Para-CR Geometry. II. Generalizations of a para-CR structure - PDF document

Para-CR Geometry. II. Generalizations of a para-CR structure Dmitri V. Alekseevsky 24th March 2009 1

An ǫ -quaternionic CR structure Summary We define the notion of ǫ -quaternionic CR struc- ture on 4 n + 3-dimensional manifold M as a triple ω = ( ω 1 , ω 2 , ω 3 ) of 1-forms, which satisfy some structure equations. Here ǫ = ± 1. It defines a decomposition TM = V M + HM of the tangent bundle into a direct sum of the horizontal subbundle HM = Ker ω and a com- plementary vertical rank 3 subbundle V and an ǫ -hypercomplex structure J = ( J 1 , J 2 , J 3 = J 1 J 2 = − J 2 J 1 ) on HM . It is a joint work with Y. Kamishima. 2

• We associate with ω a 1-parameter family of pseudo-Riemannain metrics g t . • We show that the metric g 1 is Einstein metric and • that the ǫ -quaternionic CR structure is equiv- alent to an ǫ -3-Sasakian structure subordi- nated to the pseudo-Riemannian manifold ( M, g 1 ) (which is defined as Lie algebra of Killing fields span( ξ 1 , ξ 2 , ξ 3 ) isomorphic to sp (1 , R ) for ǫ = 1 and sp (1) for ǫ = − 1 with some properties.) 3

• The cone C ( M ) = R + × M, ˆ g = dr 2 + r 2 g 1 over a ǫ -quaternionic CR manifold ( M, ω ) has a canonical ǫ -hyperKaehler structure (in particular, is Ricci-flat). • Under assumption that the Killing vectors ξ α are complete and define (almost) free action of the corresponding group K = Sp (1 , R ) or Sp (1), the orbit manifold Q = M/K has a structure of ǫ -quaternionic K¨ ahler manifold. 4

• Homogeneous manifolds with ǫ -quaternionic CR structure are described. • A simple reduction construction, which as- sociates with an ǫ -quaternionic CR man- ifold with a symmetry group G a new ǫ - quaternionic CR manifold M ′ = µ − 1 (0) /G is presented. 5

Definition of ǫ -quaternionic CR structure Let ω = ( ω 1 , ω 2 , ω 3 ) be a triple of 1-forms on a 4 n + 3-dimensional manifold M which are linearly independent, i.e . ω 1 ∧ ω 2 ∧ ω 3 � = 0. We associate with ω a triple ρ = ( ρ 1 , ρ 2 , ρ 3 ) of 2- forms by ρ 1 = dω 1 − 2 εω 2 ∧ ω 3 , ρ 2 = dω 2 + 2 ω 3 ∧ ω 1 , ρ 3 = dω 3 + 2 ω 1 ∧ ω 2 , where ε = +1 or − 1. 6

A triple J = ( J 1 , J 2 , J 3 ) of anticommuting en- domorphisms of a distribution HM is called an ε -hypercomplex structure if they satisfies ε - quaternionic relations J 2 1 = − εJ 2 2 = − εJ 2 3 = − 1 , J 3 = J 1 J 2 = − J 2 J 1 . For ε = 1, this means that J 1 is a complex structure and J 2 , J 3 = J 1 J 2 are para-complex structures. 7

Definition 1 A triple of 1-forms ω = ( ω α ) is called a ǫ -quaternionic CR structure if the associated 2 -forms ( ρ α ) are non degenerate on the distribution H = Ker ω = Ker ω 1 ∩ Ker ω 2 ∩ Ker ω 3 , have the same 3-dimensional kernel V and three fields of endomorphisms J α on H , defined by J 1 = − ε ( ρ 3 | H ) − 1 ◦ ρ 2 | H , J 2 = ( ρ 1 | H ) − 1 ◦ ρ 3 | H , J 3 = ( ρ 2 | H ) − 1 ◦ ρ 1 | H . form an ǫ -hypercomplex structure on HM . 8

Associated metric and the canonical vector fields We define 1) one-parameter family of pseudo-Riemannian metrics g t on a ǫ -quaternionic CR manifold M by g t = g t V + g H (1) where g t V = t ( ω 1 ⊗ ω 1 − εω 2 ⊗ ω 2 − εω 3 ⊗ ω 3 ) (2) � = t ε α ω α ⊗ ω α g H = ρ 1 ◦ J 1 = ρ 2 ◦ J 2 = − ερ 3 ◦ J 3 , and ε 1 = 1 , ε 2 = ε 3 = − ε . 9

2) Three vertical vector fields ξ α ∈ V M dual to the 1-forms ω α : ω β ( ξ α ) = δ αβ . Then g t ◦ ξ α = tε α ω α , (3) 10

Properties of the canonical vector fields We will denote by L X the Lie derivative in di- rection of X . (1) The vector fields ξ α preserves the decom- position TM = V ⊕ H and span a 3-dimensional Lie algebra a ε of Killing fields of the metric g t for t > 0, which is isomorphic to s p (1 , R ) for ε = 1 and s p (1) for ε = − 1. More pre- cisely, the following cyclic relations hold: [ ξ 1 , ξ 2 ] = 2 ξ 3 , [ ξ 2 , ξ 3 ] = − 2 εξ 1 , [ ξ 3 , ξ 1 ] = 2 ξ 2 . 11

(2) The vector field ξ α preserves the forms ω α and ρ α for α = 1 , 2 , 3. Moreover, the fol- lowing relations hold : L ξ 2 ω 3 = −L ξ 3 ω 2 = ω 1 , L ξ 3 ω 1 = ε L ξ 1 ω 3 = − εω 2 , L ξ 1 ω 2 = ε L ξ 2 ω 1 = ω 3 , and similar relations for ρ α . 12

Extension of the endomorphisms J α We extend endomorphisms J α of H to endo- morphisms ¯ J α of the tangent bundle TM by : ¯ ¯ J α ξ α = 0 , J α | H = J α ¯ ¯ J 1 ξ 2 = − εξ 3 , J 1 ξ 3 = εξ 2 , (4) ¯ ¯ J 2 ξ 3 = ξ 1 , J 2 ξ 1 = εξ 3 , ¯ ¯ J 3 ξ 1 = ξ 2 , J 3 ξ 2 = εξ 1 . The endomorphisms ¯ J α , α = 1 , 2 , 3 at a point x constitute the standard basis of the Lie algebra sp (1) ε ⊂ End( T x M ) where sp (1) − 1 = sp (1) , sp (1) +1 = sp (1 , R ) . 13

Integrability of extended endomorphisms ¯ J α Proposition 2 Let ( M, ω ) be an ǫ -quaternionic Then T α := Ker ω α , ¯ CR manifold. J α ) is a Levi-non-degenerate ( − ǫ α ) -CR structure. This means that T α is a contact distribution, and J α is an integrable ǫ α -complex structure, i.e. J 1 is a complex structure and J 2 , J 3 are para-complex structure. Integrability means that the Nijenhuis tensor N ( ¯ J α , ¯ J α ) T α = 0 or, equivalently, the eigendis- tributions T ± α of ¯ J α | T α are involutive. 14

Contact metric 3-structure Let ( M, g ) be a (4 n + 3)-dimensional manifold with a pseudo-Riemannian metric g of signa- ture (3 + 4 p, 4 q ). A contact metric 3-structure is ( ξ α , φ α ) , α = 1 , 2 , 3 } where ξ α are three orthonormal vec- tor fields which define contact forms η α := g ◦ ξ α , and φ α are skew-symmetric endomor- phisms with kernel Ker φ α = R ξ α such that (1) φ 2 α = − Id, φ α ( ξ α ) = 0; α | ξ ⊥ (2) φ α = φ β φ γ − ξ β ⊗ η γ = − φ γ φ β + ξ γ ⊗ η β . 15

K-contact structures A contact metric 3-structure is called a K -contact 3-structure if ξ α are Killing fields. 16

3-Sasakian structure A K -contact 3-structure is called Sasakian 3- structure if it is normal, i. e. if the following tensors N η α ( · , · ) , ( α = 1 , 2 , 3) vanish: N η α ( X, Y ) := N φ α ( X, Y )+( Xη α ( Y ) − Y η α ( X )) ξ α (5) ( ∀ X, Y ∈ TM ). Here N φ α ( X, Y ) = [ φ α X, φ α Y ] − [ X, Y ] − φ α [ φ α X, Y ] − φ α [ X, φ α Y ] is the usual Nijenhuis tensor of a field of endo- morphisms φ α . 17

Theorem The following three structures on a (4 n + 3)-dimensional manifold M are equiva- lent: contact pseudo-metric 3-structures, quater- nionic CR structures and pseudo-Sasakian 3- structures. If ω is a quaternionic CR structure, then the associated 3-Sasakian metric is � g = g 1 = ω α ⊗ ω α + ρ 1 ◦ J 1 , the Killing vectors are vertical vectors ξ α dual to 1-forms ω α and φ α = ¯ J α . The metric g is an Einstein metric. 18

ε -quaternionic K¨ ahler manifolds Recall that a (pseudo-Riemannian) quaternionic K¨ ahler manifold (respectively, para-quaternionic K¨ ahler manifold) is defined as a 4 n -dimensional pseudo-Riemannian manifold ( M, g ) with the holonomy group H ⊂ Sp (1) Sp ( p, q ) (respectively, H ⊂ Sp (1 , R ) · Sp ( n, R )). 19

This means that the manifold M admits a par- allel 3-dimensional subbundle Q (quaternionic subbundle) of the bundle of endomorphisms which is locally generated by three skew-symmetric endomorphisms J 1 , J 2 , J 3 which satisfy the quater- nionic relations (respectively, para-quaternionic relations). To unify the notations, we will call a quaternionic K¨ ahler manifold also a ( ε = − 1)-quaternionic K¨ ahler manifold and a para- quaternionic K¨ ahler manifold a ( ε = 1)-quaternionic K¨ ahler manifold. 20

Any ε -quaternionic K¨ ahler manifold is Einstein and its curvature tensor has the form R = νR 1 + W , 21

ε -quaternionic K¨ ahler manifold associated with a ε -quaternionic CR manifolds Let ( M, ω ) be a ε -quaternionic CR manifold. We will assume that the Lie algebra sp (1) ε = span( ξ α ) of vector fields is complete and gen- erates a free action of the group Sp (1) ε on M . Then the orbit space B = M/Sp (1) ε is a smooth manifold and π : M → B is a principal bundle. Moreover, the pseudo-Riemannian metric g 1 of ( M, ω ) induces a pseudo-Riemannian metric g B on B such that π : M → B is a Riemannian submersion with totally geodesic fibers. 22

Theorem 3 The space of orbit N = M/Sp (1) ε has a natural structure of ε -quaternionic K¨ ahler manifold. Conversely, The bundle of orthonormal frames over a ε -quaternionic K¨ ahler manifold N has a structure of ε -quaternionic CR manifold. 23

Examples of homogeneous ǫ -quaternionic CR manifolds of classical Lie groups: ( C n ) ε = +1, S H ′ n,n = Sp n +1 ( R ) /Sp n ( R ); ε = − 1, S p,q = Sp p +1 ,q /Sp p,q H ( A n ) ε = +1, SU p +1 ,q +1 /U p,q ; ε = − 1, SU p +2 ,q /U p,q ; ( BD n ) ε = +1, SO p +2 ,q +2 /SO p,q , ε = − 1, SO p +4 ,q /SO p,q . 24

Momentum map of a ǫ -quaternionic CR manifold with a symmetry group Let ( M, ω ) be a ǫ -quaternionic CR manifold and G be a Lie group of its authomorphisms, i.e. transformations which preserves 1-forms ω . We denote by g ∗ the dual space of the Lie algebra g of G and we will consider elements X ∈ g as vector fields on M . We define a mo- mentum map as µ : M → R 3 ⊗ g ∗ , x �→ µ x , µ x ( X ) = ω ( X x ) = ( ω 1 ( X x ) , ω 2 ( X x ) , ω 3 ( X x )) ∈ R 3 . 25