Ricci Soliton of CR -Warped Product Manifolds and Their Classiﬁcations

: In this article, we derived an equality for CR -warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a CR -warped product submanifolds in K¨ a hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classiﬁcation of CR -warped product submanifolds in the K¨ a hler manifold by using the Euler–Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor.


Introduction
The evolution equation for a one-parameter family of a Riemannian metric g is characterized

∂(g(t)) ∂t
= −Ric(g(t)), where Ric indicates the Ricci curvature.Equation ( 1) is known as the Ricci flow Equation [1].The Ricci flow equation is a nonlinear partial differential equation that is highly nonlinear and weakly parabolic.This is strictly parabolic to the group of the diffeomorphism of smooth manifold M, termed as a Gauge group [1][2][3] which has several applications in quantum physics, particle physics, and general relativity.Gauge groups such as neutrinos and leptons characterize numerous standard models in particle physics.The Yang-Mills, general relativity, and electromagnetism are our greatest theories of nature.Each is based on a gauge symmetry at its foundation.The concept of a gauge symmetry is undoubtedly complicated.However, at its core, it is just an ambiguity in the words we use to explain physics.Why should nature enjoy such uncertainty?Understanding nature as a redundant set of variables is helpful for two reasons.First, although gauge symmetry makes our explanation of physics redundant, it seems concise.Ricci flow has several applications in relativity and physics [4].The fixed point of Ricci flow is known as a Ricci soliton (for more details, see [5]), which is characterized by the following relation where L X denotes the Lie derivative and λ ∈ R can be any constant.The nature of Ricci soliton depends on the scalar λ.If λ > 0, λ < 0 and λ = 0, thus the Ricci soliton is known as expanding, shrinking and steady, respectively.In the relation (2), if we replace vector field X by gradient of some smooth function f then Equation ( 2) is transformed as follows: where ∇ 2 f is a Hessian of f .Equation ( 3) is known as gradient Ricci soliton equation, and is equivalent to The Equation ( 4) is the fundamental equation relating the Ricci tensor and Hessian tensor.As a special case, if f is a constant function on a smooth manifold M admitting gradient Ricci soliton, then (3) reduces to Ric = λg.This expression leads to M as an Einstein manifold.
Furthermore, the geometry of warped products has significant use in mathematics and physics.In physics, several solutions of the Einstein field equation have warped product structures, for example, Schwarzschild's solution and Friedmann-Lemaitre-Robertson-Walker's solution.In string theory, the RS model has a significant role which is a fivedimensional anti-de Sitter warped product manifold.Such manifolds were first realized in 1969 by R. L. Bishop and B. O'Neill when they studied manifolds of negative curvature.They proved that there does not exist any Riemannian product manifold whose curvature is negative.After that, several authors studied warped product manifolds under different circumstances.This was taking more attention at the beginning of the twenty-first century, when authors such as [6][7][8][9][10][11][12][13][14][15][16] studied relevant topics and discussed some geometric properties related to singularity theory and submanifolds theory, etc.In 2002, B. Y. Chen derived the existence of the CR-warped product of the form M T × f M ⊥ in a K ähler manifold, where M T is real submanifold and M ⊥ is a complex submanifold.After that, numerous geometers studied CR-warped product manifolds and their generalized classes in different ambient spaces [6,7,[17][18][19][20].
The warped product manifold denoted by M × f N is a product of Riemannian manifolds M 1 and M 2 , which furnished a Riemannian metric g gratifying where g M and g N are the Riemannian metric of the smooth manifolds M and N, respectively, and f : M −→ (0, ∞) bis a smooth function known as a warping function [21].If M × f N is a warped product manifold, then the following relations hold in From the above relation, we deduce that M and N are totally geodesic and totally umbilical manifolds in M × f N.
Recently, the Ricci soliton of warped products is taking more attention from the geometers.A Ricci soliton with warped product structure and gradient Ricci soliton with warped product structure were classified by different authors [22][23][24][25][26][27][28].The author of [23] derived that if warped product manifold admits a gradient Ricci soliton, then the fiber is necessarily Einstein, and the potential function depends on the base manifold.In [24], the authors considered the Ricci soliton of a warped product.They derived useful results for such a manifold and applied them to the different spacetime models.Recently, the authors of [29] derived some useful results from Sasakian manifolds that admit an almost -Ricci soliton structure, and the authors of [30] extended a -Ricci soliton to a -η-Ricci soliton in Kenmotsu manifold.Furthermore, the Ricci curvature of warped products is utilized in string theory and the general theory of relativity.In the well-known Einstein's field equations (for more details, see [31]), the Ricci tensor establishes the connection to the matter distribution in the universe.Furthermore, the Ricci tensor is a part of the curvature of spacetime, which represents gravity's general relativity and also examines the degree to which matter will tend to converge or diverge concerning time.More generally, the Riemannian curvature is not more important than the Ricci curvature in physics.Suppose the solution of the Einstein field equation is a Ricci flat Riemannian (pseudo-Riemannian) manifold.In that case, it indicates that the cosmological constant is zero (for more details, see [32][33][34]).Due to the huge application of the Ricci soliton or, more generally, the Ricci tensor in physics, we are motivated to study the Ricci soliton of CR-warped product submanifold in a complex space form.We consider the question and provide a partial answer to it.The question is that what are necessary and sufficient conditions for warped product immersions in complex space forms to be an Einstein warped product manifold with the impact of a gradient Ricci soliton?
This article is arranged as follows: Second 2 includes some necessary information about the K ähler manifold and its submanifolds.In Section 3, we derived an equality for a CR-warped product in a complex space form, forming the relationship between the gradient and Laplacian of the warping function and second fundamental form.The vector field X is considered as the gradient of the warping function in the relation (2) and derived the condition for such warped product to be Einsteinian and also derived some useful results in this article as applications of Theorem 1 and Lemma 2 into the Euler-Lagrange equation, in the Dirichlet energy and in the Ricci curvature in Section 4.

Preliminaries
From the well-known literature of complex geometry, an almost Hermitian manifold is a smooth manifold that admits an almost complex structure J and Hermitian metric g satisfying for every U 1 , U 2 ∈ Γ(T M 2n ), Γ(T M 2n ) and I indicate for the section of tangent bundle M 2n and identity transformation, respectively, (see, [6,[35][36][37]).The metric g is skew-symmetric Definition 1.A K ähler manifold [6,[35][36][37] is almost a Hermitian manifold, M 2n satisfies Here, ∇ indicates the Levi-Civita connection on M 2n .
Moreover, if the holomorphic sectional curvature of a non-flat K ähler manifold is constant, then M 2n is termed as a complex space form.In this article, we denote M(c) for a complex space form.The the curvature tensor R of M(c) is characterized by where c is holomorphic sectional curvature.Let us assume that M is an m-dimensional Riemannian submanifold of a K ähler manifold M 2n .Let us denote Γ(TM) for the section of the tangent bundle of M and Γ(TM ⊥ ) for the set of all normal vector fields of M, respectively, and also ∇ for the Levi-Civita connection on tangent bundle TM, ∇ ⊥ for the Levi-Civita connection on normal bundle TM ⊥ , respectively.Thus, the Gauss and Weingarten formulas are described as follows: ∇ for all U 1 , U 2 ∈ Γ(TM) and ξ ∈ Γ(TM ⊥ ), where A ξ and h are shape operator and second fundamental form, aided by The submanifold M is totally umbilical [6,7,19,20] if where H is the mean curvature vector described by H = 1 m trace(h).The covariant derivative of σ is computed by the following relation The following relation characterizes the Gauss and Codazzi equations for every where tU 1 = tan(J U 1 ) (resp.t ξ = tan(J ξ)) and nU 1 = nor(J U 1 ) (resp.n ξ = nor(J ξ)) are the tangential and normal parts of J U 1 (resp.J ξ), and tan and nor are orthogonal projections on TM and TM ⊥ .With the help of (9), ( 18) and (19), we have Furthermore, the covariant derivative of J , t and n are described by . By the utilization of ( 8), ( 10), ( 12), ( 13), ( 18), ( 19) and ( 22)-( 24), we obtain for every U 1 , U 2 ∈ Γ(TM).Consider a smooth function f : M −→ R, thus, the gradient and Laplacian are described by for any U 1 ∈ Γ(TM).The relation (28) can be expressed as Let H f be the Hessian of f , thus the Laplacian and Hessian are ailed by We will recall the above results for later use.

Curvature Inequality
In [36], Chen derived the general inequality for CR-warped product submanifold M T × f M ⊥ of K ähler manifold which forms a connection between the gradient of the warping function and second fundamental form via the accompanying relation where β = dim(M ⊥ ).The above relation establishes a relationship between intrinsic invariant and extrinsic invariant.He also derived the classification of such types of warped products by the solution of a special system of partial differentials in a complex space form.After some time, he derived the curvature type inequality for a CR-warped product submanifold M T × f M ⊥ in a complex space theorem which is expressed as the following: where ∆(ln f ) denotes the Laplacian of ln f .
By the inequality (1), he classified CR-warped product manifolds in complex Euclidean space satisfying the equality case of (1) up to rigid motion by the partial Serge embedding defined as φ pk a : C k * × S p −→ C ap+k , where C F * = C ∼ {0} and a, p, k ∈ N.With the help of Hopf fibration, he derived some conditions for CR-warped products in complex projective space CP n (4) and in complex hyperbolic space CH n (−4) to satisfy the equality sign in (1) (for more details, see Theorem 5.1 [35]).Thereafter, several authors studied CR-warped products in different ambient manifolds.Definition 2. Let M be a Riemannian submanifold of a K ähler manifold M 2n .Then M is real submanifold if J (TM) ⊂ TM and M is complex submanifold if J (TM) ⊂ TM ⊥ .Definition 3. A CR-submanifold of K ähler manifold M 2n whose tangent bundle decomposed as TM = D ⊕ D ⊥ , where D is a real distribution and D ⊥ is a complex distribution.Moreover, if there is a Riemannian metric on M of the form g Example 1.Let us consider 10-dimensional Euclidean space R 10 with coordinate (x 1 , • • • , x 5 , y 1 , • • • , y 5 ) and Euclidean metric g.An almost complex structure on R 10 is defined by Consider a subset M ⊂ R 10 immersed as a submanifold by the following immersion The basis spans the tangent subspace of M at each point With straightforward computation, we observed that the distribution spanned by {Z u , Z v , Z w , Z δ } and distribution spanned by Z θ are invariant and anti-invariant distribution, respectively.This shows that M is a warped product manifold with warping function Now, we recall one lemma related to CR-warped product of K ähler manifold for further use: Proof.By the application of (17), we have By the definition of (15), we obtain (∇ . By the application of covariant differentiation property into the last expression, we have With the help of Definition 10, ( 12), ( 13) and ( 14) and (39), the above expression takes the following form Similarly, we have From ( 45)-( 47), we have By using the fact that M is a CR-warped product submanifold in M(c) then, we observe that m = 2α + β and 2n } be a basis of ν.Using above frame in the relation of (48), we find 2α By the virtue of ( 40) and (41), we have 2α By the utilization of ( 49) and (50), we have (43).Since the Laplacian of some functions is a trace of Hessian, by using this fact, we obtain (44).
Then the second fundamental form satisfies Proof.By the direct use of ( 30) into (32), we obtain (51).

Applications
The study of curvatures in differential geometry and physics has great importance.The curvatures of the Riemannian (or pseudo-Riemannian) manifold are determined intrinsically and extrinsically.In curvatures, the Ricci curvature and scalar curvature are more applicable in physics.The Ricci curvature Ric and scalar curvature ρ of M is defined as where K(E q ∧ E s ) is the sectional curvature of the plane spanned by E q and E s .Let G k be the k-plane section of TM spanned by the orthonormal basis In the potential theory, Dirichlet energies have significant use.If f : M −→ R is a smooth function, then the Dirichlet energy is defined by where E( f ) and dV indicate Dirichlet energy and volume element, respectively.If R is replaced by a smooth manifold, then the Sigma model evaluates the Dirichlet energy.The Lagrange equation of the Sigma model has solutions that provide extreme Dirichlet energies.
In Lagrangian mechanics, the Lagrangian L of a mechanical system is T − V, where T is kinetic energy and V is the system's potential energy, respectively.As a generalization to smooth manifolds, the Lagrangian of the smooth function f is determined by The Euler-Lagrange equation for a Lagrangian L is ∆ f = 0. Now, we recall some useful results for further use: ).Let M be a compact, connected Riemannian manifold without boundary and f be a smooth function on M such that ∆ f ≥ 0 (∆ f ≤ 0).Then f is a constant function.
Moreover, if we apply the Green Theorem on a compact orientable Riemannian manifold without boundary, then we obtain where div(X) indicates the divergence [7] for a connected, compact Riemannian manifold with a boundary.The well-known Hopf lemma takes the following form: ).Let M be a compact, connected Riemannian manifold with boundary and f be a smooth function on M such that ∆ f = 0 on M and f ∂M = 0, then f = 0.
The above theorem is also known as the uniqueness theorem of the Dirichlet problem.The Hamiltonian, denoted by H, represents the mechanical system's total energy.On the smooth even-dimensional manifold, the Hamiltonian induces a symplectic structure.The Hamiltonian [7] on the manifold is characterized by Proof.Suppose that ln f satisfies the Euler-Lagrange equation, i.e., ∆ f = 0.This implies that Since h 2 ≥ cαβ + 2β ∇(ln f ) 2 − β∆(ln f ), therefore using the above equation, we have Using the given relation (60), the above expression is reduced to ∇(ln f ) 2 ≤ 0. However, ∇(ln f ) 2 is always positive.Therefore, we must have ∇(ln f ) = 0.This implies that f is constant; thus, the warped product is a Riemannian CR-product.This accomplishes the proof.

Corollary 1.
Let M = M T × f M ⊥ be a CR-warped product in M(c) and warping function is a solution of Euler-Lagrange equation, then M is a Riemannian CR-product Proof.By the direct application of (43) and proceeding same steps as Theorem 4, we achieve (63).
Proof.By taking integration of (32), we have Now, utilizing the relations of (57) into Equation (65), we receive By the application of ( 64) and (66), we observe M β ∇(ln f ) 2 dV ≤ 0. This implies that ∇(ln f ) 2 is negative, i.e., ∇(ln f ) 2 ≤ 0. The last expression leads to f being a constant.This completes the proof.Theorem 6.Let M = M T × f M ⊥ be a compact, orientable CR-warped product submanifold in M(c) such that ∂M = φ.Then M is a Riemannian CR-product if and only if Proof.By taking integration of (43), we have Now utilizing the relations (55) into Equation (68), we receive By the virtue of ( 69) and (67), we have M ∆(ln f )dV = 0.By applying the Hopf lemma, we achieve that f is constant.
Proof.By the utilization of (59) into (43), we have 2α By virtue of (71), we observe that the relation (70) holds if and only if ∆(ln f ) = 0. Since M is a compact, orientable Riemannian manifold, then by the application of Hopf lemma, we have f as constant.This completes the proof.

Application to Gradient Ricci Soliton
The Ricci soliton is a natural generalization of Einstein manifolds.Such manifolds are important to study warped product manifolds because any regular surface is Einsteinian, and the surface of revolution is a warped product manifold.Moreover, they apply to each other in a more general setting, which can be realized in current times.Another generalization of the Einstein manifold is an almost Ricci soliton and quasi-Einstein manifold.In this paper, we derive some characterization of Einstein manifolds with the impact of Ricci soliton.Theorem 8. Let M = M T × f M ⊥ be a CR-warped product in M(c) admitting a shrinking gradient Ricci soliton.Then the following inequality holds Additionally, the equality holds if M T is a totally geodesic submanifold and M ⊥ is totally umbilical submanifolds of M(c).
Proof.If M = M T × f M ⊥ is a CR-warped product submanifold of a complex space form M(c) admitting a shrinking gradient Ricci soliton.Then, M fulfils the following relation for every X 1 , X 2 ∈ Γ(TM T ).By using the above defined frame for M T in (73), we have By replacing E q by J E q into (74), we have By the consequence of ( 74) and (75), we have Applying (76) into the relation (51), we have This accomplishes the proof.
Theorem 9. Let M = M T × f M ⊥ be a CR-warped product in M(c) admitting a shrinking gradient Ricci soliton.Then M is Einsteinian if the following equality holds Proof.By the virtue of (44), ( 75) and (76), we receive that Suppose (78) satisfies in M then from (79), we have ∇(ln f ) 2 = 0. From the above expression, we receive that ∇(ln f ) = 0. Therefore f is constant.Therefore, by the gradient Ricci soliton equation, we easily observe that M is Einstein's warped product.This finishes the proof.
Corollary 2. Let M = M T × f M ⊥ be a CR-warped product in C n admitting a shrinking gradient Ricci soliton.Then M is Einsteinian if the following equality holds Proof.Since C n is flat space, therefore c = 0, and by the direct application of (78), we obtain the result.
Theorem 10.Let M = M T × f M ⊥ be a CR-warped product in M(c) admitting a steady gradient Ricci soliton.Then M is Einsteinian if the following equality holds Ric(E q , E q ).(81) Proof.For the steady gradient Ricci soliton λ = 0, we achieve the result by proceeding with similar steps as the proof of Theorem 9.
Theorem 11.The necessary condition for a compact CR-warped product submanifold M = M T × f M ⊥ in M(c) to be a CR-product is that Proof.Since warping function f is smooth, ln f = τ is also a smooth function, applying τ to the well known Ricci identity, we have for any vector fields X 1 , X 2 , X 3 that are tangent to M T .Because f is a smooth function and , then the curvature tensor behaves like the derivative defined by R X With the help of the property that τ is closed we can easily prove ∇ 2 d(τ)(X 2 , X 1 , X 3 ) = ∇ 2 d(τ)(X 1 , X 2 , X 3 ).Now, for an orthonormal frame {E 1 , E 2 , • • • , E 2α } on M T , we have ∇ E i E j (p) = 0, for fixed point p ∈ M T .If we describe ∇ E i X 1 = 0 for any X 1 ∈ Γ(TM T ), and considering the trace with respect to X 2 and X 3 in the relation ∇ 2 d(τ)(X 2 , X 1 , X 3 ) = ∇ 2 d(τ)(X 1 , X 2 , X 3 ), then using (83), we concede that div(Hess τ ) = Ric(∇(τ), −) − d(∆(τ)). ( Since M is a compact CR-warped product manifold with boundary then by taking the integration of (84), we have

) 4 . 1 .Theorem 4 .
Application to Euler-Lagrange Equation Let M = M T × f M ⊥ be a CR-warped product in M(c) and warping function is a solution of the Euler-Lagrange equation, then M is a Riemannian CR-product if h 2 ≥ cαβ.
orientable CR-warped product submanifold in M(c) such that ∂M = φ.Then the M is a Riemannian CR-product if and only if Hamiltonian H satisfies