On Einstein hypersurfaces of a remarkable class of Sasakian manifolds
aa r X i v : . [ m a t h . DG ] F e b On Einstein hypersurfaces of aremarkable class of Sasakian manifolds
Dario Di Pinto ∗ and Antonio Lotta ∗∗ Dipartimento di Matematica, Universit`a degli Studi di Bari AldoMoro, Via E. Orabona 4, 70125 Bari, Italy.February 10, 2021
Abstract
We present a non existence result of complete, Einstein hypersur-faces tangent to the Reeb vector field of a regular Sasakian manifoldwhich fibers onto a complex Stein manifold.
Key words : Einstein hypersurface · regular Sasakian manifold · Steinmanifold.
Mathematics Subject Classification (2020) : 53C25, 53C40, 53B25.
In [4], I. Hasegawa established that a Sasakian space form with nonconstantsectional curvature admits no Einstein hypersurfaces. The aim of this noteis to prove a new non existence result concerning Einstein hypersurfaces ofa relevant class of regular
Sasakian manifolds:
Theorem. If ( M, ϕ, ξ, η, g ) is a regular Sasakian manifold which fibersonto a complex Stein manifold, then M does not admit any complete Einsteinhypersurface tangent to ξ . ∗ e-mail: [email protected] ∗∗ e-mail: [email protected]
1e recall that a contact manifold (
M, η ) is called regular provided theReeb vector field ξ of the contact form η is, i.e. it determines a regular 1-dimensional foliation on M , so that the space B = M/ξ of maximal integralcurves of ξ is a manifold. When M carries a Sasakian metric g associated to η , yielding a Sasakian structure ( ϕ, ξ, η, g ) (we use the standard terminologyand notation according to [2]), since L ξ ϕ = 0 and L ξ g = 0, g induces in anatural way a metric g ′ on M/ξ and ϕ also descends to an almost complexstructure J . Denoting by π : M → B the canonical projection, it turns outby construction that π is a Riemannian submersion with ker( dπ ) x = R ξ x forevery x ∈ N , and dπ ◦ ϕ = J ◦ dπ and, morover ( B, J, g ′ ) is a K¨ahler manifold (see for instance [9] and [8]).Hence our assumption on the Sasakian manifold is that B , as a complexmanifold, can be realized (up to a biholomorphism) as a closed complexsubmanifold of some Euclidean space C d .For instance, according to a result due to H. Wu ([3], Theorem 4.9),it is known that every simply connected, complete K¨ahler manifold withnon-positive sectional curvature is a Stein manifold; in particular, the Her-mitian symmetric spaces of non-compact type are Stein manifolds, henceTakahashi’s Sasakian globally ϕ -symmetric spaces of non-compact type (see[10]) provide a wide class of examples of Sasakian manifolds to which ourresult applies.The proof of our result makes use of the natural CR structure of CR codimension 2 which is induced over any smooth hypersurface N ⊂ M ,under the assumption that N is everywhere tangent to the Reeb vector field ξ (see for instance [7] and section 2). We establish a basic formula relating theRicci tensor of N and the trace of a distinguished scalar Levi form of this CR structure (see (3.4)), implying that, in the Einstein case, π ( N ) is a weaklypseudoconcave real hypersurface of B . Hence a non compactness result byD. Hill and M. Nacinovich for weakly pseudoconcave CR submanifolds ofStein manifolds [5] is invoked to get the conclusion. Let’s start by recalling the definitions of CR manifolds, Levi-Tanaka formsand scalar Levi forms. In the following, given a vector bundle E over asmooth differential manifold M , we will denote by Γ( E ) the C ∞ ( M )-module2f global smooth sections of E .Let M be a smooth real manifold of dimension n , and let m, k ∈ N suchthat 2 m + k = n . If HM is a real vector subbundle of rank 2 m of thetangent bundle T M and J : HM → HM is a bundle isomorphism suchthat J = − Id , the couple ( HM, J ) is called a
CR structure on M if thefollowing properties hold for all X, Y ∈ Γ( HM ):(i) [ J X, J Y ] − [ X, Y ] ∈ Γ( HM );(ii) N J ( X, Y ) := [
J X, J Y ] − J [ J X, Y ] − J [ X, J Y ] − [ X, Y ] = 0.In this case (
M, HM, J ) is called a
CR manifold of type (m,k) and m, k arethe
CR dimension and the
CR codimension of the CR structure, respec-tively. Remark . Let S be a real submanifold of a complex manifold ( M, J ) andfor any p ∈ S set H p S := T p S ∩ J ( T p S ) . Because of the integrability of J , the couple ( HS, J | HS ) canonically definesa CR structure on S if the dimension of H p S is constant. In this case S istermed a CR submanifold of M .In particular, this condition is always satisfied when S is a real hypersurfaceof M and hence S is a CR manifold of CR codimension 1. Definition 2.2.
Let (
M, HM, J ) be a CR manifold of type ( m, k ). Givena point x ∈ M , the Levi-Tanaka form of M at x is the bilinear map L x : H x M × H x M → T x M/H x M defined by L x ( X, Y ) := p x ([ ˜ X, J ˜ Y ] x ) ∀ X, Y ∈ H x M, (2.1)where ˜ X, ˜ Y ∈ Γ( HM ) are two arbitrary extensions of X, Y and p : T M → T M/HM is the canonical projection on the quotient bundle
T M/HM .It is known that L x is well defined, i.e. the value p x ([ ˜ X, J ˜ Y ] x ) onlydepends on the values of ˜ X, ˜ Y at x , that is on X and Y .Moreover, according to (i) above, L x turns to be a vector valued symmetricHermitian form on the holomorphic tangent space H x M with respect to thecomplex structure J := J x , that is L x ( X, Y ) = L x ( J X, J Y ) , L x ( X, Y ) = L x ( Y, X ) (2.2)3or all
X, Y ∈ H x M .Given a point x on the CR manifold ( M, HM, J ), we will denote by H x M := { ω ∈ T ∗ x M | ω ( X ) = 0 ∀ X ∈ H x M } the annihilator of H x M ⊂ T x M . Then we recall the following definition. Definition 2.3.
Let (
M, HM, J ) be a CR manifold, x ∈ M and ω ∈ H x M .The Hermitian form L ω : H x M × H x M → R s.t. L ω ( X, Y ) := ωL x ( X, Y ) (2.3)is called the scalar Levi form determined by ω at x .The next lemma represents a sort of naturality property of the Levi-Tanaka form with respect a particular class of maps between CR manifoldswhich preserve the CR structures. Definition 2.4.
Let (
M, HM, J ) and (
N, HN, J ′ ) be two CR manifolds.A smooth map π : M → N is called CR map if dπ ( HM ) ⊂ HN and dπ ◦ J = J ′ ◦ dπ . Lemma 2.5.
Let ( M, HM, J ) and ( N, HN, J ′ ) be two CR manifolds havingthe same CR dimension, let π : M → N be a CR map and assume that forevery x ∈ M , ( dπ ) x : H x M → H π ( x ) N is an isomorphism. Then, given x ∈ M , the following diagram commutes: H x M × H x M L x / / π ∗ × π ∗ (cid:15) (cid:15) T x M/H x M π ∗ (cid:15) (cid:15) H y N × H y N L ′ y / / T y N/H y N where y = π ( x ) , π ∗ = ( dπ ) x and L x , L ′ y are the Levi-Tanaka forms of M and N respectively.Proof. Let us denote by p x : T x M → T x M/H x M and q y : T y N → T y N/H y N the canonical projections. As an immediate consequence of the definitionof CR map, the differential π ∗ descends to the quotient and, with abuse ofnotation, we still denote the quotient map by π ∗ : T x M/H x M → T y N/H y N .Now consider X, Y ∈ H x M : according to (2.1), L ′ y ( π ∗ X, π ∗ Y ) = q y [ Z, J ′ W ] y ,4here Z, W ∈ Γ( HN ) are two extensions of π ∗ X and π ∗ Y . Since for ev-ery a ∈ M , ( dπ ) a : H a M → H π ( a ) N is an isomorphism, we can define twoextensions ˜ X, ˜ Y ∈ Γ( HM ) of X and Y respectively by putting˜ X a := ( dπ ) − a ( Z π ( a ) ) , ˜ Y a := ( dπ ) − a ( W π ( a ) ) . It turns out that ˜ X and ˜ Y are π -related to Z and W respectively, and hence[ ˜ X, J ˜ Y ] is π -related to [ Z, J ′ W ] too, since dπ commutes with the almostcomplex structures J and J ′ . Finally, we have: π ∗ L x ( X, Y ) = π ∗ ( p x [ ˜ X, J ˜ Y ] x )= q y ( π ∗ [ ˜ X, J ˜ Y ] x )= q y [ Z, J ′ W ] y = L ′ y ( π ∗ X, π ∗ Y ) . Corollary 2.6.
In the same hypotesis and notation of the previous Lemma,for every ψ ∈ H y N one has that π ∗ L ′ ψ = L π ∗ ψ . We remark that the scalar Levi forms L ω are symmetric and hence itmakes sense to consider their index i ( L ω ), defined as the minimum betweenthe number of positive and negative eigenvalues of L ω .More specifically, we recall the following terminology from CR geometry;see for instance [6]. Definition 2.7.
Let (
M, HM, J ) be a CR manifold of type ( m, k ) and let x ∈ M . M is called pseudoconvex at x if L ω is positive definite for some ω ∈ H x M .If there exists a global section ω ∈ Γ( H M ) such that L ω is positive definiteat each point x ∈ M , M is called strongly pseudoconvex . M is said pseudoconcave at x if i ( L ω ) > ω ∈ H x M , ω = 0. M is said weakly pseudoconcave at x if L ω = 0 or i ( L ω ) > ω ∈ H x M .In this regard we recall that a Sasakian manifold ( M, ϕ, ξ, η, g ), as definedin [2], is a particular kind of strongly pseudoconvex CR manifold of hyper-surface type, i.e. of CR codimension 1. We shall refer to [2] for the notationand basic facts concerning Sasakian geometry. We only remark that in thiscase the CR structure is given by the contact distribution D = ker η = h ξ i ⊥ and the almost complex structure is J = ϕ |D . Therefore, for any x ∈ M ,5 x M is spanned by η x and, up to scaling, we have only one scalar Levi form L η x . Moreover, since M is a contact metric manifold, the identity dη ( X, Y ) = g ( X, ϕY )yields that L η x = 2 g x | H x M × H x M . We end this section by recalling the definition of Stein manifold (formore information, see for instance [3]) and a theorem due to Hill and Naci-novich [5, 6], which provides a basic restriction to the topology of CR weaklypseudoconcave submanifolds of a Stein manifold. Definition 2.8. A Stein manifold is a closed complex submanifold of C d ,for some d ≥ Theorem 2.9.
Every weakly pseudoconcave CR submanifold of a Steinmanifold cannot be compact. Let (
M, ϕ, ξ, η, g ) be a Sasakian manifold and let N be a hypersurface of M ,tangent to the Reeb vector field ξ . At each point x ∈ N , let us consider thelinear subspace of T x N defined by H x N := { X ∈ T x N | X ⊥ ξ x and ϕX ∈ T x N } . Observe that, if ν ∈ T x N ⊥ is a unit normal vector at x , then we have thefollowing orthogonal decomposition: T x N = h ξ x i ⊕ h ϕν i ⊕ H x N. It follows that HN is a subbundle of T N with constant rank and in [7] M.Munteanu proved that the couple (
HN, ϕ | HN ) defines a CR structure of CR codimension 2 on N . We remark that he assumes the orientability of N ,but this is unnecessary for our aim and the result holds true even if N isnot orientable.The CR structure ( HN, ϕ | HN ) on the hypersurface N allows us to consider,for every unit normal vector ν , the scalar Levi form L ω attached to thecovector ω ( X ) = g x ( X, ϕν ) ∀ X ∈ T x N. (3.1)6e shall denote this scalar Levi form with the symbol L ν and in the fol-lowing proposition we establish the relationship between L ν and the secondfundamental form of the hypersurface N . Proposition 3.1.
Let ( M, ϕ, ξ, η, g ) be a Sasakian manifold and let N ⊂ M be a hypersurface, tangent to ξ , with second fundamental form α . Let ν bea unit normal vector at some point x ∈ N . Then one has: L ν ( X, X ) = g x ( α ( X, X ) + α ( ϕX, ϕX ) , ν ) (3.2) for every X ∈ H x N .Proof. First we recall that Sasakian manifolds are characterized by means ofthe following identity, involving the covariant derivatives of ϕ with respectto the Levi-Civita connection (see [2]):( ∇ X ϕ ) Y = g ( X, Y ) ξ − η ( Y ) X. (3.3)Now, fix x ∈ N , X ∈ H x N and consider a smooth section in Γ( HN ) whichextends X and a local normal vector field extending ν . Then ϕX is againtangent to N . Using the fact that X , ϕX and ϕν are all orthogonal to ξ and identity (3.3), we get: L ν ( X, X ) == g x ([ X, ϕX ] , ϕν ) == g x ( ∇ X ϕX, ϕν ) − g x ( ∇ ϕX X, ϕν ) == g x ( ϕ ∇ X X, ϕν ) + g x ( ϕ ∇ ϕX X, ν ) == g x ( ∇ X X, ν ) + g x ( ∇ ϕX ϕX, ν ) == g x ( α ( X, X ) + α ( ϕX, ϕX ) , ν ) . We shall use this formula to establish an identity relating the trace (withrespect to g ) of L ν and the Ricci tensor field of N .Hereinafter we will denote with an overline the relevant geometric entitiesof the hypersurface N (Levi-Civita connection, curvature, etc.). Proposition 3.2.
Let ( M n +1 , ϕ, ξ, η, g ) be a Sasakian manifold and let N ⊂ M be a hypersurface tangent to ξ . Let x ∈ N and let ν ∈ T x N ⊥ be aunit normal vector. Then one has: Ric( ξ, ϕν ) = 12 tr( L ν ) . (3.4)7 roof. By a well known property of Sasakian manifolds (see [2]), for every X ∈ X ( M ), R ( ξ, X ) X = g ( X, X ) ξ − η ( X ) X. (3.5)Then, given X ∈ Γ( HN ), since ξ and X are normal to ϕν , it follows that R ( ξ, X, ϕν, X ) = g ( R ( ξ, X ) X, ϕν ) = 0 . (3.6)Since ϕX = −∇ X ξ is still tangent to N , we also deduce that the normalcomponent of ∇ X ξ vanishes, i.e. α ( X, ξ ) = 0. Moreover, α ( ξ, ϕν ) = g ( ∇ ϕν ξ, ν ) ν = g ( − ϕ ν, ν ) ν = ν. (3.7)Therefore, by using the Gauss formula, for every X ∈ Γ( HN ) we have that R ( ξ, X, ϕν, X ) = g ( α ( X, X ) , ν ) . (3.8)Thus, fixed a local orthonormal frame of T N of type { ξ, ϕν, E i , ϕE i } i =1 ,...,n − ,with E i , ϕE i ∈ Γ( HN ), from (3.8) and (3.2) we get:Ric( ξ, ϕν ) = n − X i =1 (cid:2) R ( ξ, E i , ϕν, E i ) + R ( ξ, ϕE i , ϕν, ϕE i ) (cid:3) = n − X i =1 g ( α ( E i , E i ) + α ( ϕE i , ϕE i ) , ν )= n − X i =1 L ν ( E i , E i ) = 12 tr( L ν ) , where the last equality follows from the fact that L ν is Hermitian and sym-metric.Now we come to the proof of our main result. Theorem 3.3. If ( M, ϕ, ξ, η, g ) is a regular Sasakian manifold which fibersonto a complex Stein manifold, then M does not admit any complete Einsteinhypersurface tangent to ξ .Proof. Assume by contradiction that M admits a complete Einstein hyper-surface N tangent to ξ , with Einstein constant c .Let ∇ be the Levi-Civita connection of N . Since ∇ ξ ξ = 0, from the Gauss8quation we deduce that ∇ ξ ξ = 0. Moreover, since ξ is a Killing vector fieldon N , the operator A ξ := −∇ ξ is skew-symmetric and henceRic( ξ, ξ ) = − div( A ξ ξ ) − tr( A ξ ) = − tr( A ξ ) ≥ . It follows that c = cg ( ξ, ξ ) = Ric( ξ, ξ ) ≥ . If c = 0, then A ξ = 0, i.e. ξ is ∇ -parallel and this leads to a contradiction.Indeed, if we consider X ∈ Γ( HN ), with X = 0, from ∇ X ξ = 0 and theGauss equation we would get − ϕX = ∇ X ξ = α ( X, ξ ) , where − ϕX ∈ Γ( HN ) is non zero and tangent to N , while α ( X, ξ ) is normal.Therefore c > N , Myers’ theorem ensuresthat N is compact.Moreover, from Proposition 3.2 we have:tr( L ν ) = 2Ric( ξ, ϕν ) = 2 cg ( ξ, ϕν ) = 0 . (3.9)Now, let π : M → M/ξ be the canonical projection, where (
M/ξ, J, g ′ ) is aStein manifold; π is a Riemannian submersion whose fibers are 1-dimensionalsubmanifolds of M tangent to ξ and dπ ◦ ϕ = J ◦ dπ. (3.10)Since at every x ∈ M , ker( dπ ) x = R ξ x , we have that π | N : N → M/ξ hasconstant rank. Hence, according to Theorem 3.5.18 in [1], S := π ( N ) is asmooth hypersurface of M/ξ and it carries a CR structure (defined as inRemark 2.1), having the same CR dimension of N . Moreover, (3.10) impliesthat π : N → S is a CR map, such that at every point x ∈ N the differential( dπ ) x : H x N → H π ( x ) S is an isomorphism.Fix a point y = π ( x ) ∈ S , with x ∈ N ; if ψ ∈ H y S , then π ∗ ψ belongsto the vector space H x N , which is spanned by ω and η , with ω as in (3.1).Actually, if π ∗ ψ = αω + βη , for some numbers α, β , evaluating at ξ we obtain β = 0 and hence π ∗ ψ = αω . Using Corollary 2.6 we get π ∗ L ′ ψ = L π ∗ ψ = α L ν and by (3.9) we conclude that tr( L ′ ψ ) = 0, so that L ′ ψ = 0 or i ( L ′ ψ ) > S is a compact weakly pseudoconcave CR hypersurface of thecomplex Stein manifold M/ξ , thus contradicting Theorem 2.9.9ith just a small change in the previous proof, we also get the followingresult.
Theorem 3.4. If ( M, ϕ, ξ, η, g ) is a regular Sasakian manifold which fiberson a complex Stein manifold, then M cannot admit any compact hypersur-face N , tangent to ξ and such that at any point of N ξ is an eigenvector ofthe Ricci operator Q of N .Proof. It suffices to note that if Qξ = αξ along N , with α ∈ C ∞ ( N ), thenone has tr( L ν ) = 2Ric( ξ, ϕν ) = 2 g ( Qξ, ϕν ) = 0 . Hence the proof ends with the same argument of the previous one.
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