The *-Ricci tensor for hypersurfaces in CP^n and CH^n
aa r X i v : . [ m a t h . DG ] N ov The *-Ricci tensor for hypersurfaces in CP n and C H n Thomas A. Ivey and Patrick J. RyanApril 2010
Abstract
We update and refine the work of T. Hamada concerning *-Einstein hypersurfacesin CP n and C H n . We also address existence questions using the methods of movingframes and exterior differential systems. The notion of *-Ricci tensor for an almost-Hermitian manifold was introduced by Tachibana[16] in 1959 and later used (along with the related concept of *-Einstein) in work on theGoldberg conjecture (see, for example, Oguro and Sekigawa [15]). These ideas also applynaturally to contact metric manifolds, and in particular, to hypersurfaces in complex spaceforms, where they were introduced by T. Hamada [4]. In this paper, we refine, clarify, andextend some of Hamada’s work, specifically the classification of *-Einstein hypersurfaces incomplex space forms. See, in particular, Theorem 4.Takagi [17], for CP n , and Montiel [13], for C H n , catalogued a specific list of real hypersur-faces, which we call “Takagi’s list” and “Montiel’s list” in [14]. These are the homogeneousHopf hypersurfaces. They have constant principal curvatures and every Hopf hypersurfacewith constant principal curvatures is an open subset of one of them.Many theorems have been published characterizing these lists or subsets of them. Forexample, the pseudo-Einstein hypersurfaces, introduced by Kon [10], form such a subset.The same subset is characterized as the the set of Hopf hypersurfaces satisfying a certaincondition on the Ricci tensor (known as pseudo-Ryan in the literature). This has beenknown for n ≥ n = 2. We also prove that the *-Einstein and pseudo-Ryan conditions areequivalent for Hopf hypersurfaces when n = 2, thus giving us three distinct characterizationsof this class of hypersurfaces.It would be of interest to find additional classes of hypersurfaces, that could be “nicely”characterized, but this seems to be a difficult problem. In this paper, we establish theexistence of a family of non-Hopf pseudo-Ryan hypersurfaces in CP and C H , and provethat (in contrast to the Hopf case), the set of non-Hopf pseudo-Ryan hypersurfaces is disjointfrom the set of non-Hopf *-Einstein hypersurfaces; see Theorem 11 and Corollary 3. We hopethat this result will lead to further refinements of these conditions that can be characterizedgeometrically. 1n §
5, we construct a family of Hopf hypersurfaces that are not *-Einstein, but satisfya weakened form of the *-Einstein condition. These examples show that the constancy ofthe *-scalar curvature is an essential assumption in the definition of the *-Einstein condi-tion, unlike the situation in the definition of “ordinary” Einstein manifold. Finally, as afurther application of our methods, in § C H with constant principal curvatures which were classified by Berndt andDiaz-Ramos [1].In what follows, all manifolds are assumed connected and all manifolds and maps areassumed smooth ( C ∞ ) unless stated otherwise. Basic notation and historical informationfor hypersurfaces in complex space forms may be found in [14]. For more on moving framesand exterior differential systems, see the monograph [2] or the textbook [5]. Throughout this paper, we will take the holomorphic sectional curvature of the complexspace form in question to be 4 c . The curvature operator e R of the space form satisfies e R ( X, Y ) = c ( X ∧ Y + J X ∧ J Y + 2 h X, J Y i J) (1)for tangent vectors X and Y (cf. Theorem 1.1 in [14]), where X ∧ Y denotes the skew-adjointoperator defined by ( X ∧ Y ) Z = h Y, Z i X − h X, Z i Y. We will denote by r the positive number such that c = ± /r . This is the same conventionas used in ([14], p. 237).A real hypersurface M in CP n or C H n inherits two structures from the ambient space.First, given a unit normal ξ , the structure vector field W on M is defined so thatJ W = ξ, where J is the complex structure. This gives an orthogonal splitting of the tangent space asspan { W } ⊕ W ⊥ . Second, we define on M the skew-symmetric (1 ,
1) tensor field ϕ which is the complexstructure J followed by projection, so that ϕX = J X − h X, W i ξ. Recall that the type (1,1) Ricci tensor of any Riemannian manifold is defined by theequation h SX, Y i = trace { Z R ( Z, X ) Y } (2)where X , Y , and Z are any tangent vectors and R is the curvature tensor. In case of aK¨ahler manifold, it is not difficult to show that h SX, Y i = (trace { J ◦ R ( X, J Y ) } ) . (3)2see [9], p. 149). This led Tachibana and others to consider, on any almost-Hermitianmanifold, the *-Ricci tensor S ∗ , which may be defined by the same formula, h S ∗ X, Y i = (trace { J ◦ R ( X, J Y ) } ) (4)and to define a space to be *-Einstein if h S ∗ X, Y i is a constant multiple of h X, Y i for alltangent vector fields X and Y . In this and subsequent sections, we follow the notation and terminology of [14]: M n − will be a hypersurface in a complex space form f M (either CP n or C H n ) having constantholomorphic sectional curvature 4 c = 0. The structures ξ , W , and ϕ are as defined in theIntroduction. The (2 n − W ⊥ is called the holomorphic distri-bution . The operator ϕ annihilates W and acts as complex structure on W ⊥ . The shapeoperator A is defined by AX = − e ∇ X ξ where e ∇ is the Levi-Civita connection of the ambient space. The Gauss equation expressesthe curvature operator of M in terms of A and ϕ , as follows: R ( X, Y ) = AX ∧ AY + c ( X ∧ Y + ϕX ∧ ϕY + 2 h X, ϕY i ϕ ) , (5)and from this we see that the Ricci tensor is given by SX = (2 n + 1) cX − c h X, W i W + m AX − A X, (6)where m = trace A . In addition, it is easy to show (see [14], p. 239) that ∇ X W = ϕAX, (7)where ∇ is the Levi-Civita connection of the hypersurface M .Following Hamada [4], we define the *-Ricci tensor S ∗ on M by h S ∗ X, Y i = (trace { ϕ ◦ R ( X, ϕY ) } ) , (8)and the *-scalar curvature ρ ∗ to be the trace of S ∗ . We say that the hypersurface M is*-Einstein if ρ ∗ is constant and h S ∗ X, Y i = ρ ∗ n − h X, Y i (9)for all X and Y in the holomorphic distribution W ⊥ .We define the function α = h A W, W i . The hypersurface is said to be Hopf if the structure vector W is a principal vector, i.e. AW = αW , and we refer to α as the Hopf principal curvature . It is important to recall that3he Hopf principal curvature is constant (see Theorem 2.1 in [14]). Of course, α need not beconstant for a non-Hopf hypersurface.We also recall the notion of pseudo-Einstein hypersurface . A real hypersurface M in CP n or C H n is said to be pseudo-Einstein if there are constants ρ and σ such that SX = ρX + σ h X, W i W for all tangent vectors X . We first note which hypersurfaces in the Takagi and Montiel lists are *-Einstein. Accordingto the standard terminology (see, for example [14], pp.254–262), the lists are broken downinto “types” A1, A2, A0, B, C, D, and E. The situation is as follows:
Theorem 1.
Among the homogeneous Hopf hypersurfaces M n − in CP n and C H n , where n ≥ (i.e. Takagi’s and Montiel’s lists), • All type A1, A0 and B hypersurfaces are *-Einstein, • A type A2 hypersurface is *-Einstein if an only if it is a tube of radius π r over CP k where ≤ k ≤ n − , • No type C, D, or E hypersurface is *-Einstein.
In other words, geodesic spheres in CP n , geodesic spheres, horospheres, and tubes over C H n − in C H n are *-Einstein, but except for that, there is just one special case. Note also,that the same classification holds locally. In other words, an open subset of a hypersurface M in the Takagi/Montiel lists is *-Einstein if and only if M is.Theorem 1 can be proved in a routine manner once we collect and verify a few facts. Wewill do this at the end of Section 3. In this section, we derive an expression for the *-Ricci tensor of a hypersurface and discussthe implications for Hopf hypersurfaces.
Theorem 2.
For a real hypersurface M n − in CP n or C H n , where n ≥ , S ∗ = − (2 ncϕ + ( ϕA ) ) . (10) Furthermore, • If M is Hopf, then S ∗ is symmetric and S ∗ W = 0 . • If M is Hopf and α = 0 , then S ∗ X = (2 n + 1) cX for all X ∈ W ⊥ , and ρ ∗ =2( n − n + 1) c . In particular, M is *-Einstein. roof. We recall that for any linear functional ψ on a finite-dimensional vector space, thetrace of the map v ψ ( v ) u is ψ ( u ). When we use the Gauss equation (5) to compute R ( X, ϕY ) ϕZ , the first term is( AX ∧ AϕY ) ϕZ = h AϕY, ϕZ i AX − h AX, ϕZ i AϕY, (11)so that trace ( AX ∧ AϕY ) ◦ ϕ = h AϕY, ϕAX i − h
AX, ϕAϕY i = − (cid:10) ( ϕA ) X, Y (cid:11) . Similarly, the other terms in the Gauss equation give( X ∧ ϕY + ϕX ∧ ϕ Y + 2 (cid:10) X, ϕ Y (cid:11) ϕ ) ϕZ = h ϕY, ϕZ i X − h X, ϕZ i ϕY + (cid:10) ϕ Y, ϕZ (cid:11) ϕX − h ϕX, ϕZ i ϕ Y + 2 (cid:10) X, ϕ Y (cid:11) ϕ Z so thattrace c ( X ∧ ϕY + ϕX ∧ ϕ Y + 2 (cid:10) X, ϕ Y (cid:11) ϕ ) ◦ ϕ = c ( h ϕY, ϕX i − (cid:10) X, ϕ Y (cid:11) + (cid:10) ϕ Y, ϕ X (cid:11) − (cid:10) ϕX, ϕ Y (cid:11) + 2 (cid:10) X, ϕ Y (cid:11) trace ϕ ) . Noting that ϕ = − ϕ and trace ϕ = − n − h S ∗ X, Y i = − (cid:10) (2 ncϕ + ( ϕA ) ) X, Y (cid:11) . (12)Now it is clear that S ∗ is symmetric if and only if ( ϕA ) = ( Aϕ ) . In case M is Hopf, wemake use of the identity ([14] p. 245) AϕA = α Aϕ + ϕA ) + cϕ (13)to reduce this condition to α ( Aϕ ) = α ( ϕ A ). Since span { W } and W ⊥ are A -invariant, wecan use the fact that ϕ is zero on W and acts as − I on W ⊥ to verify that Aϕ = ϕ A , andhence conclude that S ∗ is symmetric.Finally, since ϕAW = 0 for a Hopf hypersurface, we have S ∗ W = 0. Further, if α = 0,then applying ϕ to (13) shows that ( ϕA ) X = − cX for all X ∈ W ⊥ . This yields the desiredresults for S ∗ and ρ ∗ . In this section, we discuss the converse of Theorem 1. Must every *-Einstein Hopf hyper-surface occur in the lists of Takagi and Montiel? The answer is no, but almost. Specifically,we have,
Theorem 3.
Let M n − , where n ≥ , be a *-Einstein Hopf hypersurface in CP n or C H n whose Hopf principal curvature α is nonzero. Then M is an open subset of a hypersurfacein the lists of Takagi and Montiel. emark 1. This corrects Theorems 1.1 and 1.2 of [4], where the case α = 0 was overlooked.We will show that all Hopf hypersurfaces with α = 0 are *-Einstein. In CP n , for instance,this includes every hypersurface that is a tube of radius π r over a complex submanifold.Also, all pseudo-Einstein hypersurfaces in CP and C H are *-Einstein. Many of these havenon-constant principal curvatures; see [8] and [6].We now prove Theorem 3. Proof.
For any unit principal vector X ∈ W ⊥ with corresponding principal curvature λ , itfollows directly from (13) that ( λ − α ) AϕX = ( λα + c ) ϕX . If λ = α , then ϕX is also aprincipal vector with corresponding principal curvature ν where λν = λ + ν α + c. (14)We also note that α cannot be a principal curvature unless α + 4 c = 0.First look at the case where α + 4 c = 0. Pick a point p ∈ M where a maximal numberof eigenvalues of A (restricted to W ⊥ ) are distinct. This guarantees that the principalcurvatures have constant multiplicities in a neighborhood of p , and are therefore smooth.Let V ⊆ T p M be a principal space corresponding to a principal curvature λ . Then ϕV is aprincipal space with corresponding principal curvature ν satisfying (14). If span { V, ϕV } = W ⊥ at p , then ( ϕA ) X = − λνX for all X ∈ W ⊥ . Since M is *-Einstein, λν must beconstant near p . Since α = 0, we see from (14) that λ + ν is constant as well. Thus λ and ν are constant near p . Note that this includes the case λ = ν . On the other hand, ifspan { V, ϕV } 6 = W ⊥ at p , we can construct (at least) two such such pairs { λ, ν } and { ˜ λ, ˜ ν } .However, the *-Einstein condition guarantees that ( ϕA ) is a constant multiple of the identityon W ⊥ , so that λν = ˜ λ ˜ ν , which leads to λ + ν = ˜ λ + ˜ ν , and finally to { λ, ν } = { ˜ λ, ˜ ν } , whichis a contradiction.We now consider the case where α + 4 c = 0, and choose p as above. One possibility isthat AX = α X for all X ∈ W ⊥ . If this does not hold, suppose that λ = α is a principalcurvature at p , and that X is an associated principal vector. Then AϕX = α ϕX since (14)reduces to ( λ − α ν − α . (15)Because M is *-Einstein, ( ϕA ) Y = − λ α Y for all Y ∈ W ⊥ and λ is constant near p . Inparticular, if AY = α Y, this leads to AϕY = λϕY . Thus the principal spaces of λ and α have the same dimension and are interchanged by ϕ . Because of (15), there can be noprincipal curvatures other than λ and α . Again, M has constant principal curvatures near p . In all cases, p has a neighborhood with constant principal curvatures, which thereforemust be an open subset of some member of Takagi’s or Montiel’s lists. For the case α +4 c = 0the only possibility that can actually occur is the horosphere and only α/ W ⊥ . Thus M is an open subset of a horosphere. For the case α + 4 c = 0, the set where the principal curvature data (value and multiplicity) agree withthose at p , is open and closed and therefore is all of M . It follows that M is an open subsetof a specific member of one of these lists. 6s a consequence of Theorems 2 and 3, Proposition 2.21 of [8] and Theorem 4 of [7], wehave the following: Corollary 1.
For a hypersurface M in CP or C H , the following are equivalent.1. M is Hopf and *-Einstein;2. M is pseudo-Einstein;3. L W R W = 0 where R W is the structure Jacobi operator of M and L W is the Liederivative in the direction of the structure vector W . Remark 2.
The argument given in [4] for Theorem 3 begins as ours does, but leads toa quadratic equation with constant coefficients, that all principal curvatures on W ⊥ mustsatisfy. Unfortunately, when α = 0, all coefficients vanish, so the proof is valid only when α = 0 is assumed. We have included our alternative proof in this paper because it establishessome facts that are useful for veryifying Theorem 1, to which we now turn our attention. Proof of Theorem 1
As we have seen in the proof of Theorem 3, one can check the *-Einstein condition on aHopf hypersurface by examining the ϕ -invariant subspaces of the form span { V, ϕV } , where V ⊆ W ⊥ is a principal subspace. In the Takagi-Montiel lists, the type A hypersurfaces have ϕ -invariant principal spaces. This means that ( ϕA ) , restricted to W ⊥ , is a constant multipleof the identity for type A1 and type A0 hypersurfaces, so they must be *-Einstein. For typeA2 hypersurfaces, however, W ⊥ splits into two distinct ϕ -invariant principal subspaces,whose corresponding principal curvatures ( λ and λ , say) satisfy the quadratic equation λ = αλ + c . In order to satisfy the *-Einstein condition, we would need to have λ = λ ,which is impossible unless α = 0. For type A2 hypersurfaces in CP n , α = 0 only whenthe radius is π r , while for a type A2 hypersurface in C H n , α is nonzero for all radii. (SeeTheorems 3.9 and 3.14 in [14]).For type B hypersurfaces, W ⊥ = span { V, ϕV } , where V is a principal subspace of di-mension ( n − λ λ + c = 0, so that( ϕA ) X = cX for all X ∈ W ⊥ and hence M is *-Einstein with ρ ∗ = 2( n − n − c .However, types C, D, and E hypersurfaces cannot be *-Einstein. To see this, using thenotation of [14], p. 261, we first note that α cannot be 0 since u = π would cause the principalcurvature λ to be undefined. Further, principal curvatures λ and λ satisfy the quadraticequation λ = λα + c , and hence the corresponding principal spaces are ϕ -invariant. The*-Einstein condition would then require that λ = λ which cannot be true since α = 0.Thus we have verified Theorem 1. We now summarize the classification of *-EinsteinHopf hypersurfaces as follows: Theorem 4.
The *-Einstein Hopf hypersurfaces in CP n and C H n , where n ≥ , are precisely(i) the Hopf hypersurfaces whose Hopf principal curvature α vanishes, and ii) the open connected subsets of homogeneous Hopf hypersurfaces of types A0, A1 and B. Remark 3.
Note that geodesic spheres (type A1) of radius π r in CP n have α = 0 and thussatisfy both (i) and (ii) in Theorem 4. Other than that, there is no overlap. For furtherdetail on the structure of Hopf hypersurfaces with α = 0, see [3] and [6]. Corollary 2.
For a Hopf hypersurface M n − in CP n or C H n , where n ≥
2, ( ϕA ) cannotvanish identically. Proof.
Suppose that ( ϕA ) = 0. Then M is *-Einstein from Theorem 2. If α = 0, theresult is immediate from (13). Otherwise, Theorem 3 says that M must occur in the listsof Takagi or Montiel. However, none of the principal curvatures of these hypersurfaces (forprincipal spaces in W ⊥ ) vanish. In fact, they all satisfy identities of the form λ = αλ + c or λν + c = 0, so that the eigenvalues of ( ϕA ) on W ⊥ are all of the form − λ = 0 or − λν = c = 0. We recall the notation R ( X, Y ) · T for the action of a curvature operator on any tensor field T (see [14] p. 235). For the special case of the Ricci tensor S , R ( X, Y ) · S = R ( X, Y ) ◦ S − S ◦ R ( X, Y ) . For a Hopf hypersurface M n − , where n ≥
3, the pseudo-Einstein condition is known to beequivalent to the following: h ( R ( X , X ) · S ) X , X i = 0 (16)for all X , X , X and X in W ⊥ (see Theorem 6.30 of [14]). A hypersurface satisfying (16)is called “pseudo-Ryan” in the literature. We now discuss this condition for n = 2 and howit relates to the pseudo-Einstein and *-Einstein conditions.With this in mind, let M be a (not necessarily Hopf) hypersurface in CP or C H .Suppose that there is a point p (and hence an open neighborhood of p ) where AW = αW .Then there is a positive function β and a unit vector field X ∈ W ⊥ such that AW = αW + βX. Let Y = ϕX . Then there are smooth functions λ , µ , and ν defined near p such that withrespect to the orthonormal frame ( W, X, Y ), A = α β β λ µ µ ν . (17)Note that if, on the other hand, M is Hopf, then there still exists an orthonormal frame( W, X, Y ) near any point, such that Y = ϕX and (17) holds with β = 0; however, the choiceof ( X, Y ) is only unique up to rotation. 8sing (6), we compute the matrix of the Ricci tensor S with respect to this frame, toget, S = c + α ( λ + ν ) − β νβ − µβνβ c + λ ( ν + α ) − β − µ µα − µβ µα c + ν ( λ + α ) − µ . (18)It is easy to check that (16) is satisfied if and only if ( R ( X, Y ) · S ) X and ( R ( X, Y ) · S ) Y aremultiples of W . Proposition 5.
With
X, Y, β, µ, ν , and λ defined as above, ( R ( X, Y ) · S ) X and ( R ( X, Y ) · S ) Y are multiples of W if and only if µ ( β ν − α (4 c + λν − µ )) = 0 (19) and β ( µ − ν ) = (4 c + λν − µ )( α ( λ − ν ) − β ) . (20) This equivalence also holds at a point where AW = αW , where we take X to be any unitprincipal vector in W ⊥ , Y = ϕX , and β = µ = 0 .Proof. The X and Y components of ( R ( X, Y ) · S ) X must be computed. The Gauss equationgives the matrix of the curvature operator as R ( X, Y ) = µβ νβ − µβ λν − µ + 4 c − νβ − ( λν − µ + 4 c ) 0 . (21)From this and matrix of S , the calculation is straightforward. Also, since the calculationis pointwise, the expressions (18) and (21) and the conclusion are equally valid at a pointwhere AW = αW , when X is taken to be any unit principal vector in W ⊥ , Y = ϕX , and β = µ = 0.In particular, for a Hopf hypersurface, we have Theorem 6.
A Hopf hypersurface M in CP or C H is pseudo-Ryan if and only if it ispseudo-Einstein.Proof. We refer to the pointwise criterion for a Hopf hypersurface to be pseudo-Einstein,as given in Proposition 2.21 of [8]. Let p be any point of the Hopf hypersurface M and let X ∈ W ⊥ be a unit principal vector at p with corresponding principal curvature λ . Assumethat (16) holds at p . From (20), we have α (4 c + λν )( λ − ν ) = 0 . If α = 0, then λν = c by (14), and the pseudo-Einstein criterion is satisfied at p . If α = 0and λ = ν at p , then 4 c + λν = 0 near p . Using (14), we get − c = λ + ν α + c so that λ + ν is alsoconstant. Therefore, λ and ν are constant and a neighborhood of p is a Hopf hypersurfacewith constant principal curvatures. However, the well-known classification of such does notadmit this possibility. As seen from Theorem 4.13 of [14], such a hypersurface would haveto be in the Takagi/Montiel list and thus have λν + c = 0. Because of this contradiction, weconclude that λ = ν at p , and thus the pseudo-Einstein criterion is satisfied there. Since p was arbitrary, M must be pseudo-Einstein.Conversely, if M is pseudo-Einstein, then either α = 0 or λ = ν so that the equations inProposition 5 are satisfied, and M is pseudo-Ryan.9n view of our work in the previous section, we see that for Hopf hypersurfaces in CP and C H , the pseudo-Einstein, *-Einstein, and pseudo-Ryan conditions are equivalent. Wenow look at non-Hopf hypersurfaces.First, we improve Proposition 5. Specifically, we deduce that under the conditions of theproposition, we must have µ = 0. Proposition 7.
In the notation of Proposition 5, ( R ( X, Y ) · S ) X and ( R ( X, Y ) · S ) Y aremultiples of W if and only if µ = 0 , and β ν = − (4 c + λν )( α ( λ − ν ) − β ) . Proof.
Suppose that (19) and (20) hold and µ = 0 at some point. Then, in a neighborhoodof this point, we have β ν = α (4 c + λν − µ ) . (22)Multiplying (20) by α , we get αβ ( µ − ν ) = α (4 c + λν − µ )( α ( λ − ν ) − β ) , (23)which, upon substitution from (22), yields αβ ( µ − ν ) = β ν ( α ( λ − ν ) − β ) , (24)Cancelling β and substituting for β ν from the first equation, we get 4 cα = 0. Therefore α = ν = 0 and (23) reduces to β µ = − (4 c − µ ) β , (25)a contradiction. We conclude that µ must vanish identically. As the converse is trivial, ourproof is complete.We now turn our attention to the *-Einstein condition. In a neighborhood of a pointwhere AW = αW , using the same orthonormal frame ( W, X, ϕX ), it is easy to see from (17)that ( ϕA ) = − βν µ − λν βµ µ − λν . (26)Then, using (10), we see that ρ ∗ is locally constant if and only if µ − λν is. For a pointwhere AW = αW , we let X be a unit principal vector in W ⊥ (as before) and let Y = ϕX .Then there are numbers α , λ and ν such that equations (17) and (26) still hold at this point,but with β = µ = 0.Although we do not wish to discuss ruled hypersurfaces in depth in this paper, they areuseful for demonstrating the non-equivalence of the pseudo-Ryan and *-Einstein conditions.To be concise, we define a hypersurface in CP n or C H n to be ruled if AW ⊥ ⊆ span W. (27)Geometrically, this means that M is foliated by totally geodesic complex hypersurfaces (i.e.real codimension 2) which are orthogonal to W .10 roposition 8. For non-Hopf hypersurfaces M in CP and C H , the pseudo-Ryan and*-Einstein conditions are not equivalent. In fact, • All ruled hypersurfaces in CP and C H are *-Einstein; • No pseudo-Ryan hypersurface in CP or C H is ruled.Proof. Clearly, a ruled hypersurface satisfies ( ϕA ) = 0 and hence is *-Einstein by Theorem2 with ρ ∗ = 2( n − n + 1) c . This fact was observed by Hamada [4].It is also immediate from Corollary 2 that no Hopf hypersurface can be ruled. Therefore,any ruled hypersurface in CP or C H , must have a point p with a neighborhood in whichthe setup introduced at the beginning of this section holds with β = 0 and µ = ν = λ = 0.The equations in Proposition 7 reduce to − cβ = 0, a contradiction. Thus no pseudo-Ryanhypersurface in CP or C H is ruled. (It is easy to check that this also holds in CP n and C H n for n ≥ both pseudo-Ryan and*-Einstein. Proposition 9.
Let M be a non-Hopf hypersurface M in CP and C H that is both pseudo-Ryan and *-Einstein. Then around any point where AW = αW there is either(i) a neighborhood in which the components of the shape operator (17) with respect to thestandard basis satisfy µ = λ = 0 , βν = 0 , and β ( ν − c ) = 4 cαν, (28) or (ii) a neighborhood in which µ = 0 , σ = − λν is a nonzero constant, and β ν = (4 c − σ ) (cid:16) α (cid:16) ν + σν (cid:17) + β (cid:17) . (29) Proof.
Let σ = − λν. Since M is pseudo-Ryan, by Proposition 7 we have µ = 0 and β ν = − (4 c − σ )( α ( λ − ν ) − β ) . (30)Since M is *-Einstein, (26) shows that σ is locally constant. There are two possibilities: • If σ = 0 then λ must be identically zero. Otherwise, there is an open set where ν = 0and αλ − β = 0. But this would require that rank A ≤
1, contradicting a well-knownfact about hypersurfaces (see Proposition 2.14 of [14]). Setting σ = λ = 0 in (30)yields (28). • If σ = 0, then ν is nonvanishing, and setting λ = − σ/ν in (30) gives (29). Note thatthe constant 4 c − σ must be nonzero, since σ = 4 c in (29) would imply that ν = 0.Based on the conditions derived in the previous proposition, we can deduce Proposition 10.
If hypersurface M is non-Hopf, pseudo-Ryan and *-Einstein, then (in aneighborhood of a point where AW = αW ), we have σ = 0 and α, β, λ constant. C H , according to Berndt and Diaz-Ramos[1], this implies that M is one of the Berndt orbits – either the minimal orbit or one ofits equidistant hypersurfaces. However, these hypersurfaces do not satisfy the conditions ofProposition 7. This can be seen from their principal curvatures which are given explicitly inProposition 3.5 of [1]. On the other hand, all hypersurfaces in CP with constant principalcurvatures must be Hopf, as shown by Q.M. Wang [18]. Thus, in fact, the kind of hypersur-face envisioned in Proposition 10 does not exist and we have the following improvement ofProposition 7: Theorem 11. In CP and C H the set of non-Hopf *-Einstein hypersurfaces is disjoint fromthe set of non-Hopf pseudo-Ryan hypersurfaces. The examples we will construct in § ρ ∗ It is well-known that for a Riemannian manifold of dimension greater than 2, if h SX, Y i = ρ h X, Y i for all vector fields X and Y , then ρ must necessarily be constant. One can asksimilarly if, in the definition (9) of *-Einstein, the stipulation that ρ ∗ be constant is redun-dant.When n = 2, (i.e. for hypersurfaces of dimension 3), we find that the condition is notredundant. In fact, using Theorem 2, (17), and (26), we have the following: Proposition 12.
Every hypersurface in CP or C H satisfies h S ∗ X, Y i = ρ ∗ h X, Y i for all X and Y in W ⊥ , with ρ ∗ = 4 c + λν − µ . In particular, for a Hopf hypersurface, we have ρ ∗ = 4 c + λν . For a Hopf hypersurface with α +4 c = 0, it follows from (13) that if ρ ∗ were constant, then each of the principal curvatures λ and ν would have to be constant. Thus, we can obtain examples of Hopf hypersurfaces withnonconstant ρ ∗ by constructing examples with non-constant principal curvatures associatedto principal directions in W ⊥ . These are provided by tubes over holomorphic curves in CP or C H ; in the latter case, α + 4 c >
0. In both spaces, we can also construct Hopfhypersurfaces with nonconstant principal curvatures using Theorem 13 below, which allowsus to prescribe the principal curvature along a principal curve perpendicular to the structurevector.
Remark 4.
Our result shows that the constancy of ρ ∗ should be added to the hypothesesof Lemma 3.1 in Hamada’s paper [4].Before stating the theorem, we will introduce some necessary terminology for Frenet-typeinvariants of curves in f M = CP and C H , defined in terms of unitary frames. A unitary The usual (Riemannian) construction for Frenet frames along curves in these spaces, as set forth inseveral papers by Maeda and collaborators [11], [12], is not suitable for our purposes, as we prefer to useframes adapted to the complex structure. rame at a point p ∈ f M is an orthonormal basis ( e , e , e , e ) for T p f M such that e = J e , e = J e . (31) Definition 1.
Let I ⊂ R be an open interval, and γ : I → f M a regular unit-speed curve.We say that γ is a regular framed curve if there exists a unitary frame ( T, J T, N, J N ) definedalong γ such that γ ′ = T and the frame vectors satisfy T ′ = k J T + k N, N ′ = − k T + τ J N. (32)where the primes indicate covariant derivative with respect to γ ′ along γ , and k , k and τ are smooth functions referred to as the holomorphic curvature , transverse curvature and torsion respectively of γ . Theorem 13.
Let γ : I ֒ → f M be a regular real-analytic framed curve with zero torsion, andtransverse curvature given by an analytic function k ( s ) . For any real number α satisfying α + 4 c = 0 , there exists a Hopf hypersurface M with Hopf principal curvature α , containing γ , and for which γ is a principal curve perpendicular to the structure vector field W withprincipal curvature k . Any other Hopf hypersurface with these properties will coincide with M on an open set containing γ . This result will be proved in § k , k and τ equal to any given smoothfunctions can be shown by standard arguments about solutions of linear systems of ordinarydifferential equations (cf. Theorem 5.1 in [11]), and these arguments carry over to the real-analytic category. Thus, we can apply Theorem 13 to produce a Hopf hypersurface with azero torsion analytic curve as principal curve, with any given analytic function as principalcurvature along this curve. Let F be the unitary frame bundle of f M = CP or C H , i.e., the bundle whose fiber at apoint p is the set of orthonormal frames ( e , e , e , e ) satisfying (31). This is a principalsub-bundle of the full orthonormal frame bundle, and has structure group U (2). Let ω i and ω ij , for 1 ≤ i, j ≤
4, denote the pullbacks of the canonical forms and Levi-Civita connectionforms from the full frame bundle. If f = ( e , e , e , e ) is any local section of F , then thepullbacks of the ω i form a dual coframe, i.e., e j f ∗ ω i = δ ij . (33)As well, the connection forms have the property that e ∇ v e i = ( v f ∗ ω ji ) e j (34)for any tangent vector v . These forms satisfy the usual structure equations dω i = − ω ij ∧ ω j , dω ij = − ω ik ∧ ω kj + Φ ij , ij are the curvature 2-forms. On the unitary frame bundle, the connection forms ω ij satisfy the additional linear relations ω = ω , ω = − ω . Thus, the canonical forms ω , ω , ω , ω together with the connection forms ω , ω , ω , ω form a globally-defined coframe on the 8-dimensional manifold F .The curvature forms are as follows:Φ = c (4 ω ∧ ω + 2 ω ∧ ω ) , Φ = c (4 ω ∧ ω + 2 ω ∧ ω ) , Φ = Φ = c ( ω ∧ ω + ω ∧ ω ) , Φ = Φ = c ( ω ∧ ω + ω ∧ ω ) . For a hypersurface M ⊂ f M , we say that a section f : M → F | M is an adapted framealong M if e is normal to M . It follows from (33) that f ∗ ω = 0, and it follows from (34)that f ∗ ω i = h ij f ∗ ω j , ≤ i, j ≤ , (35)where h ij are the components of the shape operator with respect to the tangential movingframe ( e , e , e ). Furthermore, if ( W, X, Y ) is a local frame along M as constructed in § e = W , e = X and e = Y , and the entries of H are just the entries of A from (17) rearranged: H = λ µ βµ ν β α . (36) From (34) and (36), it follows that an adapted unitary frame along a Hopf hypersurface M gives a section f : M → F such that f ∗ ω = 0 and f ∗ ( ω − αω ) = 0. Thus, the image Σ of f is a three-dimensional submanifold along which the forms θ = ω , θ = ω − αω pull back to be zero. We let I be the exterior differential system on F generated by these1-forms (for a given value of the constant α ). Then there is a one-to-one correspondence be-tween Hopf hypersurfaces M equipped with an adapted unitary frame and three-dimensionalintegral submanifolds of I satisfying the independence condition ω ∧ ω ∧ ω = 0.To complete a set of algebraic generators for the ideal I , we need to compute the exteriorderivatives dθ ≡ − ω ∧ ω − ω ∧ ω ,dθ ≡ ω ∧ ω − cω ∧ ω ) + α ( ω ∧ ω − ω ∧ ω ) mod θ , θ . Let Ω , Ω be the 2-forms on the right hand sides of these equations.14or an exterior differential system, a subspace E of the tangent space at a point in theunderlying manifold is an integral element if all differential forms in the system restrict tobe zero on E . For example, let u ∈ F and let v ∈ T u F be a nonzero vector; then v spans a1-dimensional integral element if and only if v θ = v θ = 0.We define the polar space of a k -dimensional integral element as follows. Definition 2.
Let { e , · · · , e k } be a basis for an integral element E . The polar space H ( E )of E is the set of all w ∈ T u F such that ψ ( w , e , · · · , e k ) = 0 for all ( k + 1)-forms ψ ∈ I .In other words, H ( E ) contains all possible enlargements of E to an integral elementof one dimension larger (cf. [5], § v ∈ T u F is H ( v ) = { θ , θ , v Ω , v Ω } ⊥ ⊂ T u F, (37)where the ⊥ sign indicates the subspace annihilated by the 1-forms in braces. Lemma 14.
Let V u ⊂ T u F be the set of vectors v such that v θ = v θ = 0 but v ( ω ∧ ω ∧ ω ) = 0 . Then dim H ( v ) = 4 for an open dense subset of V u . For v ∈ V u , we will say that v is characteristic if dim H ( v ) > Proof.
Because T u F is 8-dimensional, the dimension of H ( v ) is 8 minus the dimension ofthe span of the 1-forms in braces in (37). This in turn depends only on the values of v Ω and v Ω . Suppose that v ω ω ω ω = pqab . (38)Then (cid:18) v Ω v Ω (cid:19) = (cid:20) a b − p − q − q + αb p − αa cb + αq − (2 ca + αp ) (cid:21) ω ω ω ω . Let R be the 2 × R has rank 2unless 0 = 2( ap + bq ) − α ( a + b ) = p + q + c ( a + b ) . (39)These equations fail to hold simultaneously on an open dense subset of V u . For vectors inthis set, v Ω and v Ω are linearly independent combinations of ω , ω , ω , ω , and hencedim H ( v ) = 4. Remark 5.
It is evident from the equations (39) that when c > a = b = p = q = 0, forming a 2-dimensional plane in T u F .(The same is true if c < | α | > /r .) However, when c < | α | ≤ /r theset of characteristic vectors is the union of the 2-dimensional plane and a 4-dimensionalsubmanifold, parametrized by a and b (not both zero) and the values of v ω and v ω .15s stated above, a Hopf hypersurface with an adapted unitary frame corresponds exactlyto a 3-dimensional integral submanifold Σ ⊂ F along which the independence condition ω ∧ ω ∧ ω = 0 holds. However, an adapted framing ( e , . . . , e ) along a given Hopf hypersurfacecan always be modified by a rotation e cos ψ e + sin ψ e , e
7→ − sin ψ e + cos ψ e foran arbitrary angle ψ . Under such changes, the corresponding section of F | M moves alongcircle within the fibers of F . On these circles the 1-form ω restricts to be nonzero butthe remaining 1-forms ω , ω , ω , ω , ω , ω , ω of the standard coframe restrict to be zero.Vector fields tangent to these circles are Cauchy characteristic vectors for I (see § is an integral manifold satisfying ω ∧ ω ∧ ω = 0,then Σ is transverse to the Cauchy characteristic circles, and the union of the circles throughΣ is a 4-dimensional integral manifold of I . Thus, M is associated to a unique I satisfying the independence condition ω ∧ ω ∧ ω ∧ ω = 0 . (40) Proposition 15.
Let α be any real number satisfying α + 4 c = 0 , and let I be the exteriordifferential system on F generated by θ and θ . Let Γ :
I ֒ → F be a real-analytic curvesuch that Γ ′ ( t ) ∈ V t ) for all t ∈ I and Γ ′ ( t ) is never characteristic. Then there exists aunique real-analytic submanifold Σ ⊂ F that contains Γ and is an integral submanifold of I satisfying the independence condition (40) .Proof. We will apply the Cartan-K¨ahler Theorem and Cartan’s Test for involutivity. (ForCartan’s Test, see Theorem 7.4.1 in [5]; for Cartan-K¨ahler see Theorem III.2.2 in [2].) Thiswill first require investigating the equations that define the set of 4-dimensional integralelements of I .At a point u ∈ F , a 4-dimensional subspace E ⊂ T u F is by definition an integral elementof I if all differential forms in I restrict to be zero on E . (We will consider only those 4-planes E that satisfy the independence condition (40).) In order that the algebraic generators of I vanish on E , the restrictions of the 1-forms ω , ω , ω , ω to E must satisfy ω = 0 , ω = αω , ω = λω + µω , ω = µω + νω , (41)for values of λ, µ, ν such that 2( λν − µ − c ) − α ( λ + ν ) = 0 . (42)This equation is obtained by substituting (41) in Ω .Using the above equations, it is easy to check that the set of integral 4-planes is a smooth2-dimensional submanifold of the Grassmannian of 4-dimensional subspaces of T u F exceptat points where λ = ν = α/ µ = 0. In that case (42) implies that α + 4 c = 0.Thus, our assumption about α guarantees that the set of integral 4-planes satisfying theindependence condition is a smooth submanifold. Since the Grassmannian has dimension16, the submanifold has codimension 14.Let u = Γ( t ) for an arbitrary t . Because v = Γ ′ ( t ) is non-characteristic and I has noalgebraic generators of degree higher than two, E = H ( v ) is the unique integral 4-planecontaining Γ ′ ( t ). 16artan’s Test for involutivity can be formulated in terms of the codimensions of thepolar spaces for a flag of integral elements terminating in E . To this end, let E = { } , E = { v } , let E ⊂ E be any 2- and 3-dimensional integral elements contained in E , andlet c k denote the codimension of H ( E k ) for k = 0 , . . .
3. Then c = 2 (because there areonly two 1-forms in the ideal), c = 4 as computed above, and c = c = 4 because I hasno additional algebraic generators. Then, because c + c + c + c = 14 coincides with thecodimension of the set of integral 4-planes, I is involutive and in particular the members ofthe flag are K¨ahler-regular integral elements. Successive applications of the Cartan-K¨ahlerTheorem give the existence and uniqueness of Σ containing Γ.Note that the image under π : F → f M of the four-dimensional integral manifold con-structed in Theorem 15 is a three-dimensional Hopf hypersurface. In the remainder of thissection we will solve a geometric initial value problem for such hypersurfaces. Proof of Theorem 13.
Let Γ : I → F be the lift of γ provided by the Frenet frame vectorssatisfying (32), with e = T , e = J T , e = N and e = − J N . We will first show that Γsatisfies the conditions of Proposition 15.Because the frame vectors e , e , e are orthogonal to γ ′ , we have Γ ∗ ω = Γ ∗ ω = Γ ∗ ω =0. Next, the Frenet equations (32) imply thatΓ ∗ ω = κ ds, Γ ∗ ω = Γ ∗ ω = 0 (43)(Here, s is an arclength coordinate along γ .) In particular, Γ is an integral curve of the1-forms θ = ω and θ = ω − αω . If we set v = Γ ′ ( s ) in (38) then a = 1, b = 0, p = κ , q = 0, and the characteristic equations (39) take the form 0 = 2 κ − α = κ + c ,which cannot simultaneously hold because of our assumption that α + 4 c = 0. Thus, Γ isnot characteristic, and by Proposition 15 there exists a unique integral manifold Σ passingthrough Γ. Then M = π (Σ) is a Hopf hypersurface containing γ . Because Γ ∗ ω = 0, γ istangent to the holomorphic distribution on M . Moreover, (43) shows that e ∇ e e = − κ e along γ , and thus γ is a principal curve in M with principal curvature κ ,Conversely, suppose M is a Hopf hypersurface containing γ , in which γ is tangent to theholomorphic distribution. Then there exists a unitary frame along M , in an open neigh-borhood of γ , such that e is tangent to γ and e is the structure vector. Moreover, if γ isprincipal in M , then e ∇ γ ′ e must be a multiple of e . Thus, the covariant derivatives of theframe vectors with respect to γ ′ satisfy the Frenet equations with τ = 0. Hence, the unitaryframe constructed along M , when viewed as a submanifold of F , passes through the curveΓ constructed above, and M must be the image of the unique integral manifold Σ throughΓ. In this section we will investigate the two possible kinds of shape operator, given by Propo-sition 9, for non-Hopf hypersurfaces in f M = CP or C H that are both *-Einstein andpseudo-Ryan. The two lemmas given in this section will prove Proposition 10. For a moredetailed explanation of the method of proof used here, see § Lemma 16.
There are no hypersurfaces in f M that satisfy condition (i) in Proposition 9. roof. Using (35) and (36), we see that adapted frames along such a hypersurface correspondto integral 3-manifolds of the Pfaffian system I generated differentially by the 1-forms θ = ω , θ = ω − βω , θ = ω − νω , θ = ω − βω − αω . We define this exterior differential system on F × R , with β, ν used as coordinates on thesecond factor, and α given in terms of these by solving (28): α = β ( ν − c )4 cν . We restrict to the open subset on which β and ν are both nonzero, and take the usualindependence condition.We compute the exterior derivatives of these generator 1-forms modulo themselves. Asusual, dθ ≡ θ , . . . , θ , while d θ θ θ + βν θ ≡ − π π π π − ( β/ν ) π π β cν (2 ν ( ν − c ) π + β ( ν + 4 c ) π ) ∧ ω ω ω mod θ , . . . , θ , (44)where π = − ν ω + 2 βν ω − ( β + c ) ω ,π = dβ − β ( ν + 4 c ) + 8 c c ω ,π = dν + ν ( β ν ( ν − c ) + 8 c (4 ν − c ))2 cβ ( ν + 4 c ) ω . In this computation, we further restrict to the open subset where ν + 4 c = 0. (For anysolution, this condition will either hold on an open subset, or ν will be locally constant; wewill consider the latter possibility below.)Inspecting the generator 2-forms given by (44) shows that, on any solution, π will be amultiple of ω , π will be a multiple of ω , and π will be a multiple of ω − ( β/ν ) ω , andfurthermore each of these multiples determines the others. More precisely, there will be afunction ρ such that the following 1-forms vanish: θ = π − ρ ω ,θ = π − ρβ ( ν + 4 c )4 cν ω ,θ = π − ρ ( ω − ( β/ν ) ω ) . To solve for this function, we add ρ as a new coordinate, and define the 1-forms θ , θ , θ onthe open subset of F × R where the nonzero conditions on β, ν hold. The Pfaffian systemgenerated by θ , . . . , θ is the prolongation of I .18e compute the exterior derivatives of the new 1-forms of the prolongation. In particular,we find dθ ∧ ω ≡ ρ ( β ν − c )2 cν ω ∧ ω ∧ ω ,dθ ∧ ( ω − ( β/ν ) ω ) ≡ ρ β ν (8 c − ν ) + 8 c (3 ν + 42 cν − c )8 c ν ( ν + 4 c ) ω ∧ ω ∧ ω modulo θ , . . . , θ . Because of our independence condition, at any point of M either ρ vanishes or both polynomials in β and ν in the numerators on the right-hand side vanish;moreover, one alternative or the other must hold on an open subset of M . Note that in thelatter case β and ν must be locally constant.Consider first the case where ρ vanishes identically on an open subset. Then the 1-forms π , π , π all vanish, as do their exterior derivatives. We compute ω ∧ dπ ≡ − ν c (cid:2) β ( ν − c ) + 2 c (cid:3) ω ∧ ω ∧ ω ,dπ ≡ β + c cν (cid:2) β ( ν + 4 c ) + 8 c (cid:3) ω ∧ ω modulo θ , . . . , θ , π , π , π . The vanishing of the expressions on the right of both equationsimplies that β + c = ν − c = 0, which is impossible.Thus we may work in a small open set where ρ = 0 and so β , ν , and α are constant. Werestrict I to a submanifold where β and ν are nonzero constants and compute d (cid:18) θ + βν θ (cid:19) ≡ − β ( ν + 4 c ) + 8 c c ω ∧ ω + β β ( ν − c ) + 2 c − cν cν ω ∧ ω modulo θ , . . . , θ . It is easy to check that the numerators of the terms on the right cannotsimultaneously vanish. This is a contradiction. Lemma 17.
Any hypersurface in f M that satisfies condition (ii) in Proposition 9 must have α, β, λ locally constant.Proof. Let U ⊂ F × R be the open subset where the coordinates α, β, ν on the second factorsatisfy β = 0 and ν = 0. On U , let I be the Pfaffian system generated by the 1-forms θ = ω , θ = ω + ( σ/ν ) ω − βω , θ = ω − νω , θ = ω − βω − αω , for a nonzero constant σ . Then an adapted framing along a hypersurface M satisfyingcondition (ii), for the given σ , generates an integral submanifold Σ of I . Moreover, Σ willlie inside the submanifold V ⊂ U determined by imposing (29) on the coordinates.We will first examine the structure equations of I on U , later passing to the restrictionof I to V. We assume that ν + 4 c = 0 and ν + σ = 0 on an open subset of Σ; we willaddress the case where ν is locally constant later. (Note that ν + σ cannot vanish on V ,since substituting ν = − σ in (29) implies that β = 0.)19he exterior derivatives of the generators of I satisfy dθ ≡ d θ θ θ ≡ − σν π − ν + σβν π π − ν + σβν π π π π π π ∧ ω ω ω mod θ , . . . , θ , (45)where π = dν − β ν ( ν − σ ) + ( ν + σ )(( c − Z ) ν + σ ( ν − α )) σβ ω ,π = βω + ( β − c − Z ) ν + σ ( ν − α ) ν ω + Z βνν + σ ω ,π = dβ − ( β + αν + c + σ + Z ) ω ,π = dα + (cid:18) β (3 ν − α ) + Z βνν + σ (cid:19) ω and Z = 4 cβ ( ν + 4( c − σ ) ν − cσ + σ )( ν + σ )( ν + 4 c )(4 c − σ ) + ( c + σ ) ν + (4 c + 20 cσ + σ ) ν + 3 cσ ( σ − c ) ν ( ν + 4 c ) . The structure equations (45) show that there is a 4-parameter family of 3-dimensional integralelements (satisfying the independence condition) at every point of U . However, we will onlyconsider those integral elements that are tangent to V .When restricted to V , the 1-forms π , . . . , π are no longer linearly independent. In fact,they satisfy a homogeneous linear relation( P π + Qπ + Rπ ) | V = 0 , (46)where P = β ( ν + (4 c + 2 σ ) ν + σ − cσ ) ν ( ν + σ ) , Q = 2 β ( ν + σ − c ) , R = ( σ − c )( ν + σ ) ν . (The value of Z is chosen so as to make the right-hand side of (46) equal to zero.)Because the pullbacks of the 2-forms in (45) to Σ ⊂ V must vanish, and the restrictionsof the π i to V satisfy (46), the pullbacks of the π i to Σ are determined up to multiple. Thatis, there exists a function ρ on Σ such that the pullbacks of these forms to Σ satisfy π = ρ ν σ (cid:18) ω − ν + σβν ω (cid:19) , (47) π = ρ ν σ ω , (48) π = ρ ( ω + Sω ) , (49) π = ρ ( Sω + T ω ) , (50)20here S, T are determined by substitution in (46), i.e., − ν ( ν + σ ) σβ P + Q + RS = 0 , ν σ + QS + RT = 0 . Note that, from now on, we will be working on V , taking α to be given by solving (29), i.e., α = β ν ( ν + σ − c )(4 c − σ )( ν + σ ) . (51)Differentiating both sides of (47) and wedging with ω − ν + σβν ω yields the integrabilitycondition[ ν ( ν + (2 σ − c ) ν − σ ) β − ( σ + ν )(4 c − σ ) (cid:0) ( σ + 6 c ) ν + (84 c − cσ − σ ) ν − c + 28 σc − cσ (cid:1) ] ρ = 0 . Similarly, differentiating both sides of (48) and wedging with ω yields the integrabilitycondition[4 ν (cid:0) ν + (4 c − σ ) ν + (8 cσ − σ ) ν − σ c + σ (cid:1) β − ( σ + ν )(4 c − σ ) (cid:0) (9 σ + 4 c ) ν + (16 c + 92 cσ − σ ) ν + 16 σ c + σ − σc (cid:1) ] ρ = 0 . Thus, either ρ = 0 on an open set in Σ, or ν is locally constant.Suppose ρ = 0. Then the 1-forms π , . . . , π vanish on Σ, and we may derive addi-tional integrability conditions as follows. By computing dπ and dπ modulo θ , . . . , θ and π , . . . , π , we obtain[ ν (cid:0) ν − c − σ (cid:1) β + (cid:0) σ + ν (cid:1) (4 c − σ ) (cid:0) ( σ + 16 c ) ν − c + 3 cσ (cid:1) ] W = 0 , (52)[(4 c − σ )( σ + ν ) (cid:0) σ + c ) ν + ( σ + 24 cσ + 8 c ) ν + 3 cσ − c σ (cid:1) (53)+ ν (cid:0) ν + 8 cν − σ − cσ + 16 c (cid:1) β ] W = 0 , where W = 4 ν (cid:0) ν + 4( c − σ ) ν + σ − cσ (cid:1) β +(4 c − σ )( ν + σ ) (cid:0) ν + (4 c + 16 σ ) ν − cσ − σ (cid:1) . Thus, either the polynomial W vanishes, or else both polynomials in square brackets in(52),(53) vanish. In the latter case, taking resultants with respect to β shows that ν must belocally constant. If W vanishes on an open set, then solving for β and differentiating W = 0modulo the θ i and π j yields another polynomial in ν which must vanish. Thus, again weconclude that ν must be locally constant.Finally we reconsider the original system I restricted to a submanifold of V on which ν is equal to a nonzero constant, and hence dν = 0. Differentiating the 1-forms of I revealsan additional integrability condition, as follows. We compute that d (cid:18) θ + ν + σβν θ (cid:19) ∧ (cid:18) ν + σβν ω − ν + σ − cσ − c ω (cid:19) = ν ( − ν + σ + 4 c ) β + ( σ − c )( ν + σ ) ( σν + 3 cσ − c + 16 cν ) βν ( σ − c ) ω ∧ ω ∧ ω . β also vanishes, we conclude that β is locally constant on solutions. It followsthat α , given by (51), and λ = − σ/ν are also locally constant.We note that, by doing further computations with this exterior differential system, onecan show that no solutions with β and ν both constant exist. We now construct an interesting class of non-Hopf hypersurfaces M in CP and C H , ob-tained by solving a certain underdetermined system of ordinary differential equations. Inparticular, this will show the existence of non-Hopf pseudo-Ryan hypersurfaces (see Theorem11 and Corollary 3).Let M be a hypersurface in C H n or CP n , with structure vector field W . At each p ∈ M we define the subspace H p ⊂ T p M as the smallest subspace that contains W and is invariantunder the shape operator A . Then M is Hopf if an only if H p is one-dimensional at each point.In what follows, we restrict to the case n = 2, and consider those hypersurfaces M where H is a smooth two-dimensional distribution on M . This means that we can locally constructan adapted orthonormal frame ( W, X, ϕX ) with respect to which the shape operator has theform A = α β β λ
00 0 ν , (54)and H is spanned by W and X at each point. Note that Y = ϕX is thus a principal vector.Our next result shows that it is relatively easy to generate examples of such hypersurfaces. Theorem 18.
Let α ( t ) , β ( t ) , λ ( t ) , ν ( t ) be analytic functions on an open interval I ⊂ R satisfying the underdetermined ODE system α ′ = β ( α + λ − ν ) ,β ′ = β + λ − λν + αν + c,λ ′ = (cid:18) (2 λ + ν ) β + ( ν − λ )( αλ − λ + c ) β (cid:19) , (55) with β ( t ) nowhere zero. Let γ ( t ) be a unit-speed analytic framed curve in f M , defined for t ∈ I , with transverse curvature ν ( t ) and zero holomorphic curvature and zero torsion. Thenthere exists a non-Hopf hypersurface M such that(i) the distribution H is rank 2 and integrable;(ii) M has a globally defined frame ( W, X, ϕX ) with respect to which the shape operatorhas the form (54) , such that α, β, λ and ν are constant along the leaves of H , and(iii) M contains γ as a principal curve to which the vector field Y = ϕX is tangent,and along which the components of A restrict to coincide with the given solution of the ODEsystem. roof. On F × R , with α, β, λ, ν as coordinates on the second factor, define the 1-forms θ = ω , θ = ω − λω − βω , θ = ω − νω , θ = ω − βω − αω , where the ω i and ω ij are pulled back to the product manifold via projection to the first factor.(We will restrict to the open subset of F × R where β = 0.) Then a non-Hopf hypersurface M equipped with an orthonormal frame with respect to which the shape operator has theform (54) can be lifted to a three-dimensional integral manifold f ( M ) of these 1-forms, byletting e = X , e = Y , e = W , e = ξ , and letting the coordinates α, β, λ, ν take the valuesof the corresponding components of A . (Note that this integral manifold also satisfies theusual independence condition ω ∧ ω ∧ ω = 0.) In what follows, we will derive necessaryconditions that this integral manifold must satisfy, if M is to satisfy the conditions (i) and(ii) of the theorem.If H is integrable then f ∗ ( dω ∧ ω ) = 0. We compute dω ∧ ω ≡ ( − ω + λω ) ∧ ω ∧ ω mod θ , θ , θ , θ . If ν is constant along the integral surfaces of H , then f ∗ ( dν ∧ ω ) = 0. We compute dθ ≡ ( ν − λ ) ω ∧ ω − βω ∧ ω + ( β − λ ( α − ν ) − c ) ω ∧ ω mod θ , . . . , θ , dν ∧ ω . So, the last two conditions imply that f ( M ) is also an integral of the 1-form θ = ω − λω − (cid:18) β + λ − αλ − cβ (cid:19) ω , Now we compute dθ ≡ ω ∧ ( dβ − ( β + λ − λν + αν + c ) ω ) + ω ∧ ( dα − β ( α + λ − ν ) ω ) mod θ , . . . , θ . Thus, the condition that α, β have nonzero derivatives only in the Y -direction implies that f ( M ) is also an integral of the 1-forms θ = dα − β ( α + λ − ν ) ω , θ = dβ − ( β + λ − λν + αν + c ) ω . Similarly, computing dθ modulo θ , . . . , θ , and using the condition that λ has a nonzeroderivative only in the Y -direction shows that f ( M ) is also an integral of θ = dλ − (cid:18) (2 λ + ν ) β + ( ν − λ )( αλ − λ + c ) β (cid:19) ω . In order to encode the condition that f ∗ ( dν ∧ ω ) = 0, we introduce a new coordinate p and define the 1-form θ = dν − pω . This, and the previous 1-forms θ i , are taken to be defined on the open set in F × R where β = 0. The framed hypersurfaces satisfying the conditions in the theorem are in one-to-one correspondence with integral manifolds (satisfying the independence condition) of thePfaffian system I defined by θ , . . . , θ . 23t is now easy to verify that this exterior differential system is involutive, with its onlynonzero Cartan character being s = 1. Moreover, the Cartan-Kahler Theorem implies thatintegral manifolds exists that pass through any non-characteristic 1-dimensional integralmanifold of I . In particular, any integral curve Γ along which ω = ω = 0 but ω = 0 isnon-characteristic. We will now show how such a curve corresponds exactly to a curve γ in f M satisfying the conditions in Theorem 18.Given γ , equipped with a unitary frame satisfying the Frenet equations (32), we constructa lift b γ into F by setting e = T , e = − J T , e = N , e = − J N . It follows that ω , ω , ω and ω = − ω pull back to be zero along b γ , and ω , ω and ω pull back to be multiples of ω that respectively are the holomorphic curvature, transverse curvature and torsion of γ .Thus, if γ has zero holomorphic curvature then b γ is an integral curve of ω . We further liftthe curve into F × R by setting α, β, λ, ν equal to the values given by the solution to theODE system, and p equal to dν/dt . Then it is easy to check that the lifted curve Γ is anintegral curve of θ , . . . , θ . Corollary 3.
Let α ( t ) , β ( t ) , λ ( t ) , ν ( t ) be analytic solutions defined for t ∈ I of the system(55), such that β is nowhere zero and β ν + (4 c + λν )( α ( λ − ν ) − β ) = 0 . Then the hypersurface M constructed by the previous theorem is a non-Hopf pseudo-Ryanhypersurface.Similarly, we can use the above theorem, together with solutions to the ODE systems,to construct non-Hopf hypersurfaces satisfying µ = 0 and any given algebraic conditioninvolving α, β, λ and ν . Theorem 18 provides a new construction for the non-Hopf hypersufaces in C H with constantprincipal curvatures. These have been classified by Berndt and Diaz-Ramos [1], who showedthat such hypersurfaces must be open subsets of homogeneous hypersurfaces. Thus, theybelong to a 1-parameter family of orbits under the action of a certain 3-dimensional groupof isometries of C H . One member of the family is a minimal hypersurface and the othersare its equidistant hypersurfaces.In Theorem 18, we take ν to be any constant in the range − /r < ν < /r and solve(55) for a constant solution. (In fact, a constant solution is possible only when ν lies in thisrange.) The shape operator can be written with respect to the frame ( W, X, ϕX ), used in §
4, as A = 1 r u − u v v u
00 0 u (56)where u = rν and v = (1 − u ) . The principal curvatures are ν and32 ν ± r r − r ν .
24n setting r = 2, we see that our result is consistent with Proposition 3.5 of [1]. References [1] J. Berndt and J.C. Diaz-Ramos,
Real hypersurfaces with constant principal curvatures inthe complex hyperbolic plane , Proc. Amer. Math. Soc. (2007), 3349–3357.[2] R. Bryant, S.-S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths,
Exterior DifferentialSystems , MSRI Publications, 1989.[3] T.E. Cecil and P.J. Ryan,
Focal sets and real hypersurfaces in complex projective space ,Trans. Amer. Math. Soc. (1982), 481–499.[4] T. Hamada,
Real hypersurfaces of complex space forms in terms of Ricci *-tensor , TokyoJ. Math (2002), 473–483.[5] T.A. Ivey and J.M. Landsberg, Cartan for Beginners: Differential geometry via movingframes and exterior differential systems , American Mathematical Society, 2003.[6] T.A. Ivey and P.J. Ryan,
Hopf hypersurfaces of small Hopf principal curvature in C H ,Geom. Dedicata (2009), 147–161.[7] T.A. Ivey and P.J. Ryan, The structure Jacobi operator for real hypersurfaces in CP and C H , Results Math. (2009), 473–488.[8] H.S. Kim and P.J. Ryan, A classification of pseudo-Einstein hypersurfaces in CP , Differ-ential Geom. Appl. (2008), 106–112.[9] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry II ,Wiley, New York,1969.[10] Kon, M.: Pseudo-Einstein real hypersurfaces in complex space forms. J. DifferentialGeom. , 339–354 (1979)[11] S. Maeda and Y. Ohnita, Helical geodesic immersions into complex space forms , Geom.Dedicata (1989), 93–114.[12] S. Maeda and T. Adachi, Holomophic helices in a complex space form , Proc. Amer. Math.Soc. (1997), 1197–1202.[13] S. Montiel,
Real hypersurfaces of a complex hyperbolic space , J. Math. Soc. Japan (1985), 515–535.[14] R. Niebergall and P.J. Ryan, Real hypersurfaces in complex space forms , pp. 233–205 in
Tight and taut submanifolds (ed. S.-S. Chern and T.E. Cecil), MSRI Publications, 1997.[15] T. Oguro and K. Sekigawa,
Four-dimensional almost K¨ahler Einstein and *-Einstein mani-folds , Geom. Dedicata (1998), 91–112.2516] S. Tachibana, On almost-analytic vectors in almost-K¨ahlerian manifolds , Tohoku Math. J. (1959), 247–265.[17] R. Takagi, Real hypersurfaces in a complex projective space with constant principal curva-tures , J. Math. Soc. Japan (1975), 43–53.[18] Q.M. Wang, Real hypersurfaces with constant principal curvatures in complex projectivespaces I , Sci. Sinica Ser. A26