From 21th Century, many authors have investigated real hypersurfaces in Hermitian symmetric spaces with rank 1 or rank 2 of compact type. For the case of rank 2, real hypersurfaces in the complex two-plane Grassmannians G2(Cn+2) or in the complex quadric Qn were extensively studied by many authors (Lee-Suh [19],[20] and [21], Pérez [26], Pérez-Suh [27], Pérez-Suh-Watanabe [28], Suh [32], [33], [34] and [35], and Suh-Hwang-Woo [36]).

Motivated by the study of rank 2, in the class of Hermitian symmetric spaces with rank 1 of compact type, we can give the example of the complex projective space CPn=SUn+1/S(U1·Un) (see Kobayashi-Nomizu [17]). It is geometrically different from the case of rank 2, which has a Kähler structure and a Fubini-Study metric g of constant holomorphic sectional curvature 4 (see Cecil-Ryan [7], Djorić-Okumura [12], Romero [29], [30], and Smyth [31]).

Recently, Yamabe solitons and Ricci solitons on almost co-Kähler manifolds and three dimensional N(k)-contact manifolds have been investigated by Chaubey-De-Suh [9] and [11]. Moreover, the study of the Yamabe flow was initiated in the work of Hamilton [14], Morgan-Tian [22] and Perelman [25] as a geometric method to construct Yamabe metrics on Riemannian manifolds.

Let g(t) be a Riemannian metric which is time dependent on a Riemannian manifold M. It is said to be evolved by the Yamabe flow if the metric g satisfies

on M, where γ denotes the scalar curvature on M. From such a view point, in this paper we want to give a complete classification of Yamabe solitons and gradient Yamabe solitons on Hopf real hypersurfaces in the complex projective space CPn.

On the other hand, it is well known that there exist two focal submanifolds of real hypersurfaces in Hermitian symmetric spaces of compact type and only one focal submanifold in Hermitian symmetric spaces of non-compact type (see Cecil and Ryan [7] and Helgason [13]). Since the complex projective space CPn is a Hermitian symmetric space of compact type, any real hypersurface has two focal submanifolds (see Djorić-Okumura [12], Pérez [26]). Among them we consider two kinds of real hypersurfaces in CPn with isometric Reeb flow or contact hypersurfaces. In CPn, Cecil-Ryan [7], and Okumura [23] gave a classification of real hypersurfaces with isometric Reeb flow as follows:

Theorem A. Let M be a real hypersurface in the complex projective space CPn, n≥3. Then the Reeb flow on M is isometric if and only if M is an open part of a tube of radius 0<r<π2 around a totally geodesic CPk⊂CPn for some k∈{0,⋯,n-1} or a tube of radius π2-r over CPℓ, where k+ℓ=n-1.

When a real hypersurface M in the complex projective space CPn satisfies the formula Aφ+φA=kφ, k≠0 and constant, we say that M is a contact real hypersurface in CPn. In the papers due to Blair [2] and Yano-Kon [39], they introduce the classification of contact real hypersurfaces in CPn as follows:

Theorem B. Let M be a connected orientable real hypersurface in the complex projective space CPn, n≥3. Then M is a contact real hypersurface if and only if M is congruent to an open part of a tube of radius 0<r<π4 around an n-dimensional real projective space RPn or a tube of radius π4-r over Qn-1, where 0<r<π4.

Motivated by these results, in this paper we give some characterizations of real hypersurfaces in the complex projective space CPn regarding a family of geometric flows. Indeed, we know that a solution of the Ricci flow equation ∂∂tg(t)=-2Ric(g(t)) is given by

where Ω is a constant and LV denotes the Lie derivative along the direction of the vector field V (see Chaubey-Suh-De [10], Jeong-Suh [15], Morgan-Tian [22], Perelman [25], Wang [37] and [38]). Then this solution (M,V,Ω,g) is said to be a Ricci soliton with potential vector field V and Ricci soliton constant Ω.

As a generalization of the Ricci flow, the Ricci-Bourguignon flow (see Bourguignon [3] and [4], Catino-Cremaschi-Djadli-Mantegazza-Mazzieri [6]) is given by

This family of geometric flows with θ=0 reduces to the Ricci flow ∂∂tg(t)=-2Ric(g(t)), g(0)=g0. If the constant θ=12, it is said to be Einstein flow. Its critical point of the Einstein flow

implies that the Einstein gravitational tensor Ric(g(t))-12γg(t) vanishes. For a four-dimensional space time M4, this is equivalent to the vanishing Ricci tensor by virtue of dγ=2div(Ric). In this case, M4 becomes vacuum. That is, g(t)=g(0), the metric is constant along the time (see O'Neill [24]). For θ=1n, the tensor Ric-γng is said to be traceless Ricci tensor, and for θ=12(n-1), it is said to be the Schouten tensor.

Now let us introduce a Ricci-Bourguignon soliton (M,V,Ω,θ,γ,g) which is a solution of the Ricci-Bourguignon flow as follows:

for any tangent vector fields X and Y on M, where Ω is a soliton constant, θ any constant and γ the scalar curvature on M, and LV denotes the Lie derivative along the direction of the vector field V (see Bourguignon [3], [4], and Morgan-Tian [22]). Then (M,g) is said to be a Ricci-Bourguignon soliton with potential vector field V and Ricci-Bourguignon soliton constant Ω.

If the Ricci operator Ric of a real hypersurface M in CPn satisfies

for smooth functions a, b on M, then M is said to be pseudo-Einstein. Then we introduce a complete classification of pseudo-Einstein Hopf real hypersurfaces in the complex projective space CPn due to Cecil-Ryan [7] as follows:

Theorem C. Let M be a pseudo-Einstein real hypersurface in the complex projective space CPn, n≥3. Then M is locally congruent to one of the following:

• (i)

  a geodesic hypersphere,

• (ii)

  a tube of radius r around a totally geodesic CPk, 0<k<n-1, where 0<r<π2 and cot2r=kn-k-1,

• (iii)

  a tube of radius r around a complex quadric Qn-1 where 0<r<π4 and cot22r=n-2.

Let M be a Hopf hypersurface in the complex projective space CPn. Then we have

for the shape operator A with the Reeb function α=g(Aξ,ξ) on M in G2(Cn+2). When we consider a tensor field J for any vector field X on M , which is a Kähler structure on the tangent space TzM, z∈M, then JX is given by

where φX=(JX)T is the tangential component of the vector field JX, η(X)=g(ξ,X), ξ=-JN, and N denotes a unit normal vector field on M.

In this paper we introduce a new notion named generalized pseudo-anti commuting property for the Ricci tensor of a real hypersurface M in the complex projective space CPn as follows:

for a smooth function f on M in CPn (see Ki-Suh [16], and Yano-Kon [39]).

It is known that Einstein and pseudo-Einstein real hypersurfaces M in the complex projective space CPn satisfy the condition of generalized pseudo-anti commuting Ricci tensor, that is, Ricφ+φRic=fφ, where f denotes a smooth function on M in CPn (see Besse [1], Cecil-Ryan [7] and Kon [18]). In Yano-Kon [39], real hypersurfaces of type (B) in the complex projective space CPn, which are characterized by Aφ+φA=kφ, k≠0, also satisfy the formula of generalized pseudo-anti commuting Ricci tensor in (1.2).

Let us define a pseudo Ricci-Bourguignon soliton (M,V,η,Ω,θ,γ,g) as follows:

for any tangent vector fields X and Y on M, where Ω is said to be a pseudo Ricci-Bourguignon soliton constant, the functions θ and ψ are any constants and γ the scalar curvature on M, and LV denotes the Lie derivative along the direction of the vector field V.

When the function ψ identically vanishes, the pseudo Ricci-Bourguignon soliton (M,V,η,Ω,θ,γ,g) is said to be a Ricci-Bourguignon soliton (M,V,Ω,θ,γ,g). We also say that the pseudo Ricci-Bourguignon soliton is shrinking, steady, and expanding according to the pseudo Ricci-Bourguignon soliton constant function Ω>0, Ω=0, and Ω<0 respectively.

Now in this paper, by using the notion of generalized pseudo-anti commuting Ricci tensor (1.2), we give a theorem as follows:

Theorem 1. Let M be a Hopf pseudo Ricci-Bourguignon soliton (M,ξ,η,Ω,θ,γ,g) in the complex projective space CPn, n≥3. Then M is pseudo-Einstein and locally congruent to one of the following:

• (i)

  a geodesic hypersphere, Ω+θγ=2{(n-1)cot2(r)+n}, and ψ=2n,

• (ii)

  a tube of radius r around a totally geodesic CPk, 0<k<n-1, where 0<r<π2, cot2r=kn-k-1, Ω+θγ=2n, and ψ=2.

Let us denote by Df the gradient vector field of the function f on a real hypersurface M in the complex projective space CPn defined by g(Df,X)=g(gradf,X)=X(f) for any tangent vector field X on M. Now let us consider the gradient pseudo Ricci-Bourguignon soliton (M,Df,η,Ω,θ,γ,g). It is a generalization of gradient Einstein soliton derived from a generalized Ricci potential for a Riemannian manifold (M,g) (see Catino-Mazzieri [5], Cernea-Guan [8]). It is defined by

where Hess(f) is defined by Hess(f)=∇Df and for any tangent vector fields X and Y on M

Then a gradient pseudo Ricci-Bourguignon soliton in CPn can be defined by

for any vector field X tangent to M in CPn. Then first by Theorem C and Theorem 1 we can assert a classification theorem of gradient pseudo Ricci-Bourguignon solitons in CPn as follows:

Theorem 2. Let M be a real hypersurface in CPn with isometric Reeb flow, n≥3. If it admits the gradient pseudo Ricci-Bourguignon soliton (M,Df,η,Ω,θ,γ,g), then M is pseudo-Einstein and locally congruent to one of the following

• (i)

  a geodesic hypersphere, Ω+θγ=2{(n-1)cot2(r)+n}, and ψ=2n,

• (ii)

  a tube of radius r around a totally geodesic CPk, 0<k<n-1, where 0<r<π2, cot2r=kn-k-1, Ω+θγ=2n, and ψ=2.

Next by virtue of Theorem B let us consider a contact real hypersurface in the complex projective space CPn. Then we can assert a classification of gradient pseudo Ricci-Bourguignon soliton in CPn as follows:

Theorem 3. Let M be a contact real hypersurface in the complex projective space CPn, n≥3. If it admits the gradient pseudo Ricci-Bourguignon soliton (M,Df,η,Ω,θ,γ,g), then M is pseudo-Einstein and locally congruent to a tube of radius r around a complex quadric Qn-1 where 0<r<π4 and cot2(2r)=n-2. Moreover, the soliton constants are given by Ω+θγ=2n and ψ=2(2n-1).

Let (M¯,g,J) be a Kähler manifold and R¯ the Riemannian curvature tensor of (M¯,g). Since ∇¯J=0, we immediately see that

holds for all X,Y,Z∈Tp(M¯), p∈M¯. From the curvature identities in Kobayashi and Nomizu [17] we also get

Let G2J(TM¯) be the Grassmann bundle over M¯ consisting of all 2-dimensional J-invariant linear subspaces V of TpM¯, p∈M. Thus every V∈G2J(TM¯) is a complex line in the corresponding tangent space of M¯. The restriction of the section curvature function K to G2J(TM¯) is called the holomorphic sectional curvature function on M¯ and K(V) is called the holomorphic sectional curvature of M¯ with respect to V∈G2J(TM¯).

A Kähler manifold M is said to have constant holomorphic sectional curvature if the holomorphic sectional curvature function is constant. Now we want to introduce the following.

Theorem 2.1. A Kähler manifold (M¯,g,J) has constant holomorphic sectional curvature c∈R if and only if its Riemannian curvature tensor R¯ is of the form

for any vector fields X,Y and Z on M¯.

The complex vector space Cn (n∈N) is in a canonical way an n-dimensional complex manifold. For p∈Cn denote by πp:TpCn→Cn the canonical isomorphism. We define a Riemannian metric g on Cn by

for all u,v∈TpCn and p∈Cn, where <·,·> is the real part of the standard Hermitian inner product on Cn, that is,

The metric g is called the canonical Riemannian metric on Cn. The complex structure J on Cn is given by the equation πp(Ju)=iπp(u). It is easy to verify that (Cn,g,J) is a Kähler manifold. In fact, (Cn,g,J) a complex Euclidean space with vanishing constant holomorphic sectional curvature. The Kähler manifold (Cn,g,J) is known to be the n-dimensional complex Euclidean space.

We define an equivalence relation ∼ on Cn+1∖{0} by z1∼z2 if and only if there exists λ∈C∖{0} so that z2=z1λ. We denote the quotient space Cn+1∖{0})/∼ by CPn. By construction, the points in CPn are in one-to-one correspondence with the complex lines through 0∈Cn+1. We equip CPn with the quotient topology with respect to the canonical projection τ:Cn+1∖{0}→CPn. Then CPn is a compact Hausdorff space and τ is a continuous map. There exists a unique complex manifold structure on CPn so that τ is a holomorphic submersion. In this way CPn becomes an n-dimensional complex manifold (CPn,J). For z∈Cn+1∖{0} we also write [z]=τ(z)∈CPn (see Kobayashi and Nomizu [17]).

Let S2n+1 be the unit sphere in Cn+1 and denote by π the restriction of τ to S2n+1. We consider S2n+1 with the Riemannian metric induced from Cn+1, which is the standard metric on S2n+1 turning it into a real space form with constant sectional curvature 1. The map π:S2n+1→CPn is a surjective submersion whose fibers are 1-dimensional circles. There exists a unique Riemannian metric g on CPn so that π becomes a Riemannian submersion. In such a way, the map π:S2n+1→CPn is known as the Hopf map from S2n+1 onto CPn and the Riemannian metric g is known as the Fubini-Study metric on CPn. The manifold (CPn,J,g) is a Kähler manifold and called the n-dimensional complex projective space. The complex projective space (CPn,J,g) is a complex space form with constant holomorphic sectional curvature 4.

By virtue of Theorem 2.1, the Riemannian curvature tensor R¯ of CPn can be given for any vector fields X, Y and Z in Tp(CPn), p∈CPn as follows:

Let M be a real hypersurface in the complex projective space CPn and denote by (φ,ξ,η,g) the induced almost contact metric structure. Note that ξ=-JN, where N is a (local) unit normal vector field of M. Then the vector field ξ is said to be the Reeb vector field on M in CPn. The tangent bundle TM of M splits orthogonally into TM=C⊕Rξ, where C=ker(η) is the maximal complex subbundle of TM. The structure tensor field φ restricted to C coincides with the complex structure J restricted to C, and φξ=0.

In different way, the complex projective space CPn is defined by using the fibration

which is said to be a Riemannian submersion. Then naturally we can consider the following diagram for a real hypersurface in the complex projective space CPn as follows:

We now assume that M is a Hopf hypersurface. Then we have

where A denotes the shape operator of M in CPn and the smooth function α is defined by α=g(Aξ,ξ) on M. When we consider the transformed vector field JX by the Kähler structure J on CPn for any vector field X on M in CPn, we may write

Then by using Kähler structure ∇¯J=0, we get the following

where ∇¯ and ∇ denote the Levi-Civita connections of M¯ and M respectively.

Now we consider the equation of Codazzi

By the equation of Gauss, the curvature tensor R(X,Y)Z for a real hypersurface M in CPn induced from the curvature tensor R¯ in (2.1) of CPn can be described in terms of the almost contact structure tensor φ and the shape operator A of M in CPn as follows:

for any vector fields X,Y,Z∈TzM, z∈M. From this, contracting Y and Z on M in CPn, we get the Ricci operator of a real hypersurface M in CPn as follows:

Then by contracting the Ricci operator in (3.2) the scalar curvature γ of M in CPn is given by

where the function h denotes the trace of the shape operator A of M in CPn.

Putting Z=ξ in the Codazzi equation, we get

Since we have assumed that M is Hopf in CPn, differentiating Aξ=αξ gives

From this, the left side of (3.4) becomes

Putting X=ξ in (3.4) and (3.5) and using the almost contact structure of (M,g), we have

Inserting this formula into (3.4) and (3.5) implies the following for any vector fields X and Y on M

By virtue of this equation, we can assert the following

Lemma 3.1. Let M be a Hopf real hypersurface in CPn, n≥3. Then we have

for any tangent vector field X on M.

By using the formulas given in section 3 we want to introduce an important lemma due to Okumura [23] and Yano-Kon [39] as follows:

Lemma 3.2. Let M be a Hopf real hypersurface in CPn. Then the Reeb function α is constant. Moreover, if X∈C is a principal curvature vector of M with principal curvature λ, then 2λ≠α and φX is a principal curvature vector of M with principal curvature αλ+22λ-α. on M, where C denotes the orthogonal complement of the Reeb vector field ξ on M.

Now by using (3.2) and (3.3), we introduce an important proposition due to Cecil-Ryan [7], Djorić-Okumura [12] as follows:

Proposition 3.3. Let M be the tube of radius 0<r<π2 around the totally geodesic CPk, k∈{1,⋯,n-2} in CPn, which is said to be of type (A2). Then the following statements hold:

• (1)

  M is a Hopf hypersurface.

• (2)

  The principal curvatures and corresponding principal curvature spaces of M are given by

  principal curvature	eigenspace	multiplicity
  λ=cot(r)	Tλ	2ℓ
  μ=-tan(r)	Tμ	2k
  α=2cot(2r)	Tα=RJN	1

  where ℓ=n-k-1.

• (3)

  The shape operator A commutes with the structure tensor field φ as

  Aφ=φA.

• (4)

  The trace h of the shape operator A and its square h2 becomes the following respectively

  h=(2ℓ+1)cot(r)-(2k+1)tan(r),

  h2=(2ℓ+1)2cot2(r)+(2k+1)2tan2(r)-2(2ℓ+1)(2k+1).

• (5)

  The trace of the matrix A2 is given by

  TrA2=(2ℓ+1)cot2(r)+(2k+1)tan2(r)-2.

• (6)

  The scalar curvature γ of the tube M is given by

  γ=4(n-1)n-8kℓ+2(2ℓ+1)ℓcot2(r)+2(2k+1)ktan2(r).

Remark 3.4. For k=0, M is pseudo Einstein, that is, a geodesic hypersphere, which is said to be of type (A1) such that

Now let M be a tube of radius r, 0<r<π4, over the real projective space RPn, which is said to be of type (B) and a contact real hypersurface in the complex projective space CPn. It also can be regarded as a tube of radius π4-r over the complex quadric Qn-1.

The tube of radius r around totally geodesic and totally real projective space RPn has therefore three distinct constant principal curvatures 2tan(2r), -cot(r), and tan(r). It also can be regarded as a tube of radius π4-r over a totally geodesic complex quadric Qn-1. Then by (3.2) and (3.3), we want to give an important proposition due to Cecil-Ryan [7] as follows:

Proposition 3.5. Let M be the tube of radius 0<r<π4 around the complex quadric Qn-1 in CPn. Then the following statements hold:

• (1)

  M is a Hopf hypersurface.

• (2)

  The principal curvatures and corresponding principal curvature spaces of M are

  principal curvature	eigenspace	multiplicity
  λ=-cot(π4-r)	Tλ	n-1
  μ=tan(π4-r)	Tμ	n-1
  α=2cot(2r)	RJN	1

• (3)

  The shape operator A and the structure tensor field φ satisfy

  Aφ+φA=kφ, k≠0:const.

• (4)

  The trace h of the shape operator A and its square h2 becomes the following respectively

  h=TrA=2cot(2r)-2(n-1)tan(2r), h2=4cot2(2r)+4(n-1)2tan2(2r)-8(n-1).

• (5)

  The trace of the matrix A2 is given by

  TrA2=4cot2(2r)+4(n-1)tan2(2r).

• (6)

  The scalar curvature γ of the tube M is given by

  γ=4(n-1)2+4(n-1)(n-2)tan2(2r).

• (7)

  For cot2(r)=n-2, M is pseudo-Einstein such that

  Ric(X)=2nX-2(2n-1)η(X)ξ.

Let us introduce a pseudo Ricci-Bourguignon soliton (M,ξ,η,Ω,θ,γ,g) which is a solution of the pseudo Ricci-Bourguignon flow defined by

Then it is given by the following

for any tangent vector fields X and Y on M, where Ω is a pseudo Ricci-Bourguignon soliton constant, ψ and θ any constants and γ the scalar curvature on M, and LV denotes the Lie derivative along the direction of the vector field V (see Morgan-Tian [22]). Then by virtue of the Lie derivative, we have

Then the formula (4.1) can be given by

From this, by applying the structure tensor φ to both sides, we get the following two formulas

and

By using the almost contact structure (φ,ξ,η,g) in the right side above, we know that the generalized pseudo anti-commuting property holds as follows:

Now we want to introduce an important proposition due to Ki-Suh [16], and Yano-Kon [39], which will be used in the proof of our Theorem 1 as follows:

Proposition 4.1 Let M be a connected complete Hopf real hypersurface in the complex projective space CPn. If M satisfies the generalized pseudo-anti commuting property, then M is locally congruent to a geodesic hypersphere in the class of type (A1), a pseudo-Einstein hypersurface in the class of type (A2), or M is locally congruent to of type (B).

Among real hypersurfaces of type (A2) satisfying the generalized pseudo-anti commuting property (4.2) is only pseudo-Einstein. Then it is exactly the second case in Theorem C. That is M is locally congruent to a tube of radius r around a totally geodesic CPk, 0<k<n-1, where 0<r<π2 and cot2(r)=kn-k-1.

Now geodesic hyperspheres and pseudo-Einstein real hypersurfaces are included in the class of type (A1) and A2 respectively. So by Theorem A, they are characterized by the commuting shape operator. That is, Aφ=φA. Accordingly, from the notion of pseudo Ricci-Bourguignon soliton (M,ξ,η,Ω,θ,γ,g) of M, (4.1) becomes

This means that those hypersurfaces are pseudo-Einstein. Then by virtue of Theorem C there exist three kind of pseudo-Einstein real hypersurfaces in complex projective space CPn such that

• (i)

  a geodesic hypersphere,

• (ii)

  a tube of radius r around a totally geodesic CPk, 0<k<n-1, where 0<r<π2 and cot2r=kn-k-1,

• (iii)

  a tube of radius r around a complex quadric Qn-1 where 0<r<π4 and cot22r=n-2.

For the case (i) it can be easily verified that a geodesic hypersphere in Remark 0.4 satisfies the following

for any vector fields X on M in CPn. Then Ω+θγ=2n+2(n-1)cot2(r), and ψ=2n.

Now let us check the second case (ii) whether it satisfies a pseudo Ricci-Bourguignon soliton for cot2r=kn-k-1. Then for the Reeb vector field ξ we have the following

where ℓ=n-k-1 and cot2(r)=kn-k-1. Then the coefficient a+b is given by

Moreover, by using cot2(r)=kn-k-1 and Proposition 0.3, for any vector fields X∈Tλ, λ=cot(r) and Y∈Tμ, μ=tan(r), we have the following formulas, respectively

and

Then the Ricci operator of pseudo-Einstein hypersurfaces satisfying the pseudo Ricci-Bourguignon soliton becomes

where soliton constants are given by Ω+θγ=a=2n and ψ=-b=2, respectively.

Finally, in the third case (iii) let us check that a tube of radius r around the complex quadric Qn-1 with cot2(2r)=n-2 in the complex projective space CPn could satisfy the pseudo Ricci-Bourguignon soliton.

In order to do this, first we should check that this tube is pseudo-Einstein for cot2(2r)=n-2. In fact, it is characterized by Aφ+φA=kφ, where k=0 : constant. Moreover, by Proposition 0.5, the principal curvature are given by λ=-cot(π4-r), μ=tan(π4-r) and α=2cot(2r). So k=λ+μ=-4α. For any X∈Tλ the vector field φX∈Tμ. Then by (3.2) we know the following for any X∈Tλ

and for any Y∈Tμ

Then from cot2(2r)=n-2 we can verify the following

where by Proposition 0.5, we have used the following from cot2(2r)=n-2

Moreover, Ric(ξ)={2(n-1)+(hα-α2)}ξ. So for a pseudo-Einstein real hypersurface in CPn we may put

where by Proposition 0.5 and cot2(2r)=n-2, the constant a+b is given by

and by the property of contact hypersurfaces, we know that λ+μ=-4α. So by virtue of (4.4), it follows that h=λ+μ. Then the constant a is given by

By the above two constants a and a+b, another constant b becomes b=-2(2n-1). Then if the third case satisfies the pseudo Ricci-Bourguignon soliton, we can assert the following