Clairaut Submersion

In this chapter, we give the detailed study about the Clairaut submersion. The fundamental notations are given. Clairaut submersion is one of the most interesting topics in differential geometry. Depending on the condition on distribution of submersion, we have different classes of submersion such as anti-invariant, semi-invariant submersions etc. We describe the geometric properties of Clairaut anti-invariant submersions and Clairaut semi-invariant submersions whose total space is a Kähler, nearly Kähler manifold. We give condition for Clairaut anti-invariant submersion to be a totally geodesic map and also study Clairaut anti-invariant submersions with totally umbilical fibers. We also give the conditions for the semi-invariant submersions to be Clairaut map and also for Clairaut semi-invariant submersion to be a totally geodesic map. We also give some illustrative example of Clairaut anti-invariant and semi-invariant submersion.

In 1735, A.C. Clairaut [25] obtained the very important result in the theory of surfaces, which is Clairaut's theorem and stated that for any geodesic α on a surface of revolution S, the function r sin θ is constant along α, where r is the distant from a point on the surface to the rotation axis and θ is the angle between α and the meridian through α. Bishop [26] introduced the idea of Riemannian submersions and gave a necessary and sufficient conditions for a Riemannian submersion to be Clairaut. Allison [27] considered Clairaut semi-Riemannian submersions and showed that such submersions have interesting applications in the static space-times.
Kähler manifolds [32,33] have an especially rich geometric and topological structure because of Kähler identity. Kähler manifolds are very important in differential geometry, which has applications in several different fields such as supersymmetric gauge theory and superstring theory in theoretical physics, signal processing in information geometry. The simplest example of Kähler manifold is a complex Euclidean space ℂ n with the standard Hermitian metric.
Nearly Kähler manifolds introduced by Gray and Hervella [32], are the geometrically interesting class among the sixteen classes of almost Hermitian manifolds. The geometrical meaning of nearly Kähler condition is that the geodesics on the manifolds are holomorphically planar curves. Gray [2] studied nearly Kähler manifolds broadly and gave example of a non-Kählerian nearly Kähler manifold, which is 6-dimensional sphere.
Motivated by this, the authors [34] studied Clairaut anti-invariant submersions from nearly Kähler manifolds onto Riemannian manifolds with some examples and obtained conditions for Clairaut Riemannian submersion to be totally geodesic map. The authors investigated conditions for the Clairaut anti-invariant submersions to be a totally umbilical map. The authors [34] studied Clairaut semi-invariant submersions from Kähler manifolds onto Riemannian manifolds with some examples. The authors also obtained conditions for Clairaut semi-invariant Riemannian submersion to be totally geodesic map and investigated conditions for the semi-invariant submersion to be a Clairaut map.

Almost complex manifold
An almost complex structure on a smooth manifold M is a smooth tensor field φ of type 1, 1 ð Þ such that φ 2 ¼ ÀI. A smooth manifold equipped with such an almost complex structure is called an almost complex manifold. An almost complex manifold M, φ ð Þendowed with a chosen Riemannian metric g satisfying for all X, Y ∈ TM, is called an almost Hermitian manifold. An almost Hermitian manifold M is called a nearly Kähler manifold [2] if then M is known as Kähler manifold [33]. Every Kähler manifold is nearly Kähler but converse need not be true. i. π has maximal rank.

Riemannian submersion
ii. The differential π * preserves the lengths of horizontal vectors.
For each q ∈ N, π À1 q ð Þ is an m À n ð Þ-dimensional Riemannian submanifold of M. The submanifolds π À1 q ð Þ, q ∈ N, are called fibers. A vector field on M is called vertical if it is always tangent to fibers. A vector field on M is called horizontal if it is always orthogonal to fibers. A vector field X on M is called basic if X is horizontal and π-related to a vector field X 0 on N, that is, π * X p ¼ X 0 π * p ð Þ for all p ∈ M: We denote the projection morphisms on the distributions kerπ * and kerπ * ð Þ ⊥ by V and H, respectively. The sections of V and H are called the vertical vector fields and horizontal vector fields, respectively. So The second fundamental tensors of all fibers π À1 q ð Þ, q ∈ N gives rise to tensor field T and A in M defined by O'Neill [1] for arbitrary vector field E and F, which is where V and H are the vertical and horizontal projections.
To discuss geodesics, we need a linear connection. We denote the Levi-Civita connection on M by∇ and the adapted connection of the submersion by ∇. From Eqs. (3) and (4), we have It is easily seen that for p ∈ M, U ∈ V p and X ∈ H p the linear operators are skew-symmetric, that is, for all E, F ∈ T p M: We also see that the restriction of T to the vertical distribution Tj kerπ * Âkerπ * is exactly the second fundamental form of the fibers of π. Since T U is skew-symmetric, therefore π has totally geodesic fibers if and only if T 0.
Let π : M, g m À Á ! N, g n À Á be a smooth map between Riemannian manifolds. Then the differential π * of π can be observed as a section of the bundle Hom TM, π À1 TN ð Þ!M, where π À1 TN is the bundle which has fibers π À1 TN ð Þ x ¼ T f x ð Þ N. Hom TM, π À1 TN ð Þhas a connection ∇ induced from the Riemannian connection ∇ M and the pullback connection ∇ N [36,37]. Then the second fundamental form of π is given by We also know that π is said to be totally geodesic map [36]

Clairaut submersion from Riemannian manifold
Let S be a revolution surface in  3 with rotation axis L. For any p ∈ S, we denote by r p ð Þ the distance from p to L. Given a geodesic α : J ⊂  ! S on S, let θ t ð Þ be the angle between α t ð Þ and the meridian curve through α t ð Þ, t ∈ I. A well-known Clairaut's theorem [25] named after Alexis Claude de Clairaut, says that for any geodesic on S, the product r sin θ is constant along α, i.e., it is independent of t. For proof, see [38, p.183]. In the theory of Riemannian submersions, Bishop [26] introduced the notion of Clairaut submersion in the following way: is called a Clairaut submersion if there exists a positive function r on M, which is known as the girth of the submersion, such that, for any geodesic α on M, the function r ∘ α ð Þsin θ is constant, where, for any t, θ t ð Þ is the angle between _ α t ð Þ and the horizontal space For further use, we are stating one important result of Bishop.
Bishop also gave the following necessary and sufficient condition for a Riemannian submersion to be a Clairaut submersion, which is Theorem 1.2 [26] Let π : M, g ð Þ! N, g n À Á be a Riemannian submersion with connected fibers. Then, π is a Clairaut submersion with r ¼ e f if and only if each fiber is totally umbilical and has the mean curvature vector field H ¼ Àgradf , where gradf is the gradient of the function f with respect to g.
Proof: Let π : M ! N be a Riemannian submersion. For a geodesic h in M, we Let θ s ð Þ be the angle between _ h s ð Þ and the horizontal space at h s ð Þ. Then Differentiating (12), we get Using Theorem 1.1, (13) becomes Since T U is skew-symmetric, so form above equation, we have Now, π is a Clairaut submersion with r ¼ e f if and only if d ds e f ∘ h sinθ À Á ¼ 0: Using (12,15) in d ds e f ∘ h sinθ Consider any geodesic h on M with initial vertical tangent vector, so gradf turns out to be horizontal. Therefore, the function f is constant on any fiber, the fibers being connected. Therefore (17) reduces to where U 1 , U 2 are vertical vector fields and using the fact that T is symmetric for vertical vector fields, we obtain holds for all vertical vector fields U 1 , U 2 :.
Since the restriction of T to the vertical distribution Tj kerπ * Âkerπ * is exactly the second fundamental form of the fibers of π: It means that any fiber is totally umbilical with mean curvature vector field H ¼ Àgradf : Conversely, suppose the fibers are totally umbilic with normal curvature vector field H ¼ Àgradf so that we have Since gradf is orthogonal to fibers, so Since (18) holds. so r ∘ h ð Þsinθ is constant along any geodesic h. Example 1.1 [24] Consider the warped product manifold The fibers of the first projection p 1 : M 1 Â f M 2 ! M 1 are totally umbilical with mean curvature vector field

Anti-invariant Riemannian submersion
Þbe an almost Hermitian manifold and N be a Riemannian manifold with Riemannian metric g n . Suppose that there exists a Riemannian submersion π : M ! N, such that the vertical distribution kerπ * is anti-invariant with respect to φ, i.e., φkerπ * ⊆ kerπ ⊥ * . Then, the Riemannian submersion π is called an anti-invariant Riemannian submersion. We will briefly call such submersions as anti-invariant submersions.
Let π be an anti-invariant Riemannian submersion from nearly Kähler manifold M, φ, g m À Á onto Riemannian manifold N, g n À Á . For any arbitrary tangent vector fields U and V on M, we set where P U V, Q U V denote the horizontal and vertical part of ∇ U φ ð ÞV, respectively. Clearly, if M is a Kähler manifold then P ¼ Q ¼ 0.
If M is a nearly Kähler manifold then P and Q satisfy Consider where μ is the complementary distribution to φkerπ * in kerπ * ð Þ ⊥ and φμ ⊂ μ.

Anti-invariant Clairaut submersions from nearly Kähler manifolds
In this section, we give new Clairaut conditions for anti-invariant submersions from nearly Kähler manifolds after giving some auxiliary results.
Proof: Let π be an anti-invariant submersion from a nearly Kähler manifold Taking the covariant derivative of this and using (2), we have Since U s ð Þ and X s ð Þ are the vertical and horizontal parts of the tangent vector Using (5)- (8) in (29), we get By (31), we have since P and Q are skew-symmetric, so Using (32) and equating the vertical and horizontal part of (30), we obtain Now, π is a Clairaut submersion with r ¼ e f if and only if d ds e f sin θ Therefore by using (36), we get the result. Theorem 1.5 [34] Let π be an Clairaut anti-invariant submersion from a nearly Kähler manifold M, φ, g ð Þonto a Riemannian manifold N, g n À Á with r ¼ e f . Then for X ∈ kerπ * ð Þ ⊥ , W ∈ kerπ * and φW is basic. Proof: Let π be an anti-invariant submersion from a nearly Kähler manifold M, φ, g ð Þonto a Riemannian manifold N, g n À Á with r ¼ e f . We know that any fiber of Riemannian submersion π is totally umbilical if and only if for all V, W ∈ Γ kerπ * ð Þ, where H denotes the mean curvature vector field of any fiber in M. By using Theorem 1.2 and (37), we have Let X ∈ μ and V, W ∈ Γ kerπ * ð Þ, then by using (1) and (2), we have By using (1), we have where Y, Z ∈ TM. Taking covariant derivative of above, we get using (23), we get Using (40), we have Using (5), (38), (41) in (39), we have Since φW is basic, so because A is skew-symmetric. By using (42), we get the result. Theorem 1.6 [34] Let π be a Clairaut anti-invariant submersion from a nearly Kähler manifold M, φ, g ð Þonto a Riemannian manifold N, g n À Á with r ¼ e f and gradf ∈ φkerπ * . Then either f is constant on φkerπ * or the fibers of π are 1-dimensional.
Proof: Using (5) and (38), we have By use of (1) and (23) in (43), we get By using (5), we obtain Now, using (38), we get Interchange V with W in (45), we have By (45) and (46), we have Therefore either f is constant on φkerπ * or V ¼ aW, where a is constant (by using Schwarz's Inequality for equality case). Corollary 1.1 [34] Let π be a Clairaut anti-invariant submersion from a nearly Kähler manifold M, φ, g ð Þonto a Riemannian manifold N, g n À Á with r ¼ e f and gradf ∈ φkerπ * . If dim kerπ * ð Þ> 1, then the fibers of π are totally geodesic if and only if A φW φX þ Q W φX ¼ 0 for W ∈ kerπ * such that φW is basic and X ∈ μ.
Proof: By Theorem 1.5 and Theorem 1.6, we get the result. Corollary 1.2 [34] Let π be an Clairaut Lagrangian submersion from a nearly Kähler manifold M, φ, g ð Þonto a Riemannian manifold N, g n À Á with r ¼ e f . Then either the fibers of π are 1-dimensional or they are totally geodesic.
Consider the Koszul formula for Levi-Civita connection ∇ for  4 for all X, Y, Z ∈  4 . By simple calculations, we obtain ∇ e i e j ¼ 0 for all i, j ¼ 1, 2, 3, 4: Hence Therefore fibers of π are totally geodesic. Thus π is Clairaut trivially.
ii. Consider a map π : Then by direct calculations, we have Thus, we can say that π is an anti-invariant Riemannian submersion. Since the fibers of π are 1-dimensional, therefore fibers are totally umbilical. By using Koszul formula, we obtain ∇ e i e j ¼ 0 for all i, j ¼ 1, 2, 3, 4: Hence Now, for the function f ¼ ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the gradient of f with respect to g is given by k k 2 gradf . By using Theorem 1.2, we can say that π is an proper Clairaut Remark: From all results of this section, we can easily find conditions for anti-invariant Clairaut Submersions from Kähler manifolds.

Semi-invariant Clairaut submersions from Kähler manifolds
In this section, we give new Clairaut conditions for semi-invariant submersions from Kähler manifolds after giving some auxiliary results.
Proof: Let π be a semi-invariant submersion from a Kähler manifold M, φ, g ð Þ onto a Riemannian manifold N, g n À Á . Since φ 2 _ h ¼ À _ h. Taking the covariant derivative of this and using (2), we have Since U s ð Þ and X s ð Þ are the vertical and horizontal parts of the tangent vector field _ h s Using (5)- (8) in (51), we get Equating the vertical and horizontal part of (52), we obtain Consider a map π :  6 , φ, g À Á !  3 , g 1 À Á defined by Then by direct calculations, we have Author details