Hypo contact and Sasakian SU ( 2 ) -structures in 5-dimensions - PowerPoint PPT Presentation

Hypo contact Anna Fino Hypo contact and Sasakian SU ( 2 ) -structures in 5-dimensions structures on Lie groups Sasakian structures Sasaki-Einstein structures Hypo structures η -Einstein structures Hypo-contact structures Classification “Workshop on CR and Sasakian Geometry”, Consequences New metrics with holonomy Luxembourg – 24 - 26 March 2008 SU ( 3 ) Sasakian structures on Lie groups 3-dimensional Lie groups General results 5-dimensional Lie groups SU ( n ) -structures in ( 2 n + 1 ) -dimensions Generalized Killing spinors Contact SU ( n ) -structures Examples Contact reduction Anna Fino 1 Dipartimento di Matematica Università di Torino

Hypo contact SU ( 2 ) -structures in 5 -dimensions Anna Fino SU ( 2 ) -structures in Definition 5-dimensions Sasakian structures An SU ( 2 ) -structure ( η, ω 1 , ω 2 , ω 3 ) on N 5 is given by a 1-form η Sasaki-Einstein structures Hypo structures and by three 2-forms ω i such that η -Einstein structures Hypo-contact structures ω i ∧ ω j = δ ij v , v ∧ η � = 0 , Classification i X ω 3 = i Y ω 1 ⇒ ω 2 ( X , Y ) ≥ 0 , Consequences New metrics with holonomy SU ( 3 ) where i X denotes the contraction by X . Sasakian structures on Lie groups 3-dimensional Lie groups General results Remark 5-dimensional Lie groups SU ( n ) -structures in The pair ( η, ω 3 ) defines a U ( 2 ) -structure or an almost contact ( 2 n + 1 ) -dimensions metric structure on N 5 , i.e. ( η, ξ, ϕ, g ) such that Generalized Killing spinors Contact SU ( n ) -structures Examples ϕ 2 = − Id + ξ ⊗ η, Contact reduction η ( ξ ) = 1 , g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) . 2

Hypo contact SU ( 2 ) -structures in 5 -dimensions Anna Fino SU ( 2 ) -structures in Definition 5-dimensions Sasakian structures An SU ( 2 ) -structure ( η, ω 1 , ω 2 , ω 3 ) on N 5 is given by a 1-form η Sasaki-Einstein structures Hypo structures and by three 2-forms ω i such that η -Einstein structures Hypo-contact structures ω i ∧ ω j = δ ij v , v ∧ η � = 0 , Classification i X ω 3 = i Y ω 1 ⇒ ω 2 ( X , Y ) ≥ 0 , Consequences New metrics with holonomy SU ( 3 ) where i X denotes the contraction by X . Sasakian structures on Lie groups 3-dimensional Lie groups General results Remark 5-dimensional Lie groups SU ( n ) -structures in The pair ( η, ω 3 ) defines a U ( 2 ) -structure or an almost contact ( 2 n + 1 ) -dimensions metric structure on N 5 , i.e. ( η, ξ, ϕ, g ) such that Generalized Killing spinors Contact SU ( n ) -structures Examples ϕ 2 = − Id + ξ ⊗ η, Contact reduction η ( ξ ) = 1 , g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) . 2

Hypo contact SU ( 2 ) -structures in 5 -dimensions Anna Fino SU ( 2 ) -structures in Definition 5-dimensions Sasakian structures An SU ( 2 ) -structure ( η, ω 1 , ω 2 , ω 3 ) on N 5 is given by a 1-form η Sasaki-Einstein structures Hypo structures and by three 2-forms ω i such that η -Einstein structures Hypo-contact structures ω i ∧ ω j = δ ij v , v ∧ η � = 0 , Classification i X ω 3 = i Y ω 1 ⇒ ω 2 ( X , Y ) ≥ 0 , Consequences New metrics with holonomy SU ( 3 ) where i X denotes the contraction by X . Sasakian structures on Lie groups 3-dimensional Lie groups General results Remark 5-dimensional Lie groups SU ( n ) -structures in The pair ( η, ω 3 ) defines a U ( 2 ) -structure or an almost contact ( 2 n + 1 ) -dimensions metric structure on N 5 , i.e. ( η, ξ, ϕ, g ) such that Generalized Killing spinors Contact SU ( n ) -structures Examples ϕ 2 = − Id + ξ ⊗ η, Contact reduction η ( ξ ) = 1 , g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) . 2

Hypo contact Sasakian structures Anna Fino An almost contact metric structure ( η, ξ, ϕ, g ) on N 2 n + 1 is said contact metric if 2 g ( X , ϕ Y ) = d η ( X , Y ) . On N 5 the pair ( η, ω 3 ) SU ( 2 ) -structures in 5-dimensions Sasakian structures defines a contact metric structure if d η = − 2 ω 3 . Sasaki-Einstein structures Hypo structures ( η, ξ, ϕ, g ) is called normal if η -Einstein structures Hypo-contact structures N ϕ ( X , Y ) = ϕ 2 [ X , Y ] + [ ϕ X , ϕ Y ] − ϕ [ ϕ X , Y ] − ϕ [ X , ϕ Y ] , Classification Consequences New metrics with holonomy SU ( 3 ) satisfies the condition N ϕ = − d η ⊗ ξ . Sasakian structures on Lie groups Definition (Sasaki) 3-dimensional Lie groups General results A Sasakian structure on N 2 n + 1 is a normal contact metric 5-dimensional Lie groups SU ( n ) -structures in structure. ( 2 n + 1 ) -dimensions Generalized Killing spinors Contact SU ( n ) -structures Examples Theorem (Boyer, Galicki) Contact reduction A Riemannian manifold ( N 2 n + 1 , g ) has a compatible Sasakian structure if and only if the cone N 2 n + 1 × R + equipped with the g = dr 2 + r 2 g is Kähler. conic metric ˜ 3

Hypo contact Sasakian structures Anna Fino An almost contact metric structure ( η, ξ, ϕ, g ) on N 2 n + 1 is said contact metric if 2 g ( X , ϕ Y ) = d η ( X , Y ) . On N 5 the pair ( η, ω 3 ) SU ( 2 ) -structures in 5-dimensions Sasakian structures defines a contact metric structure if d η = − 2 ω 3 . Sasaki-Einstein structures Hypo structures ( η, ξ, ϕ, g ) is called normal if η -Einstein structures Hypo-contact structures N ϕ ( X , Y ) = ϕ 2 [ X , Y ] + [ ϕ X , ϕ Y ] − ϕ [ ϕ X , Y ] − ϕ [ X , ϕ Y ] , Classification Consequences New metrics with holonomy SU ( 3 ) satisfies the condition N ϕ = − d η ⊗ ξ . Sasakian structures on Lie groups Definition (Sasaki) 3-dimensional Lie groups General results A Sasakian structure on N 2 n + 1 is a normal contact metric 5-dimensional Lie groups SU ( n ) -structures in structure. ( 2 n + 1 ) -dimensions Generalized Killing spinors Contact SU ( n ) -structures Examples Theorem (Boyer, Galicki) Contact reduction A Riemannian manifold ( N 2 n + 1 , g ) has a compatible Sasakian structure if and only if the cone N 2 n + 1 × R + equipped with the g = dr 2 + r 2 g is Kähler. conic metric ˜ 3

Hypo contact Sasakian structures Anna Fino An almost contact metric structure ( η, ξ, ϕ, g ) on N 2 n + 1 is said contact metric if 2 g ( X , ϕ Y ) = d η ( X , Y ) . On N 5 the pair ( η, ω 3 ) SU ( 2 ) -structures in 5-dimensions Sasakian structures defines a contact metric structure if d η = − 2 ω 3 . Sasaki-Einstein structures Hypo structures ( η, ξ, ϕ, g ) is called normal if η -Einstein structures Hypo-contact structures N ϕ ( X , Y ) = ϕ 2 [ X , Y ] + [ ϕ X , ϕ Y ] − ϕ [ ϕ X , Y ] − ϕ [ X , ϕ Y ] , Classification Consequences New metrics with holonomy SU ( 3 ) satisfies the condition N ϕ = − d η ⊗ ξ . Sasakian structures on Lie groups Definition (Sasaki) 3-dimensional Lie groups General results A Sasakian structure on N 2 n + 1 is a normal contact metric 5-dimensional Lie groups SU ( n ) -structures in structure. ( 2 n + 1 ) -dimensions Generalized Killing spinors Contact SU ( n ) -structures Examples Theorem (Boyer, Galicki) Contact reduction A Riemannian manifold ( N 2 n + 1 , g ) has a compatible Sasakian structure if and only if the cone N 2 n + 1 × R + equipped with the g = dr 2 + r 2 g is Kähler. conic metric ˜ 3

Hypo contact Sasakian structures Anna Fino An almost contact metric structure ( η, ξ, ϕ, g ) on N 2 n + 1 is said contact metric if 2 g ( X , ϕ Y ) = d η ( X , Y ) . On N 5 the pair ( η, ω 3 ) SU ( 2 ) -structures in 5-dimensions Sasakian structures defines a contact metric structure if d η = − 2 ω 3 . Sasaki-Einstein structures Hypo structures ( η, ξ, ϕ, g ) is called normal if η -Einstein structures Hypo-contact structures N ϕ ( X , Y ) = ϕ 2 [ X , Y ] + [ ϕ X , ϕ Y ] − ϕ [ ϕ X , Y ] − ϕ [ X , ϕ Y ] , Classification Consequences New metrics with holonomy SU ( 3 ) satisfies the condition N ϕ = − d η ⊗ ξ . Sasakian structures on Lie groups Definition (Sasaki) 3-dimensional Lie groups General results A Sasakian structure on N 2 n + 1 is a normal contact metric 5-dimensional Lie groups SU ( n ) -structures in structure. ( 2 n + 1 ) -dimensions Generalized Killing spinors Contact SU ( n ) -structures Examples Theorem (Boyer, Galicki) Contact reduction A Riemannian manifold ( N 2 n + 1 , g ) has a compatible Sasakian structure if and only if the cone N 2 n + 1 × R + equipped with the g = dr 2 + r 2 g is Kähler. conic metric ˜ 3

Hypo contact Sasakian structures Anna Fino An almost contact metric structure ( η, ξ, ϕ, g ) on N 2 n + 1 is said contact metric if 2 g ( X , ϕ Y ) = d η ( X , Y ) . On N 5 the pair ( η, ω 3 ) SU ( 2 ) -structures in 5-dimensions Sasakian structures defines a contact metric structure if d η = − 2 ω 3 . Sasaki-Einstein structures Hypo structures ( η, ξ, ϕ, g ) is called normal if η -Einstein structures Hypo-contact structures N ϕ ( X , Y ) = ϕ 2 [ X , Y ] + [ ϕ X , ϕ Y ] − ϕ [ ϕ X , Y ] − ϕ [ X , ϕ Y ] , Classification Consequences New metrics with holonomy SU ( 3 ) satisfies the condition N ϕ = − d η ⊗ ξ . Sasakian structures on Lie groups Definition (Sasaki) 3-dimensional Lie groups General results A Sasakian structure on N 2 n + 1 is a normal contact metric 5-dimensional Lie groups SU ( n ) -structures in structure. ( 2 n + 1 ) -dimensions Generalized Killing spinors Contact SU ( n ) -structures Examples Theorem (Boyer, Galicki) Contact reduction A Riemannian manifold ( N 2 n + 1 , g ) has a compatible Sasakian structure if and only if the cone N 2 n + 1 × R + equipped with the g = dr 2 + r 2 g is Kähler. conic metric ˜ 3