Sasakian immersions of Sasaki-Ricci solitons into Sasakian space forms
aa r X i v : . [ m a t h . DG ] F e b SASAKIAN IMMERSIONS OF SASAKI-RICCI SOLITONS INTO SASAKIAN SPACEFORMS
G. PLACINIA
BSTRACT . Let ( g, X ) be a Sasaki-Ricci soliton on a Sasakian manifold S . We prove that if ( S, g ) admits a local Sasakian immersion in a Sasakian space form S ( N, c ) of constant φ -sectional cur-vature c , then S is η -Einstein and its η -Einstein constants are rational. Moreover, if c ≤ − , S islocally equivalent to the Sasakian space form S ( n, c ) and its η -Einstein constants are determinedby c . Further results are obtained in the compact setting, i.e. when c > − , under additionalhypotheses.
1. I
NTRODUCTION AND STATEMENTS OF THE MAIN RESULTS
Since the publication of [3] Sasakian geometry has received growing interested. This is possi-bly due to the connections with physics and the abundance of structures that concur in a Sasakianmanifold. This abundance allows one to study Sasakian geometry from several points of view.A Sasakian manifold S is a contact manifold endowed with a compatible metric g and holomor-phic structure transverse to the Reeb foliation. The cases where the metric satisfies some addi-tional property are widely studied. An instance of such properties is given by Sasaki-Einsteinmetrics. Sasaki-Einstein manifolds have drown the attention of many mathematicians in the lastfew decades. A generalization of Sasaki-Einstein metrics is represented by η -Einstein metrics, i.e.,transversally Einstein metrics. Namely, Sasakian manifolds that satisfy the relation Ric g = λg + νη ⊗ η for some constants λ, ν ∈ R , where Ric g is the Ricci tensor of the metric and η denotes the contactform. We refer the reader to [3, 4] for an introduction. A further generalization of such metricsis given by Sasaki-Ricci solitons (SRS for short). These have been introduced in [10] as specialsolutions of the Sasaki-Ricci flow of [18]. Since then SRS have received growing interest, see forinstance [16, 19].Sasakian geometry seats between two K¨ahler geometries. Namely, the K¨ahler structure of thecone and the one transverse to the Reeb foliation. For this reason it is often referred to as the odddimensional counterpart to K¨ahler geometry. It is then natural to try to extend known results forK¨ahler manifolds to the Sasakian setting. Such an instance is given by the problem of immersingcertain classes of K¨ahler manifolds into complex space forms. This translates in the Sasakiansetting to Question:
Which Sasakian manifolds admit a Sasakian immersion into a Sasakian space form?
The question above is clearly too general so that it needs to be specialized to specific classesin order to be answered. For instance the problem has been studied when the immersion is takento be CR, i.e. to preserve the underlying CR structure but not necessarily the metric. In fact,Ornea and Verbitsky [15] proved that any compact Sasakian manifold admits a CR embedding
Date : February 5, 2021; © G. Placini 2020. n a Sasakian manifold diffeomorphic to a sphere. The case of η -Einstein manifolds admitting aSasakian immersion has been, in some instances only partially, solved in [1, 8, 12]. On the otherhand, this question has not been addressed for Sasaki-Ricci solitons. Our goal in this article is tocharacterize SRS that admit a (local) immersion into a Sasakian space form.In light of the K¨ahler/Sasaki parallel, our results extend those of [13] to the Sasakian setting.The first of these is the following Theorem 1.
Let S be a (2 n + 1) -dimensional complete regular Sasakian manifold endowed with aSasaki-Ricci soliton. Suppose there exists a neighbourhood U p of a point p ∈ S and an immersion ψ : U p −→ S ( N, c ) into a Sasakian space form S ( N, c ) with c ≤ − .Then S is Sasaki equivalent to S ( n, c ) / Γ for a discrete subgroup Γ of the group of Sasakiantranformations of S ( n, c ) . In particular, S is η -Einstein and its η -Einstein constants ( λ, ν ) aredetermined by c .If additionally U p = S , then Γ = 0 and, up to a Sasakian transformation of S ( n, c ) , ψ is of theform ψ ( z, t ) = ( z, , t + a ) for a constant a . In contrast with the K¨ahler analogue of Theorem 1 proved in [13], we are not assuming theexistence of a global immersion of S into a Sasakian space form. Furthermore, while in [13] theEinstein constant is a rational multiple of the holomorphic curvature, in the Sasakian setting theEinstein constants are completely determined by the φ -sectional curvature c , cf. (9). Namely, wehave Ric g = (cid:18) n + 12 ( c + 3) − (cid:19) g + (cid:18) n + 2 − n + 12 ( c + 3) (cid:19) η ⊗ η . Our second result addresses the problem in the case where the Sasakian space form is compact,i.e. the standard sphere S N +1 . Very little is known in this setting, compared to the non-compactcase, even in the Sasakian manifold is assumed to be eta -Einstein, see [8]. For this reason we statea general result and several corollaries under additional or different hypotheses. Theorem 2.
Let S be a (2 n + 1) -dimensional complete regular Sasakian manifold endowed with aSasaki-Ricci soliton. Suppose there exists a neighbourhood U p of a point p ∈ S and an immersion ψ : U p −→ S N +1 into the standard Sasakian sphere. Then ( S, g ) is a η -Einstein Sasakian manifoldwhose η -Einstein constants ( λ, ν ) are given by λ = 4 µ − for some µ ∈ Q . Theorem 1 and Theorem 2 have interesting consequences when investigating which SRS admita global immersion into a Sasakian space forms. It is natural to consider first η -Einstein Sasakianmanifolds, as these are trivially SRS. Combining the results in the compact and non-compact caseswe see that the existence of an immersion into certain Sasakian space forms constrains the possiblevalues of the η -Einstein constants ( λ, ν ) . Corollary 1.
A complete, η -Einstein manifold with rational η -Einstein constants cannot be im-mersed in a Sasakian space form S ( N, c ) of irrational φ -sectional curvature c ∈ R \ Q . The assumptions of completeness ensures that the leaf space is a manifold, i.e., that the Sasakianmanifold S fibers over a K¨ahler manifold. Next we concentrate on some consequences of Theo-rem 2. Firstly, let us consider the case where S is compact and the codimension of the immersionis arbitrary, cf. Theorems and in [8]. orollary 2. Let S be a (2 n + 1) -dimensional compact Sasakian manifold endowed with a Sasaki-Ricci soliton. Suppose there exists an immersion ψ : S −→ S N +1 into the standard Sasakiansphere. Then ( S, g ) is either a η -Einstein Sasakian manifold with rational η -Einstein constants ( λ > n, ν ) or it is Sasaki equivalent to S n +1 . Notice that in this case S is necessarily regular since it can be immersed into a regular Sasakianmanifold. Lastly we turn our attention to the case of small codimension to prove the two followingcorollaries, cf. part ( i ) of the main theorem of [12] and [8, Theorem 2]. Corollary 3.
Let S be a (2 n + 1) -dimensional complete (not necessarily compact) Sasakian man-ifold endowed with a Sasaki-Ricci soliton. Suppose there exists an immersion ψ : S −→ S n +3 into the standard Sasakian sphere. Then ( S, g ) is Sasaki equivalent to S n +1 or the Boothby-Wangbundle over the complex quadric Q n ∈ C P n +1 . Corollary 4.
Let S be a (2 n + 1) -dimensional compact Sasakian manifold endowed with a Sasaki-Ricci soliton. Suppose there exists an immersion ψ : S −→ S n +5 into the standard Sasakiansphere. Then ( S, g ) is Sasaki equivalent to S n +1 or the Boothby-Wang bundle over the complexquadric Q n ∈ C P n +1 . We believe that these results can be generalized to arbitrary codimension. Such a generalizationof Corollary 3 and Corollary 4 is related to an analogous conjecture on Sasakian immersions of η -Einstein manifolds, see [8] and referecences therein.2. S ASAKIAN MANIFOLDS AND S ASAKI -R ICCI SOLITONS
Sasakian manifolds and immersions.
We begin with some definitions and known results,for a more exhaustive treatment we refer to the monograph by Boyer and Galicki [3]. All manifoldsare assumed to be smooth, connected and oriented.A
K-contact structure ( S, η, φ, R, g ) on a manifold S consists of a contact form η and an endo-morphism φ of the tangent bundle T S satisfying the following properties: • φ = − Id + R ⊗ η where R is the Reeb vector field of η , • φ |D is an almost complex structure compatible with the symplectic form d η on D = ker η , • The Reeb vector field R is Killing with respect to the metric g ( · , · ) = d η ( φ · , · ) + η ( · ) η ( · ) .Given such a structure one can consider the almost complex structure I on the Riemannian cone (cid:0) S × (0 , ∞ ) , t g + d t (cid:1) given by • I = φ on D = ker η , • I ( R ) = t∂ t .A Sasakian structure is a K-contact structure ( S, η, φ, R, g ) such that the associated almost com-plex structure I is integrable. We call a manifold S Sasakian if it is equipped with a Sasakianstructure.A Sasakian manifold is called regular (respectively quasi-regular , irregular ) if its Reeb foliationis such. Every regular compact Sasakian manifold is a Boothby-Wang fibration S over a projectivemanifold ( K, ω ) with ω representing an integral class ([2, 3]), that is, the principal S -bundle π : S −→ K with Euler class [ ω ] and connection -form η such that π ∗ ( ω ) = d η . This is notnecessarily true in the non-compact case. Nevertheless one can prove a similar statement under ome additional conditions, cf. the proof of Theorem 1. In general the Reeb foliation F of aSasakian structure is transversally K¨ahler. This endows the space of leaves with a K¨ahler structure.Two Sasakian manifolds ( S , η , φ , R , g ) and ( S , η , φ , R , g ) are equivalent if there existsa diffeomorphism f : S −→ S such that f ∗ η = η and f ∗ g = g . If this holds then f also preserves the endomorphism φ and the Reeb vector field. A Sasakianequivalence from a Sasakian manifold ( S, η, φ, R, g ) to itself is called a Sasakian transformation of ( S, η, φ, R, g ) .In this article we discuss Sasakian immersions into Sasakian space forms. Given two Sasakianmanifolds ( S , η , φ , R , g ) and ( S , η , φ , R , g ) , a Sasakian immersion of S in S is an im-mersion ψ : S −→ S such that ψ ∗ η = η , ψ ∗ g = g ,ψ ∗ R = R and ψ ∗ ◦ φ = φ ◦ ψ ∗ . Sasakian η -Einstein manifolds and Sasaki-Ricci solitons. The Riemannian properties ofSasakian manifold, in particular Sasaki-Einstein and η -Einstein metrics, have received great atten-tion from many authors, partially due to their connection to physics. We recall now the definitionsand main properties of these structures with a particular focus on the relation with the transverseK¨ahler geometry, we refer the interested reader to [3, 4]On a Sasakian manifold ( S, η, φ, R, g ) the tangent bundle splits canonically as T S = D ⊕ T F where D = ker η and T F denotes the tangent to the Reeb foliation F . The transverse K¨ahlergeometry is given by ( D , φ | D , d η ) . When the space of leaves of the Reeb foliation is a K¨ahlermanifold ( K, J, ω ) we have a fibration π : S −→ K such that π ∗ ω = d η and π ∗ ◦ φ = J ◦ π ∗ . In virtue of this the metric decomposes as(1) g = g T ⊕ η ⊗ η where g T ( · , · ) = d η ( · , φ · ) . With an abuse of notation we write g T for both the transverse metricand the metric on K . It follows from (1) that the Riemannian properties of S can be expressed interms of those of the transverse K¨ahler geometry and of the contact form η . For instance, the Riccitensor of g is given by(2) Ric g = Ric g T − g. A Sasakian manifold ( S, η, φ, R, g ) is said to be η -Einstein if the Ricci tensor satisfies(3) Ric g = λg + νη ⊗ η for some constants λ, ν ∈ R . It follows from (2) and (3) that a Sasakian manifold is η -Einstein withconstants ( λ, ν ) if, and only if, its transverse geometry is K¨ahler-Einstein with Einstein constant λ + 2 . Since on a K-contact manifold the Ricci tensor satisfies Ric(
R, X ) = 2 nη ( X ) we have that λ + ν = 2 n . Therefore, a Sasakian η -Einstein manifold is Einstein if and only if λ = 2 n , that is, ifit has a transverse K¨ahler-Einstein geometry with Einstein constant n + 2 .Generalizing η -Einstein manifolds are Sasaki-Ricci solitons. In order to introduce them weneed to recall known facts about the transverse K¨ahler geometry. On a Sasakian manifold S of imension n + 1 there exists a covering { U α } with foliated charts ϕ α : U α −→ ϕ α ( U α ) ⊂ R × C n .Denote by π α the following map π α = π C n ◦ ϕ α : U α −→ V α ⊂ C n . The Sasakian structure is transversally holomorphic, that is, the maps π α ◦ π − β : V α ∩ V β −→ V α ∩ V β are biholomorphisms. A basic p -form α on S is a p -form such that ι R α = 0 , L R α = 0 . It is easy to see that the exterior derivative d sends basic forms to basic forms. Therefore wedenote it by d B when we want to emphasize that it is restricted to basic forms. Suppose now that ( x, z , . . . , z n ) are local coordinates in U α . If a basic form α can be written locally as α = a i ,...,i p + q d z i ∧ · · · ∧ d z i p ∧ d¯ z i p +1 ∧ · · · ∧ d¯ z i p + q , then α is said to be a basic ( p, q ) -form. One can show that such a local form is also of type ( p, q ) in any chart U β with U α ∩ U β = ∅ . Therefore we have well defined operators ∂ B , respectively ∂ B ,of degree (1 , , resp. (0 , , such that d B = ∂ B + ∂ B . Definition 2.1.
A complex vector field X on a Sasakian manifold S is called Hamiltonian holo-morphic if it satisfies the following conditions(a) the vector field d π α ( X ) is holomorphic on V α ,(b) the function u X : = iη ( X ) is such that ∂u X = − iι X d η . In [18] Smoczyk, Wang and Zhang introduced the Sasaki-Ricci flow(4) dd t g T ( t ) = − (cid:0) Ric g T ( t ) − λg T ( t ) (cid:1) with the aim of proving the existence of η -Einstein metrics. In order to study the Sasaki-Ricci flowon positive Sasakian manifolds, Futaki, Ono and Wang [10] defined Sasaki-Ricci solitons as a pair ( g, X ) consisting of a Sasakian metric g and a Hamiltonian holomorphic vector field X such that(5) Ric g T = (2 n + 2) g T + L X g T . The equation (5) is commonly used in literature to define Sasaki-Ricci solitons, see for instance[16, 19]. On the other hand, on a complex manifold M a K¨ahler-Ricci soliton (KRS for short) isdefined to be a pair ( g, X ) where g is a K¨ahler metric and X is a holomorphic vector field on M satisfying(6) Ric g = λg + L X g with λ ∈ R . Although a compact K¨ahler manifold does not admit non-trivial KRS with λ ≤ ,these cases are widely studied on open K¨ahler manifolds. In analogy with the K¨ahler setting wegeneralize the above definition and give the following Definition 2.2. A Sasaki-Ricci soliton (SRS in short) on a Sasakian manifold S is a pair ( g, X ) consisting of the Sasakian metric g and a Hamiltonian holomorphic vector field X such that (7) Ric g T = λg T + L X g T for some λ ∈ R . If a manifold S is endowed wit a SRS, with an abuse of notation we will simply say that S isa SRS. One can easily construct examples of SRS on open Sasakian manifolds, also in the casewhere λ ≤ , as bundles over certain gradient KRS, see for instance [6, 7, 9]. emark 1. By definition a Sasaki-Ricci soliton ( X, g ) on a regular Sasakian manifold is a KRSon the transverse K¨ahler geometry. In particular, if the space of leaves of the Reeb foliation is asmooth manifold K , then it has a canonically induced KRS. Sasakian space forms.
Let ( S, η, φ, R, g ) be a Sasakian manifold. If Sec is the ordinaryRiemannian sectional curvature of g , then the φ -sectional curvature H of g is defined by H ( X ) = Sec( X, φX ) for all vector fields X of unit length orthogonal to R .A Sasakian space form S ( n, c ) is a Sasakian manifold of dimension n + 1 with constant φ -sectional curvature H ≡ c . Tanno [20] proved that there are three types of Sasakian space forms,namely, those with H ≡ c < − , = − and > − . These are analogous to complex spaceforms, that is, complex manifolds of constant holomorphic sectional curvature. Indeed, under theBoothby-Wang correspondence, constant φ -sectional curvature c corresponds precisely to constantholomorphic transverse sectional curvature c + 3 .Explicitly Tanno proved that every Sasakian space form is a quotient of one of the followingthree by a subgroup of Sasakian transformations. • If c > − , S ( n, c ) is Sasaki equivalent to the Sasakian sphere S n +1 ( c ) . This is theBoothby-Wang bundle over C P n ( c + 3) . • If c = − , S ( n, c ) is Sasaki equivalent to R n +1 ( − . The transverse K¨ahler structure isthe standard one on C n . • If c < − , S ( n, c ) is Sasaki equivalent to B n +1 C ( c + 3) × R where the transverse K¨ahlerstructure is that of the hyperbolic complex space B n +1 C ( c + 3) of constant holomorphicsectional curvature c + 3 .Finally we make note of the Ricci tensor for the transverse K¨ahler structures of constant holo-morphic sectional curvature c + 3 :(8) Ric g T = n + 12 ( c + 3) g T which in turn implies(9) Ric g = (cid:18) n + 12 ( c + 3) − (cid:19) g + (cid:18) n + 2 − n + 12 ( c + 3) (cid:19) η ⊗ η .
3. P
ROOF OF THE MAIN RESULTS
Proof of Theorem 1.
Let ( X, g ) be a Sasaki-Ricci soliton on a Sasakian manifold S . Suppose thereexists a neighbourhood U p of a point p ∈ S and an immersion ψ : U p −→ S ( N, c ) into a Sasakianspace form S ( N, c ) with c ≤ − .We cannot conclude that S is an S -bundle over a K¨ahler manifold because S is not necessarilycompact. Nevertheless, the Reeb foliation still defines a fibration π : S −→ K over a K¨ahlermanifold because S is regular and complete, see [17]. ow ψ covers a K¨ahler immersion into a definite complex space form K ( N, c + 3) of dimension N and constant holomorphic curvature c + 3 , see [8, 11]. Thus we get the commutative diagram U p S ( N, c ) V x K ( N, c + 3) ψπ π ′ φ where x = π ( p ) and V x = π ( U p ) ⊂ K .Moreover, the space of leaves K of the Reeb fibration is a K¨ahler manifold equipped with aK¨ahler–Ricci soliton (d π ( X ) , g T ) , cf. Remark 1. The existence of the Ricci soliton ( g T , d π ( X )) implies that the K¨ahler metric g T is real-analytic, (see [14, Corollary 1.3]),. Therefore K is acomplex manifold equipped with a real-analytic K¨ahler metric which admits a local immersion V p −→ K ( N, c +3) into a complex space form. Then a classical result of Calabi [5] implies that forevery point y ∈ K there exists a neighbourhood V y and a K¨ahler immersion V y −→ K ( N, c + 3) .Hence, the main result of [13] implies that ( V y , g T ) is K¨ahler–Einstein. As reviewed in Section 2,this is equivalent to π − ( V y ) being η -Einstein. Now ( S, g ) is η -Einstein because the sets π − ( V y ) cover S .Therefore the thesis of Theorem 1 follows from a result of Bande, Cappelletti–Montano and Loi[1] on immersions of η -Einstein manifolds into Sasakian space forms. (cid:3) Proof of Theorem 2.
Let ( X, g ) be a Sasaki-Ricci soliton on a Sasakian manifold S . Suppose thereexists a neighbourhood U p of a point p ∈ S and an immersion ψ : U p −→ S N +1 into the standardSasakian sphere.Following the arguments in the proof of Theorem 1 we see that S is the total space of a fibration π : S −→ K over a K¨ahler manifold K . Moreover, ψ covers a K¨ahler immersion U p S N +1 V x C P Nψπ π ′ φ where V x = π ( U p ) and π ′ is the standard Hopf bundle.Hence, every point of K admits a neighbourhood which can be immersed in C P N . This impliesthat K is a K¨ahler -Einstein manifold with Einstein constant λ + 2 = 4 µ for some rational number µ . We conclude that S is η -Einstein with constants ( λ, ν ) given by λ = 4 µ − . (cid:3) Proof of Corollary 2.
Since S is admits a global Sasakian immersion into a regular Sasakian man-ifold, it is itself regular, cf. [8, Proposition 1]. Thus S is a Boothby-Wang bundle π : S −→ K over a compact K¨ahler manifold K which is endowed with the induced KRS.Moreover, K admits a K¨ahler immersion in C P N which is covered by ψ : S S N +1 K C P N . ψπ π ′ φ gain by [13] K is K¨ahler -Einstein with rational Einstein constant λ + 2 . Therefore S is a η -Einstein manifold with constants ( λ, ν ) which admits a Sasakian immersion in S N +1 .Now [8, Theorem 3] implies that λ ≥ n . Moreover, if λ = 2 n , then S is Sasaki-Einstein and[8, Theorem 1] implies that S is Sasakian equivalent to S n +1 . (cid:3) Remark 2.
With the definition of SRS given in [10]
Theorem 1 and Corollary 2 imply that acompact SRS immersed in a Sasakian space form is necessarily Sasakian equivalent to a standardsphere S n +1 .Proof of Corollary 3 and Corollary 4. Notice that the Sasakian structure on S is regular becuse itadmits an immersion into a regular Sasakian manifold.We want to prove now that S fibers over a K¨ahler manifold K . As discussed in the proofs abovethis follows from the classical result of Boothby and Wang [2] when S is compact and from [17]when S is complete.The same line of arguments as in the proof of Theorem 1 shows that S is η -Einstein. NowCorollary 3 follows from the main result of [12] while Corollary 4 is a consequence of [8, Theo-rem 2] (cid:3) R EFERENCES [1] G. Bande, B. Cappelletti–Montano and A. Loi, η -Einstein Sasakian immersions in non-compact Sasakian spaceforms , Ann. Mat. Pura Appl. (4) (2020), no. 6, 2117–2124.[2] W. M. Boothby and H. C. Wang, On contact manifolds , Ann. of Math. (2) (1958) 721–734.[3] C. Boyer and K. Galicki, Sasakian geometry, Oxford Mathematical Monographs. Oxford University Press, Ox-ford, 2008.[4] C. Boyer, K. Galicki and P. Matzeu, On eta-Einstein Sasakian geometry , Comm. Math. Phys. (2006), no. 1,177–208.[5] E. Calabi,
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Sasakian manifolds with constant φ -holomorphic sectional curvature , Tohoku Math. J. (2) (1969),501–507.D IPARTIMENTO DI M ATEMATICA E I NFORMATICA , U
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