Preliminaries Main results How to obtain Sasaki-Einstein and paraSasaki-Einstein metrics from contact metric ( κ, µ )-spaces Ver´ onica Mart´ ın-Molina Joint work with A. Carriazo (University of Sevilla) and B. Cappelletti Montano (Universit` a degli Studi di Cagliari) Department of Geometry and Topology University of Sevilla PADGE 2012
Preliminaries Main results Index Preliminaries Contact metric Geometry Paracontact metric Geometry Main results
Preliminaries Main results Almost contact metric structure M 2 n +1 is said to be an almost contact manifold if there exists on it a triple ( ϕ, ξ, η ) satisfying ϕ 2 = − I + η ⊗ ξ, η ( ξ ) = 1 .
Preliminaries Main results Almost contact metric structure M 2 n +1 is said to be an almost contact manifold if there exists on it a triple ( ϕ, ξ, η ) satisfying ϕ 2 = − I + η ⊗ ξ, η ( ξ ) = 1 . It follows that ϕξ = 0 , η ◦ ϕ = 0 , rank( ϕ ) = 2 n .
Preliminaries Main results Almost contact metric structure M 2 n +1 is said to be an almost contact manifold if there exists on it a triple ( ϕ, ξ, η ) satisfying ϕ 2 = − I + η ⊗ ξ, η ( ξ ) = 1 . It follows that ϕξ = 0 , η ◦ ϕ = 0 , rank( ϕ ) = 2 n . Given an almost contact manifold ( M , ϕ, ξ, η ), we define an almost complex structure J on M × R as � � � � X , f d ϕ X − f ξ, η ( X ) d J = dt dt
Preliminaries Main results Almost contact metric structure M 2 n +1 is said to be an almost contact manifold if there exists on it a triple ( ϕ, ξ, η ) satisfying ϕ 2 = − I + η ⊗ ξ, η ( ξ ) = 1 . It follows that ϕξ = 0 , η ◦ ϕ = 0 , rank( ϕ ) = 2 n . Given an almost contact manifold ( M , ϕ, ξ, η ), we define an almost complex structure J on M × R as � � � � X , f d ϕ X − f ξ, η ( X ) d J = dt dt An almost contact manifold will be called normal if the almost complex structure J is integrable. This condition is equivalent to N ϕ := [ ϕ, ϕ ] + 2 d η ⊗ ξ = 0 .
Preliminaries Main results Every almost contact manifold ( M , ϕ, ξ, η ) admits a compatible metric, i.e. a Riemannian metric g satisfying g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) .
Preliminaries Main results Every almost contact manifold ( M , ϕ, ξ, η ) admits a compatible metric, i.e. a Riemannian metric g satisfying g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) . M is said to be an almost contact metric manifold with structure ( ϕ, ξ, η, g ).
Preliminaries Main results Every almost contact manifold ( M , ϕ, ξ, η ) admits a compatible metric, i.e. a Riemannian metric g satisfying g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) . M is said to be an almost contact metric manifold with structure ( ϕ, ξ, η, g ). We define the fundamental 2-form as the 2-form Φ on M such that Φ ( X , Y ) = g ( X , ϕ Y ).
Preliminaries Main results Every almost contact manifold ( M , ϕ, ξ, η ) admits a compatible metric, i.e. a Riemannian metric g satisfying g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) . M is said to be an almost contact metric manifold with structure ( ϕ, ξ, η, g ). We define the fundamental 2-form as the 2-form Φ on M such that Φ ( X , Y ) = g ( X , ϕ Y ). If Φ = d η , then η is a contact form and we say that ( M , ϕ, ξ, η, g ) is a contact metric manifold.
Preliminaries Main results Every almost contact manifold ( M , ϕ, ξ, η ) admits a compatible metric, i.e. a Riemannian metric g satisfying g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) . M is said to be an almost contact metric manifold with structure ( ϕ, ξ, η, g ). We define the fundamental 2-form as the 2-form Φ on M such that Φ ( X , Y ) = g ( X , ϕ Y ). If Φ = d η , then η is a contact form and we say that ( M , ϕ, ξ, η, g ) is a contact metric manifold. A normal contact metric manifold is called a Sasakian manifold.
Preliminaries Main results A contact metric manifold is said to be η -Einstein if Ric = ag + b η ⊗ η, where a and b are functions on M . This notion is a generalisation of the concept of Einstein metric.
Preliminaries Main results A contact metric manifold is said to be η -Einstein if Ric = ag + b η ⊗ η, where a and b are functions on M . This notion is a generalisation of the concept of Einstein metric. A contact metric manifold ( M , ϕ, ξ, η, g ) is said to be a contact metric ( κ, µ )-space [BKP95] if it satisfies that R ( X , Y ) ξ = κ ( η ( Y ) X − η ( X ) Y ) + µ ( η ( Y ) hX − η ( X ) hY ) , where κ, µ ∈ R .
Preliminaries Main results Given a constant c > 0, a D c -homothetic deformation on a contact metric manifold ( M , ϕ, ξ, η, g ) is the following change of the structure ξ ′ := 1 ϕ ′ := ϕ, η ′ := c η, g ′ := cg + c ( c − 1) η ⊗ η. c ξ,
Preliminaries Main results Given a constant c > 0, a D c -homothetic deformation on a contact metric manifold ( M , ϕ, ξ, η, g ) is the following change of the structure ξ ′ := 1 ϕ ′ := ϕ, η ′ := c η, g ′ := cg + c ( c − 1) η ⊗ η. c ξ, The D c -homothetic deformations preserve the contact metric and Sasakian structures. Moreover, if ( ϕ, ξ, η, g ) is a ( κ, µ )-structure, then the deformed structure ( ϕ ′ , ξ ′ , η ′ , g ′ ) is a ( κ ′ , µ ′ )-structure with κ ′ = κ + c 2 − 1 µ ′ = µ + 2 c − 2 , . c 2 c
Preliminaries Main results Boeckx gave in [B00] a local classification of the non-Sasakian contact metric ( κ, µ )-spaces using the number I M := 1 − µ 2 √ 1 − κ. I M is an invariant of the contact metric ( κ, µ )-structures, up to D -homothetic deformations.
Preliminaries Main results Boeckx gave in [B00] a local classification of the non-Sasakian contact metric ( κ, µ )-spaces using the number I M := 1 − µ 2 √ 1 − κ. I M is an invariant of the contact metric ( κ, µ )-structures, up to D -homothetic deformations. Two non-Sasakian contact metric ( κ, µ )-spaces ( M 1 , ϕ 1 , ξ 1 , η 1 , g 1 ) and ( M 2 , ϕ 2 , ξ 2 , η 2 , g 2 ), are locally isometric, up to D -homothetic defomations, if and only if I M 1 = I M 2 .
Preliminaries Main results Boeckx gave in [B00] a local classification of the non-Sasakian contact metric ( κ, µ )-spaces using the number I M := 1 − µ 2 √ 1 − κ. I M is an invariant of the contact metric ( κ, µ )-structures, up to D -homothetic deformations. Two non-Sasakian contact metric ( κ, µ )-spaces ( M 1 , ϕ 1 , ξ 1 , η 1 , g 1 ) and ( M 2 , ϕ 2 , ξ 2 , η 2 , g 2 ), are locally isometric, up to D -homothetic defomations, if and only if I M 1 = I M 2 . Remark A geometric interpretation of the invariant I M and Boeckx’s deformation can be seen in [CM09].
Preliminaries Main results Almost paracontact metric structure A manifold M 2 n +1 is said to be almost paracontact if there exists a triple ( � ϕ, ξ, η ) satisfying ϕ 2 = I − η ⊗ ξ , η ( ξ ) = 1, (i) � (ii) the eigenspaces D + and D − corresponding to the eigenvalues 1 and − 1 of � ϕ are both of dimension n .
Preliminaries Main results Almost paracontact metric structure A manifold M 2 n +1 is said to be almost paracontact if there exists a triple ( � ϕ, ξ, η ) satisfying ϕ 2 = I − η ⊗ ξ , η ( ξ ) = 1, (i) � (ii) the eigenspaces D + and D − corresponding to the eigenvalues 1 and − 1 of � ϕ are both of dimension n . We define an almost paracomplex structure � J on M × R as � � � � X , f d ϕ X + f ξ, η ( X ) d � J = � , dt dt
Preliminaries Main results Almost paracontact metric structure A manifold M 2 n +1 is said to be almost paracontact if there exists a triple ( � ϕ, ξ, η ) satisfying ϕ 2 = I − η ⊗ ξ , η ( ξ ) = 1, (i) � (ii) the eigenspaces D + and D − corresponding to the eigenvalues 1 and − 1 of � ϕ are both of dimension n . We define an almost paracomplex structure � J on M × R as � � � � X , f d ϕ X + f ξ, η ( X ) d � J = � , dt dt ϕ, ξ, η ) is normal if � We say that the paracontact structure ( � J is integrable. This condition is equivalent to N � ϕ := [ � ϕ, � ϕ ] − 2 d η ⊗ ξ = 0 .
Preliminaries Main results Almost paracontact metric structure A manifold M 2 n +1 is said to be almost paracontact if there exists a triple ( � ϕ, ξ, η ) satisfying ϕ 2 = I − η ⊗ ξ , η ( ξ ) = 1, (i) � (ii) the eigenspaces D + and D − corresponding to the eigenvalues 1 and − 1 of � ϕ are both of dimension n . We define an almost paracomplex structure � J on M × R as � � � � X , f d ϕ X + f ξ, η ( X ) d � J = � , dt dt ϕ, ξ, η ) is normal if � We say that the paracontact structure ( � J is integrable. This condition is equivalent to N � ϕ := [ � ϕ, � ϕ ] − 2 d η ⊗ ξ = 0 . A normal paracontact metric manifold is said to be paraSasakian.
Preliminaries Main results An almost paracontact manifold is said to be a almost paracontact metric if it has a semi-Riemannian metric � g such that � g ( � ϕ X , � ϕ Y ) = − � g ( X , Y ) + η ( X ) η ( Y ) The signature of this semi-Riemannian metric is ( n , n + 1) and it satisfies automatically the condition (ii) of the definition of almost paracontact structure.
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