CONTACT HORIZONTALLY CONFORMAL SUBMERSIONS

Using the notion of horizontally conformai submersion, we generalize the contact metric submersions and obtain classification theorems for this submersion when the total manifold has some special almost contact structures.


Introduction
Let (M,gM) and (B,gB) be Riemannian manifolds and F : M -> B be a smooth submersion. Then F is called a Riemannian submersion if gM(X,Y)=gB(F*X,F*Y) for every X, Y G T((A;erF*)-L ), where * is symbol for the tangent map. The theory of Riemannian submersions was initiated by O'Neill in [12] and it has been used widely in differential geometry to investigate the geometry of manifolds. In [7] (see also, [5], [6] and [8]), Chinea introduced almost contact metric submersion between two almost contact manifolds with compatible metrics as a Riemannian submersion which is in addition an almost contact map. Then he showed that various properties of the total space are preserved. For Riemannian submersions between various manifolds, see: [5], [9] and [14].

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B. Sahin for every X, Y G r((kerF*)-1 ), where g M and g B are the Riemannian metrics of M and B, respectively. It is obvious that every Riemannian submersion is a particular horizontally conformai submersion with A = 1. We note that horizontally conformai submersions are special horizontally conformai maps which were introduced independently by Fuglede [10] and Ishihara [11]. We also note that a horizontally conformai submersion F : M -> B is said to be horizontally homothetic if the gradient of its dilation A is vertical, i.e., (1. 2) H(gradX) = 0 at p G M, where H is the projection on the horizontal space (ker. A vector field X on M is said to be projectable if there exists a vector field X' on B such that F*X p = for all p G M. In this case X' and X are called F-related. As it is well known, the vector field X is called a basic vector field. In this paper, we consider horizontally conformai submersion between almost contact metric manifolds and show that vertical kerF* and horizontal (kerF*) 1 -spaces of a contact horizontally conformai submersion are invariant with respect to the almost contact structure of the total manifold M. Also we obtain that if M is a normal almost contact metric manifold and B is an almost metric manifold, then B is also normal if and only if F is a special horizontally homothetic map. Moreover, we investigate the contact character of the base manifold when the total manifold is almost Sasakian, cosymplectic or Kenmotsu.
We have seen from above results that the geometry of contact horizontally conformai submersions is quite different from the geometry of almost contact metric submersions. For example, if M is a Sasakian manifold and B is an almost contact metric manifold, then the almost contact metric submersion F : M -> B implies that B is also a Sasakian manifold. But in the contact horizontally conformai situation, this is not true even for additional condition.

Preliminaries
In this section, we give brief information for almost contact manifolds. Our main reference is Blair's book [2], We also mention the second fundamental form of a map only as much as we need to carry out our work on contact horizontally conformai submersions.
An odd dimensional Riemannian manifold (M, g) is called almost contact metric manifold if there is a (1,1) tensor field <j >, a vector field called the characteristic vector field and its 1-form 77 such that for X,Y € T(TM). It follows that = 0 and 77 o <j) = 0. An almost contact metric manifold M is said to have a normal contact structure if N ( f ) + d-q®^ = 0, where N^ is the Nijenhuis tensor field of (p and it is defined by (2.4) Nt

Y) = g(X,(f>Y).
An almost contact metric manifold M is called almost cosymplectic if <¿77 = 0 and d<& = 0. A normal almost cosymplectic manifold is called cosymplectic. Let M be an almost contact metric manifold, if $ = drj, then M is called a contact metric manifold. A normal contact metric manifold is called a Sasakian manifold. Equivalently an almost contact metric manifold is a Sasakian manifold if and only if (2.5) (

V x <l>)Y = g(X,Y)S-rj(Y)X.
Moreover, a c-Sasakian manifold [9], c € R, is an almost contact metric manifold which is normal and satisfies drj = c$. An almost contact metric manifold is c-Sasakian if and only if the following formula holds Besides Sasakian manifolds, another well known almost contact metric manifolds are Kenmotsu manifolds and they are characterized by the following tensor equation

B. Sahin
Hom(TM,F~lTB) has a connection V induced from the Levi-Civita connection V M and the pullback connection. Then the second fundamental form of F is given by where V F is the pullback connection along F. It is known that the second fundamental form is symmetric.

Contact horizontally conformai submersions
In this section, we consider contact horizontally conformai submersion between almost contact metric manifolds and check the contact structure of the base manifold when the total manifold has a special contact structure. First recall that a submersion F ( or a map) between almost contact manifolds (M,<£ m ,£ m ,T7 m ) and (B, It is easy to see [4] that belongs to horizontal distribution (kerF*) 1 -when F is a submersion.
It is clear that every contact submersion is a special contact horizontally conformai submersion with A = 1. REMARK 1. We note that contact horizontally conformai submersions were already studied in [4] by Burel under the name of semi-conformal (4> M ,<j> B )holomorphic submersion. He investigates the harmonicity of this map in that paper. Our objective is to obtain classification theorems when the total space has some geometric structures. For the notations, we follow [1] and [9].
Let kerF* p be the kernel space of F* and denote its orthogonal complementary space in T p M by (kerF*) 1 -at p G M. Then one can observe that vertical distribution kerF» is <p M -invariant, see [4], Then invariant kerF* implies that g M (ct> M X,V) = -g M {X,<j> M V) = 0 for X G T^kerF^) and V G T(kerF^). This shows that (kerF*) 1 -is also invariant and any fibre of the contact horizontally conformai submersion is an invariant submanifold. Thus, we have the following result. For any submersion F : M -> B between Riemannian manifolds, the restriction of the differential F* p to the horizontal space (kerF^p) 1 -maps that space isomorphically on to TF^B. Denote its inverse by", then for any vector Z £ is called the horizontal lift of Z. If Z is a vector field on an open subset V of B, then the horizontal lift of Z is horizontal vector field Z on F" 1^) such that F* We denote the space (kerF*) 1 ' -span{£ M } by T>. Then, we say that a contact horizontally conformal submersion is V-homothetic if X(A) = 0 for every X € r(P). Now, we can state and prove our first classification theorem for contact horizontally conformal submersions.

{-X(X)Vm(Y) + Y(X)Vm(X)}^B=0.
Since B is an almost contact metric manifold, we have £B ^ 0. Then, for X = and Y G r(X>), above equation gives Y(A) = 0, which shows that F is V-homothetic. Conversely, suppose that F is T>-homothetic, then for X,Y G T(V) we  In a similar way, we have the following theorem:

Let (M,(f)M,£M,r]M,gM) be a Sasakian manifold and
(B,4>b,^b,TJB) be an almost contact metric manifold. Suppose that F :

Sasakian manifold if and only if F is T>-homothetic.
Proof. Let  Using (3.4) and (3.6) in the above equation, we get (

3.7) 4>B(F.X,F,Y) = ^{-X(X)r]M{Y)-\-Y(X)r)M(X)+2dr]B(F*X,F*Y)}.
Unauthenticated Download Date | 12/12/19 6:11 PM B. Sahin Prom Theorem 3.1, we know that B is normal if and only if F is V-homothetic. Thus it is enough to show that $B = A d'qB for manifold B. Since F is V-homothetic, for X,Y £ T(£>), from (3.7), we get For X G T(V) and Y = J£,m, we also have which shows that B is j-Sasakian.
When the total manifold of a contact horizontally conformai submersion is almost cosymplectic, we have the following strong result.
Finally, we investigate contact character of the base manifold of a contact horizontally conformai submersion when the total manifold is Kenmotsu manifold. for X , Y 6 r((A;eri ? *)-L ). Hence, using again (3.2) and (3.1), we get

F * { V x < j > M Y ) -F m V M x Y ) = j { g B ( c f > B F * X , F * Y ) Z B -( F + Y Ŵ X } .
Considering (2.9) and (3.1) we write

< j > M Y ( l n \ ) F * X -g M ( X , < f > M Y ) F * ( g r a d I n X ) = Y ( l n \ ) F * ( < f > M X ) ~ 9 m ( X I Y ) F i f ( 4 > M g r a d l n \ )
for X , Y e T((A;erF») x ). Then for 7 = ^andlG T{T>), we derive