Riemannian sub-mersion. If the total space M is a para-Sasakian manifold, then the base space B is a locally semi-Riemannian product manifold.
Proof. Let X and Y be basic vector …elds on M; related to X0 and Y0 on B.
Since M is a para–Sasakian manifold, we have
(rX')Y = g(X; Y ) + (Y )X
= g(X; Y ) : Since is a semi-Riemannian submersion, we get
((rX')Y ) = g(X; Y ) = 0:
Then, by using ' = F ; we obtain ((rX')Y ) = (r0X0F )Y0= 0;
which proves the assertion.
Theorem 3.3.Let : M ! B be a para-contact product semi-Riemannian sub-mersion. If the total space M is a quasi para-Sasakian manifold, then the base space B is a locally semi-Riemannian product manifold.
Proof. Let X and Z be basic vector …elds on M; related to X0 and Z0 on B.
Since (5), we have
2g((rX')X; Z) = d (X; X; Z) d (X; 'X; 'Z) N(1)(X; Z; 'X)
+N(2)(X; Z) (X) 2d ('Z; X) (X) + 2d ('X; X) (Z): Since M is a quasi-para-Sasakian manifold, we obtain
Since vanishes on the horizontal distribution, we have g((rX')X; Z) = 0:
Thus, we deduce that
((rX')X) = 0 = (r0X0F )X0;
which shows that the base space is a locally semi-Riemannian product manifold. Proposition 3.5.Let : (M2m+1; '; ; ; g) ! (B2n; F; g0) be a para-contact
prod-uct semi-Riemannian submersion from an almost para-contact metric manifold M onto a semi-Riemannian product manifold B: Then, the …bres are almost para-contact metric manifolds.
Proof. Denoting by F the …bres, it is clear that dimF = 2(m n) + s = 2r + 1; where r = m n: We de…ne an almost para-contact structure (^g; ^'; ^; ^); by setting ' = ^'; = ^and = ^: Then, we get
^
'2U = '2U = U (U ) ; for U 2 (V).
On the other hand, for U; V 2 (V) we obtain ^
g(^'V; ^'U ) = g('V; 'U ) =^ ^g(V; '2U ) = ^g(V; U (U ) ) = ^g(V; U ) + ^(U )^(V );
which gives the proof of assertion.
We now check the properties of the tensor …elds T and A for a para-contact product semi-Riemannian submersion, we will see that such tensors have extra properties for such submersions.
Proposition 3.6. Let : M ! B be a para-contact product semi-Riemannian submersion from a para- cosymplectic manifold M onto a semi-Riemannian product manifold B; and let X and Y be horizontal vector …elds. Then, we have
(i) AX'Y = 'AXY;
(ii) A'XY = 'AXY:
Proof. (i) Let X and Y be horizontal vector …elds, and U vertical. Since M is a Para cosymplectic manifold, we have
(rX )(U; Y ) = g((rX')Y; U )
Thus, since the vertical and the horizontal distributions are invariant, from (17) we obtain
g(AX'Y 'AXY; U ) = 0:
Then, we have
AX'Y = 'AXY:
In a similar way, we obtain (ii).
For the tensor …eld T we have the following.
Proposition 3.7. Let : M ! B be a para-contact product semi-Riemannian submersion from a para-cosymplectic manifold M onto a semi-Riemannian product manifold B; and let U and V be vertical vector …elds. Then, we have
(i) TU'V = 'TUV;
(ii) T'UV = 'TUV:
We now investigate the integrability of the horizontal distribution H:
Theorem 3.4.Let : M ! B be a para-contact product semi-Riemannian sub-mersion from an almost para-cosymplectic manifold M onto a semi-Riemannian product manifold B. Then, the horizontal distribution is integrable.
Proof. Let X and Y be basic vector …elds. It su¢ ces to prove that v([X; Y ]) = 0; for basic vector …elds on M: Since M is an almost para–cosymplectic manifold, it implies d (X; Y; V ) = 0; for any vertical vector V: Then, one obtains
X( (Y; V )) Y ( (X; V )) + V ( (X; Y )) ([X; Y ]; V ) + ([X; V ]; Y ) ([Y; V ]; X) = 0:
Since [X; V ]; [Y; V ] are vertical and the two distributions are ' invariant, the last two and the …rst two terms vanish. Thus, one gets
g([X; Y ]; 'V ) = V (g(X; 'Y )):
On the other hand, if X is basic then h(rVX = h(rXV ) = AXV; thus we have
V (g(X; 'Y )) = g(rVX; 'Y ) + g(rV'Y; X)
= g(AXV; 'Y ) + g(A'YV; X):
Since, A is skew-symmetric and alternating operator, we get V (g(X; 'Y )) = 0: This proves the assertion.
Since for a quasi-para-Sasakian manifold d = 0; applying Theorem 3.4, we have the following result.
Corollary 3.1 Let : M ! B be a para-contact product semi-Riemannian sub-mersion from a quasi-para-Sasakian manifold M onto a semi-Riemannian product manifold B. Then, the horizontal distribution is integrable.
Corollary 3.2.Let : M ! B be a para-contact product semi-Riemannian sub-mersion from a para–cosymplectic manifold M onto a semi-Riemannian product manifold B. Then, the horizontal distribution is completely integrable.
Theorem 3.5. Let : M ! B be a para-contact product semi-Riemannian sub-mersion from an almost para–cosymplectic manifold M onto a semi-Riemannian product manifold B with dimVp 2; 8p 2 M. If X horizontal vector …eld is an
in…nitesimal automorphism of ' tensor …eld, then TVX = 0, for any V 2 (V),
if and only if (rXV ) = ([X; V ]).
Proof. Let W and V be vertical vector …elds on M; X horizontal. Since M is an almost para–cosymplectic manifold, it implies d = 0: Then, we obtain
d (W; 'V; X) = W ( ('V; X)) 'V ( (W; X)) + X( (W; 'V )) ([W; 'V ]; X) + ([W; X]; 'V ) (['V; X]; W ) = 0:
Since [W; 'V ] is vertical and the two distributions are ' invariant, the …rst two terms vanish. Thus, we get
X( (W; 'V )) + ([W; X]; 'V ) (['V; X]; W ) = 0: By direct computations, one obtains:
0 = X(g(W; V ) (V )g(W; )) + g([W; X]; V (V ) ) g(['V; X]; 'W ) 0 = g(W; rVX + [X; V ]) + g(rWX; V ) g('[X; 'V ]; W ) (V )(g(rX ; W ) + g(rWX; )) X( (V )) (W ): Using (15) we derive 0 = g(TVX; W ) + g(TWX; V ) 2g(T X; W ) (V ) + (W )( ([X; V ]) X( (V )) (V ) ([X; ])): (18) Moreover, we have ([X; V ]) X( (V )) (V ) ([X; ]) = (rVX) g(rX ; V ) + (V ) (r X) = g(TV ; X) g(rX ; V ) (V )g(T ; X): Substituting in (18), we obtain 0 = 2g(TVX; W ) 2g(T X; W ) (V ) + (W )(g(TV ; X) + (V )g(T X; ) g(rX ; V )): (19)
Now, assume that TVX = 0; for any X 2 (V): Then (19) implies g(rX ; V ) = 0;
for any V and we have
([X; V ]) = g(rXV; ) g(rVX; )
= X( (V )) g(rX ; V ) g(TVX; )
= (rXV ):
On the other hand, for any X 2 (H) and V 2 (V), the hypothesis ([X; V ]) = (rXV ) implies g(TVX; ) = g(rVX; ) = g(rXV + [V; X]; ) = 0. So, (19)
reduces to
0 = 2g(TVX; W ) (W )g(rX ; V );
for any V; W 2 (V): Thus, for any vertical vector …eld W orthogonal to ; we get g(TVX; W ) = 0: Since g(TVX; ) = 0; one has TVX = 0; V 2 (V) and the proof
is completed.
From Theorem 3.5, we have the following result.
Corollary 3.3 Let : M ! B be a para-contact product semi-Riemannian sub-mersion from a quasi-para-Sasakian manifold M onto a semi-Riemannian product manifold B with dimVp 2; 8p 2 M. If X horizontal vector …eld is an
in…nites-imal automorphism of ' tensor …eld, then TVX = 0, for any V 2 (V), if and
only if (rXV ) = ([X; V ]).
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Current address : Department of Mathematics, Dicle University, 21280, Diyarbak¬r, Turkey. E-mail address : ygunduzalp@dicle.edu.tr
