CONFORMAL SEMI-INVARIANT RIEMANNIAN MAPS TO KÄHLER MANIFOLDS

As a generalization of CR-submanifolds and semi-invariant Riemannian maps, we introduce conformal semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds. We give non-trivial examples, investigate the geometry of foliations, and obtain decomposition theorems by using the existence of conformal Riemannian maps. We also investigate the harmonicity of such maps and find necessary and sufficient conditions for conformal anti-invariant Riemannian maps to be totally geodesic.


Introduction
Let (M , g, J) be an almost Hermitian manifold with almost complex structure J. CR-submanifolds of almost Hermitian manifolds were introduced by Bejancu as a generalization of holomorphic submanifolds and totally real submanifolds. Let  [4] and [5].
Riemannian maps as a generalization of isometric immersions and Riemannian submersions were defined by Fischer in [6]. Such maps have been studied widely by many authors, see monograph [13]. On the other hand, as a generalization of holomorphic submanifolds and totally real submanifolds, invariant Riemannian maps and anti-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds were introduced in [15]. Semi-invariant Riemannian maps were introduced in [14] and it was shown that such maps include CR-submanifolds (therefore holomorphic immersions and totally real immersions), invariant Riemannian maps, and anti-invariant Riemannian maps. Recently, Riemannian maps have been investigated for various manifolds; see [8,9,10,11,12,13].
As a generalization of Riemannian maps, conformal Riemannian maps have been defined in [16] and it is shown that conformal submersions and conformal immersions are particular conformal Riemannian maps. Recently, conformal antiinvariant Riemannian maps and conformal slant Riemannian maps from Riemannian manifolds to Kähler manifolds have been introduced and studied in [2] and [1], respectively.
In this paper, we introduce and study conformal semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of CR-submanifolds of almost Hermitian manifolds and semi-invariant Riemannian maps to almost Hermitian manifolds. We first present the notion of conformal semiinvariant Riemannian maps supported by examples. Then by using the existence of conformal semi-invariant Riemannian maps, we obtain a decomposition theorem. We also observe that conformal semi-invariant maps allow us to obtain new conditions for a map to be harmonic. The total geodesicity of conformal semi-invariant maps is also studied.

Preliminaries
In this section, we recall some basic materials from [3,19]. A 2n-dimensional Riemannian manifold (M, g, J M ) is called an almost Hermitian manifold if there exists a tensor field J of type (1, 1) on M such that J 2 = −I and where I denotes the identity transformation of T p M . Consider an almost Hermitian manifold (M, g, J) and denote by ∇ the Levi-Civita connection on M with respect to g. Then M is called a Kähler manifold if J is parallel with respect to ∇, i.e.
Let (M, g M ) and (N, g N ) be Riemannian manifolds and suppose that ϕ : M → N is a smooth map between them. Then the differential ϕ * of ϕ can be viewed a section of the bundle Hom(T M, ϕ −1 T N ) → M , where ϕ −1 T N is the pullback bundle which has fibres (ϕ −1 T N ) p = T ϕ(p) N , p ∈ M . Hom(T M, ϕ −1 T N ) has a connection ∇ induced from the Levi-Civita connection ∇ M and the pullback connection. Then the second fundamental form of ϕ is given by where ∇ ϕ is the pullback connection. It is known that the second fundamental form is symmetric. A smooth map ϕ : (M, g M ) → (N, g N ) is said to be harmonic if trace(∇ϕ * ) = 0. On the other hand, the tension field of ϕ is the section τ (ϕ) of Γ(ϕ −1 T N ) defined by where {e 1 , . . . , e m } is the orthonormal frame on M . Then it follows that ϕ is harmonic if and only if τ (ϕ) = 0; for details, see [3].
We denote by ∇ 2 both the Levi-Civita connection of (M 2 , g 2 ) and its pullback along F . Then according to [7], for any vector field X on M 1 and any section V of (range where A V F * X is the tangential component (a vector field along F ) of ∇ 2 X V . It is easy to see that A V F * X is bilinear in V and F * and A V F * X at p depends only on V p and F * p X p . By direct computations, we obtain Here we have the following definition from [17].
We now recall the definition of conformal Riemannian maps from [16] as follows. Let (M m , g M ) and (N n , g N ) be Riemannian manifolds and F : (M m , g M ) → (N n , g N ) a smooth map between them. Then we say that F is a conformal Riemannian map at p ∈ M if 0 < rank F * p ≤ min{m, n} and F * p maps the horizontal space H(p) = (ker(F * p )) ⊥ conformally onto range(F * p ), i.e., there exists a number λ 2 (p) = 0 such that Finally, we recall that, in [16], the second author of the present paper showed that the second fundamental form (

Conformal semi-invariant Riemannian maps
We present the following definition for conformal semi-invariant Riemannian maps as a generalization of CR-submanifolds and semi-invariant Riemannian maps. Definition 3.1. Let F be a conformal Riemannian map from a Riemannian manifold (M 1 , g 1 ) to an almost Hermitian manifold (M 2 , g 2 , J). Then we say that F is a conformal semi-invariant Riemannian map at p ∈ M if there is a subbundle The following examples are our motivation to introduce and study conformal semi-invariant Riemannian maps. The theory of CR-submanifolds has been studied widely in differential geometry, however this subject is an active research area of differential geometry, see for instance [18].   This proposition is obvious from [16,Theorem 5.2], and therefore we omit its proof.
As an application of Theorem 3.1, we give the following example of proper conformal semi-invariant Riemannian map.

Example 3.4. Consider the map
which is the composition of the conformal submersion π : (R 5 , g = e x5 (dx 2 followed by the CR-immersion It is easy to verify that F is a conformal semi-invariant Riemannian map with λ 2 = e −x5 with respect to the compatible almost complex structure J on R 4 . Let F be a conformal semi-invariant Riemannian map from a Riemannian manifold (M 1 , g 1 ) to an almost Hermitian manifold (M 2 , g 2 , J). Then for F * (X) ∈ Γ(range F * ), X ∈ Γ((ker F * ) ⊥ ), we write JF * (X) = φF * (X) + ωF * (X), (3.1) where φF * (X) ∈ Γ(D 1 ) and ωF * (X) ∈ Γ(JD 2 ). On the other hand, for V ∈ Γ((range F * ) ⊥ ), we have JV = BV + CV, where BV ∈ Γ(D 2 ) and CV ∈ Γ(µ). Here µ is the complementary orthogonal distribution to ω(D 2 ) in (range F * ) ⊥ . It is easy to see that µ is invariant with respect to J. For the geometry of the leaves of D 1 , we have the following.
In a similar way, we get the following theorem for D 2 .
We now investigate the geometry of the leaves of (range F * ) and (range F * ) ⊥ . First, we give the following result.

Theorem 3.4. Let F be a conformal semi-invariant Riemannian map from a
Riemannian manifold (M 1 , g 1 ) to a Kähler manifold (M 2 , g 2 , J). Then any two conditions below imply the third: (i) (range F * ) defines a totally geodesic foliation on M 2 .
(ii) F is a horizontally homothetic conformal Riemannian map.
In a similar way, we obtain the following theorem: Now, we give necessary and sufficient conditions for a conformal semi-invariant Riemannian map to be totally geodesic. We recall that a differentiable map F between Riemannian manifolds (M 1 , g 1 ) and (M 2 , g 2 ) is called a totally geodesic map if (∇F * )(X, Y ) = 0 for all X, Y ∈ Γ(T M 1 ).
In the sequel we are going to investigate the harmonicity of conformal semiinvariant Riemannian maps. We first have the following general result.