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Preliminaries Main results

How to obtain Sasaki-Einstein and paraSasaki-Einstein metrics from contact metric (κ, µ)-spaces

Ver´

• nica 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

M2n+1 is said to be an almost contact manifold if there exists on it a triple (ϕ, ξ, η) satisfying ϕ2 = −I + η ⊗ ξ, η(ξ) = 1.

Preliminaries Main results

Almost contact metric structure

M2n+1 is said to be an almost contact manifold if there exists on it a triple (ϕ, ξ, η) satisfying ϕ2 = −I + η ⊗ ξ, η(ξ) = 1. It follows that ϕξ = 0, η ◦ ϕ = 0, rank(ϕ) = 2n.

Preliminaries Main results

Almost contact metric structure

M2n+1 is said to be an almost contact manifold if there exists on it a triple (ϕ, ξ, η) satisfying ϕ2 = −I + η ⊗ ξ, η(ξ) = 1. It follows that ϕξ = 0, η ◦ ϕ = 0, rank(ϕ) = 2n. Given an almost contact manifold (M, ϕ, ξ, η), we define an almost complex structure J on M × R as J

• X, f d

dt

• =
• ϕX − f ξ, η (X) d

dt

Preliminaries Main results

Almost contact metric structure

M2n+1 is said to be an almost contact manifold if there exists on it a triple (ϕ, ξ, η) satisfying ϕ2 = −I + η ⊗ ξ, η(ξ) = 1. It follows that ϕξ = 0, η ◦ ϕ = 0, rank(ϕ) = 2n. Given an almost contact manifold (M, ϕ, ξ, η), we define an almost complex structure J on M × R as J

• X, f d

dt

• =
• ϕX − f ξ, η (X) d

dt

• An almost contact manifold will be called normal if the almost

complex structure J is integrable. This condition is equivalent to Nϕ := [ϕ, ϕ] + 2dη ⊗ 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 (ϕX, ϕY ) = 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 (ϕX, ϕY ) = 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 (ϕX, ϕY ) = 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, ϕY ).

Preliminaries Main results

Every almost contact manifold (M, ϕ, ξ, η) admits a compatible metric, i.e. a Riemannian metric g satisfying g (ϕX, ϕY ) = 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, ϕY ). If Φ = dη, 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 (ϕX, ϕY ) = 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, ϕY ). If Φ = dη, 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 + bη ⊗ 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 + bη ⊗ 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 Dc-homothetic deformation on a contact metric manifold (M, ϕ, ξ, η, g) is the following change of the structure ϕ′ := ϕ, ξ′ := 1 c ξ, η′ := cη, g′ := cg + c(c − 1)η ⊗ η.

Preliminaries Main results

Given a constant c > 0, a Dc-homothetic deformation on a contact metric manifold (M, ϕ, ξ, η, g) is the following change of the structure ϕ′ := ϕ, ξ′ := 1 c ξ, η′ := cη, g′ := cg + c(c − 1)η ⊗ η. The Dc-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 + c2 − 1 c2 , µ′ = µ + 2c − 2 c .

Preliminaries Main results

Boeckx gave in [B00] a local classification of the non-Sasakian contact metric (κ, µ)-spaces using the number IM := 1 − µ

2

√1 − κ. IM 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 IM := 1 − µ

2

√1 − κ. IM is an invariant of the contact metric (κ, µ)-structures, up to D-homothetic deformations. Two non-Sasakian contact metric (κ, µ)-spaces (M1, ϕ1, ξ1, η1, g1) and (M2, ϕ2, ξ2, η2, g2), are locally isometric, up to D-homothetic defomations, if and only if IM1 = IM2.

Preliminaries Main results

Boeckx gave in [B00] a local classification of the non-Sasakian contact metric (κ, µ)-spaces using the number IM := 1 − µ

2

√1 − κ. IM is an invariant of the contact metric (κ, µ)-structures, up to D-homothetic deformations. Two non-Sasakian contact metric (κ, µ)-spaces (M1, ϕ1, ξ1, η1, g1) and (M2, ϕ2, ξ2, η2, g2), are locally isometric, up to D-homothetic defomations, if and only if IM1 = IM2.

Remark

A geometric interpretation of the invariant IM and Boeckx’s deformation can be seen in [CM09].

Preliminaries Main results

Almost paracontact metric structure

A manifold M2n+1 is said to be almost paracontact if there exists a triple ( ϕ, ξ, η) satisfying (i) ϕ2 = I − η ⊗ ξ, η(ξ) = 1, (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 M2n+1 is said to be almost paracontact if there exists a triple ( ϕ, ξ, η) satisfying (i) ϕ2 = I − η ⊗ ξ, η(ξ) = 1, (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

• J
• X, f d

dt

• =
• ϕX + f ξ, η(X) d

dt

• ,

Preliminaries Main results

Almost paracontact metric structure

A manifold M2n+1 is said to be almost paracontact if there exists a triple ( ϕ, ξ, η) satisfying (i) ϕ2 = I − η ⊗ ξ, η(ξ) = 1, (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

• J
• X, f d

dt

• =
• ϕX + f ξ, η(X) d

dt

• ,

We say that the paracontact structure ( ϕ, ξ, η) is normal if J is

• integrable. This condition is equivalent to

N

ϕ := [

ϕ, ϕ] − 2dη ⊗ ξ = 0.

Preliminaries Main results

Almost paracontact metric structure

A manifold M2n+1 is said to be almost paracontact if there exists a triple ( ϕ, ξ, η) satisfying (i) ϕ2 = I − η ⊗ ξ, η(ξ) = 1, (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

• J
• X, f d

dt

• =
• ϕX + f ξ, η(X) d

dt

• ,

We say that the paracontact structure ( ϕ, ξ, η) is normal if J is

• integrable. This condition is equivalent to

N

ϕ := [

ϕ, ϕ] − 2dη ⊗ 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(

ϕX, ϕY ) = − 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.

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(

ϕX, ϕY ) = − 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. We define the fundamental 2-form of an almost paracontact manifold as Φ(X, Y ) = g(X, ϕY ).

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(

ϕX, ϕY ) = − 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. We define the fundamental 2-form of an almost paracontact manifold as Φ(X, Y ) = g(X, ϕY ). If dη = Φ, η is a contact form and (M, ϕ, ξ, η, g) is called a paracontact metric manifold.

Preliminaries Main results

Example ([IVZ10])

The standard example of paraSasakian manifolds are the hyperboloids H2n+1

n+1 (1) = {(x0, y0, . . . , xn, yn) ∈ R2n+2 |

x2

0 + . . . + x2 n − y2 0 − . . . − y2 n = 1}

and the hyperbolic Heisenberg group H2n+1 = R2n × R.

Preliminaries Main results

Example ([IVZ10])

The standard example of paraSasakian manifolds are the hyperboloids H2n+1

n+1 (1) = {(x0, y0, . . . , xn, yn) ∈ R2n+2 |

x2

0 + . . . + x2 n − y2 0 − . . . − y2 n = 1}

and the hyperbolic Heisenberg group H2n+1 = R2n × R. The concepts of η-Einstein metric and Dc-homothetic deformations can be introduced in the field of paracontact geometry, with the only difference that the homothety constant c can now be any non-zero real number because the metric does not need to be positive definite.

Preliminaries Main results

Theorem

Let (M, ϕ, ξ, η, g) be a non-Sasakian contact metric (κ, µ)-space such that IM = ±1.

Preliminaries Main results

Theorem

Let (M, ϕ, ξ, η, g) be a non-Sasakian contact metric (κ, µ)-space such that IM = ±1. (i) If |IM|>1, then M admits a Sasakian structure, compatible with the contact form η, given by ¯ φ := ǫ 1 (1 − κ)√αLξh ◦ h, ¯ g := −dη(·, ¯ φ·) + η ⊗ η, where α = (2 − µ)2 − 4(1 − κ) and ǫ =sign(IM).

Preliminaries Main results

Theorem

Let (M, ϕ, ξ, η, g) be a non-Sasakian contact metric (κ, µ)-space such that IM = ±1. (i) If |IM|>1, then M admits a Sasakian structure, compatible with the contact form η, given by ¯ φ := ǫ 1 (1 − κ)√αLξh ◦ h, ¯ g := −dη(·, ¯ φ·) + η ⊗ η, where α = (2 − µ)2 − 4(1 − κ) and ǫ =sign(IM). (ii) If |IM|<1, then M admits a paraSasakian structure, compatible with the contact form η, given by

• φ :=

1 (1 − κ)√−αLξh ◦ h,

• g := dη(·,

φ·) + η ⊗ η.

Preliminaries Main results

Theorem

Let (M, ϕ, ξ, η, g) be a non-Sasakian contact metric (κ, µ)-space such that IM = ±1.

Preliminaries Main results

Theorem

Let (M, ϕ, ξ, η, g) be a non-Sasakian contact metric (κ, µ)-space such that IM = ±1. (i) If |IM|>1, then we know the writing of the Levi-Civita connection and the curvature tensor R of the Sasakian structure (M, ¯ φ, ξ, η, ¯ g). The Ricci tensor of M2n+1 is Ric = (ǫn√α − 2)¯ g + (−ǫn√α + 2n + 2)η ⊗ η, so the Sasakian structure is always η-Einstein.

Preliminaries Main results

Theorem

Let (M, ϕ, ξ, η, g) be a non-Sasakian contact metric (κ, µ)-space such that IM = ±1. (ii) If |IM|<1, then we know the writing of the Levi-Civita connection and the curvature tensor R of the paraSasakian structure (M, φ, ξ, η, ¯ g). The Ricci tensor of M2n+1 is

• Ric = (−n

√ −α + 3) g + (n √ −α − 2n − 3)η ⊗ η, so the paraSasakian structure is always η-Einstein.

Preliminaries Main results

• The Sasakian metric is Einstein if and only if

−ǫn√α + 2n + 2 = 0.

Preliminaries Main results

• The Sasakian metric is Einstein if and only if

−ǫn√α + 2n + 2 = 0. If IM > 1, this condition is equivalent to (2 − µ + 2λ)(2 − µ − 2λ) = 4

• 1 + 1

n 2 .

Preliminaries Main results

• The Sasakian metric is Einstein if and only if

−ǫn√α + 2n + 2 = 0. If IM > 1, this condition is equivalent to (2 − µ + 2λ)(2 − µ − 2λ) = 4

• 1 + 1

n 2 . If IM < −1, this condition is never true.

Preliminaries Main results

• The Sasakian metric is Einstein if and only if

−ǫn√α + 2n + 2 = 0. If IM > 1, this condition is equivalent to (2 − µ + 2λ)(2 − µ − 2λ) = 4

• 1 + 1

n 2 . If IM < −1, this condition is never true.

• The paraSasakian metric is Einstein if and only if

n√−α − 2n − 3 = 0, which is equivalent to (2 − µ + 2λ)(2 − µ − 2λ) = −

• 2 + 3

n 2 .

Preliminaries Main results

Corollary

The tangent sphere bundle T1N of a space form N of constant sectional curvature c = 1 admits a canonical η-Einstein Sasakian metric g or paraSasakian metric g, if c > 0 or c < 0.

Preliminaries Main results

Corollary

The tangent sphere bundle T1N of a space form N of constant sectional curvature c = 1 admits a canonical η-Einstein Sasakian metric g or paraSasakian metric g, if c > 0 or c < 0. Their Ricci tensors are Ric = 2(2n√c − 1)g + 2(−2n√c + n + 1)η ⊗ η,

• Ric = (−4n

√ −c + 3) g + (4n √ −c − 2n − 3)η ⊗ η, where dim(N) = n + 1.

Preliminaries Main results

Corollary

The tangent sphere bundle T1N of a space form N of constant sectional curvature c = 1 admits a canonical η-Einstein Sasakian metric g or paraSasakian metric g, if c > 0 or c < 0. Their Ricci tensors are Ric = 2(2n√c − 1)g + 2(−2n√c + n + 1)η ⊗ η,

• Ric = (−4n

√ −c + 3) g + (4n √ −c − 2n − 3)η ⊗ η, where dim(N) = n + 1. Moreover,

• g is Einstein if and only if dim(N) > 2 and c = 1

4

• 1 + 1

n

2,

g is Einstein if and only if c = − 1

16

• 2 + 3

n

2.

Preliminaries Main results

Lastly, we observe that the behaviour of the (κ, µ)-spaces in the positive and negative cases is very different.

Preliminaries Main results

Lastly, we observe that the behaviour of the (κ, µ)-spaces in the positive and negative cases is very different.

Corollary

Every contact metric (κ, µ)-space (M, ϕ, ξ, η, g) satisfying IM > 1 admits a compatible Sasaki-Einstein metric.

Preliminaries Main results

Lastly, we observe that the behaviour of the (κ, µ)-spaces in the positive and negative cases is very different.

Corollary

Every contact metric (κ, µ)-space (M, ϕ, ξ, η, g) satisfying IM > 1 admits a compatible Sasaki-Einstein metric.

Corollary

Given a contact metric (κ, µ)-space (M, ϕ, ξ, η, g) satisfying |IM| < 1, M admits a paraSasaki-Einstein structure if and only if (2 − µ + 2λ)(2 − µ − 2λ) = − 1 n2 .

Preliminaries Main results

• D. E. Blair, T. Koufogiorgos, B. J. Papantoniou, Contact

metric manifolds satisfyng a nullity condition. Israel J. Math. 91(1995), 189-214.

• E. Boeckx, A full classification of contact metric (κ, µ)-spaces.

Illinois J. Math. 44(2000), 212-219.

• B. Cappelletti Montano, The foliated structure of contact

metric (κ, µ)-spaces. Illinois J. Math. 53(2009), 1157–1172.

• B. Cappelletti Montano, A. Carriazo, V. Mart´

ın-Molina Sasaki-Einstein and paraSasaki-Einstein metrics from (κ, µ)-structures. arXiv:1109.6248

• S. Ivanov, D. Vassilev and S. Zamkovoy, Conformal

paracontact curvature and the local flatness theorem. Geom. Dedicata 144(2010), 79-100.

Preliminaries Main results

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