XVII Geometrical Seminar

On the classification of Lorentzian Sasakispace forms

Letizia Brunetti

joint work with A.M. Pastore

University of Bari

Zlatibor, September 2012

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 1 / 15

Indefinite structuresLet M be a (2n+ s)-dimensional manifold.

A g.f.f -structure on M isgiven by

a (1, 1)-tensor field ϕ of constant rank such that

there exist global vector fields ξ1, . . . , ξs and 1-forms η1, . . . , ηs s.t.

ϕ2 = −I +s∑i=1

ηi ⊗ ξi and ηi(ξj) = δij

which imply ϕ(ξi) = 0 and ηi ◦ ϕ = 0, for any i ∈ {1, . . . , s}.A semi-Riemannian metric g on M is compatible with (ϕ, ξi, η

i),i ∈ {1, . . . , s}, if

g(ϕX,ϕY ) = g(X,Y )−s∑i=1

εiηi(X)ηi(Y ), ∀X,Y ∈ X(M2n+s),

being εi = ±1 according to whether ξi is spacelike or timelike.

(M2n+s, ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, is an indefinite g.f.f -manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 2 / 15

Indefinite structuresLet M be a (2n+ s)-dimensional manifold. A g.f.f -structure on M isgiven by

a (1, 1)-tensor field ϕ of constant rank such that

there exist global vector fields ξ1, . . . , ξs and 1-forms η1, . . . , ηs s.t.

ϕ2 = −I +s∑i=1

ηi ⊗ ξi and ηi(ξj) = δij

which imply ϕ(ξi) = 0 and ηi ◦ ϕ = 0, for any i ∈ {1, . . . , s}.A semi-Riemannian metric g on M is compatible with (ϕ, ξi, η

i),i ∈ {1, . . . , s}, if

g(ϕX,ϕY ) = g(X,Y )−s∑i=1

εiηi(X)ηi(Y ), ∀X,Y ∈ X(M2n+s),

being εi = ±1 according to whether ξi is spacelike or timelike.

(M2n+s, ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, is an indefinite g.f.f -manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 2 / 15

Indefinite structuresLet M be a (2n+ s)-dimensional manifold. A g.f.f -structure on M isgiven by

a (1, 1)-tensor field ϕ of constant rank such that

there exist global vector fields ξ1, . . . , ξs and 1-forms η1, . . . , ηs s.t.

ϕ2 = −I +s∑i=1

ηi ⊗ ξi and ηi(ξj) = δij

which imply ϕ(ξi) = 0 and ηi ◦ ϕ = 0, for any i ∈ {1, . . . , s}.A semi-Riemannian metric g on M is compatible with (ϕ, ξi, η

i),i ∈ {1, . . . , s}, if

g(ϕX,ϕY ) = g(X,Y )−s∑i=1

εiηi(X)ηi(Y ), ∀X,Y ∈ X(M2n+s),

being εi = ±1 according to whether ξi is spacelike or timelike.

(M2n+s, ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, is an indefinite g.f.f -manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 2 / 15

Indefinite structuresLet M be a (2n+ s)-dimensional manifold. A g.f.f -structure on M isgiven by

a (1, 1)-tensor field ϕ of constant rank such that

there exist global vector fields ξ1, . . . , ξs and 1-forms η1, . . . , ηs s.t.

ϕ2 = −I +

s∑i=1

ηi ⊗ ξi and ηi(ξj) = δij

which imply ϕ(ξi) = 0 and ηi ◦ ϕ = 0, for any i ∈ {1, . . . , s}.

A semi-Riemannian metric g on M is compatible with (ϕ, ξi, ηi),

i ∈ {1, . . . , s}, if

g(ϕX,ϕY ) = g(X,Y )−s∑i=1

εiηi(X)ηi(Y ), ∀X,Y ∈ X(M2n+s),

being εi = ±1 according to whether ξi is spacelike or timelike.

(M2n+s, ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, is an indefinite g.f.f -manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 2 / 15

Indefinite structuresLet M be a (2n+ s)-dimensional manifold. A g.f.f -structure on M isgiven by

a (1, 1)-tensor field ϕ of constant rank such that

there exist global vector fields ξ1, . . . , ξs and 1-forms η1, . . . , ηs s.t.

ϕ2 = −I +

s∑i=1

ηi ⊗ ξi and ηi(ξj) = δij

which imply ϕ(ξi) = 0 and ηi ◦ ϕ = 0, for any i ∈ {1, . . . , s}.A semi-Riemannian metric g on M is compatible with (ϕ, ξi, η

i),i ∈ {1, . . . , s},

if

g(ϕX,ϕY ) = g(X,Y )−s∑i=1

εiηi(X)ηi(Y ), ∀X,Y ∈ X(M2n+s),

being εi = ±1 according to whether ξi is spacelike or timelike.

(M2n+s, ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, is an indefinite g.f.f -manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 2 / 15

Indefinite structuresLet M be a (2n+ s)-dimensional manifold. A g.f.f -structure on M isgiven by

a (1, 1)-tensor field ϕ of constant rank such that

there exist global vector fields ξ1, . . . , ξs and 1-forms η1, . . . , ηs s.t.

ϕ2 = −I +

s∑i=1

ηi ⊗ ξi and ηi(ξj) = δij

which imply ϕ(ξi) = 0 and ηi ◦ ϕ = 0, for any i ∈ {1, . . . , s}.A semi-Riemannian metric g on M is compatible with (ϕ, ξi, η

i),i ∈ {1, . . . , s}, if

g(ϕX,ϕY ) = g(X,Y )−s∑i=1

εiηi(X)ηi(Y ), ∀X,Y ∈ X(M2n+s),

being εi = ±1 according to whether ξi is spacelike or timelike.

(M2n+s, ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, is an indefinite g.f.f -manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 2 / 15

Indefinite structuresLet M be a (2n+ s)-dimensional manifold. A g.f.f -structure on M isgiven by

a (1, 1)-tensor field ϕ of constant rank such that

there exist global vector fields ξ1, . . . , ξs and 1-forms η1, . . . , ηs s.t.

ϕ2 = −I +

s∑i=1

ηi ⊗ ξi and ηi(ξj) = δij

which imply ϕ(ξi) = 0 and ηi ◦ ϕ = 0, for any i ∈ {1, . . . , s}.A semi-Riemannian metric g on M is compatible with (ϕ, ξi, η

i),i ∈ {1, . . . , s}, if

g(ϕX,ϕY ) = g(X,Y )−s∑i=1

εiηi(X)ηi(Y ), ∀X,Y ∈ X(M2n+s),

being εi = ±1 according to whether ξi is spacelike or timelike.

(M2n+s, ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, is an indefinite g.f.f -manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 2 / 15

Indefinite structuresLet M be a (2n+ s)-dimensional manifold. A g.f.f -structure on M isgiven by

a (1, 1)-tensor field ϕ of constant rank such that

there exist global vector fields ξ1, . . . , ξs and 1-forms η1, . . . , ηs s.t.

ϕ2 = −I +

s∑i=1

ηi ⊗ ξi and ηi(ξj) = δij

which imply ϕ(ξi) = 0 and ηi ◦ ϕ = 0, for any i ∈ {1, . . . , s}.A semi-Riemannian metric g on M is compatible with (ϕ, ξi, η

i),i ∈ {1, . . . , s}, if

g(ϕX,ϕY ) = g(X,Y )−s∑i=1

εiηi(X)ηi(Y ), ∀X,Y ∈ X(M2n+s),

being εi = ±1 according to whether ξi is spacelike or timelike.

(M2n+s, ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, is an indefinite g.f.f -manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 2 / 15

Indefinite S-structures

An indefinite g.f.f -manifold (M,ϕ, ξi, ηi, g) is said to be an

indefinite S-manifold, if

it is normal, i.e. N = [ϕ,ϕ] + 2dηi ⊗ ξi = 0

dηi = Φ for any i ∈ {1, . . . , s} where Φ(·, ·) = g(·, ϕ·).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 3 / 15

Indefinite S-structures

An indefinite g.f.f -manifold (M,ϕ, ξi, ηi, g) is said to be an

indefinite S-manifold, if

it is normal, i.e. N = [ϕ,ϕ] + 2dηi ⊗ ξi = 0

dηi = Φ for any i ∈ {1, . . . , s} where Φ(·, ·) = g(·, ϕ·).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 3 / 15

Indefinite S-structures

An indefinite g.f.f -manifold (M,ϕ, ξi, ηi, g) is said to be an

indefinite S-manifold, if

it is normal,

i.e. N = [ϕ,ϕ] + 2dηi ⊗ ξi = 0

dηi = Φ for any i ∈ {1, . . . , s} where Φ(·, ·) = g(·, ϕ·).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 3 / 15

Indefinite S-structures

An indefinite g.f.f -manifold (M,ϕ, ξi, ηi, g) is said to be an

indefinite S-manifold, if

it is normal, i.e. N = [ϕ,ϕ] + 2dηi ⊗ ξi = 0

dηi = Φ for any i ∈ {1, . . . , s} where Φ(·, ·) = g(·, ϕ·).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 3 / 15

Indefinite S-structures

An indefinite g.f.f -manifold (M,ϕ, ξi, ηi, g) is said to be an

indefinite S-manifold, if

it is normal,

i.e. N = [ϕ,ϕ] + 2dηi ⊗ ξi = 0

dηi = Φ for any i ∈ {1, . . . , s} where

Φ(·, ·) = g(·, ϕ·).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 3 / 15

Indefinite S-structures

An indefinite g.f.f -manifold (M,ϕ, ξi, ηi, g) is said to be an

indefinite S-manifold, if

it is normal,

i.e. N = [ϕ,ϕ] + 2dηi ⊗ ξi = 0

dηi = Φ for any i ∈ {1, . . . , s} where Φ(·, ·) = g(·, ϕ·).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 3 / 15

The problem

Is it always possible to consider aLorentz metric on a manifold?

and

How may a Lorentz metric beconstructed on a manifold starting

from a Riemannian metric?

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 4 / 15

The problem

Is it always possible to consider aLorentz metric on a manifold?

and

How may a Lorentz metric beconstructed on a manifold starting

from a Riemannian metric?

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 4 / 15

The problem

Is it always possible to consider aLorentz metric on a manifold?

and

How may a Lorentz metric beconstructed on a manifold starting

from a Riemannian metric?

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 4 / 15

Solutions and conditions

O’Neill1: let (M, g) be a Riemannian manifold and U be a unit vectorfield. Then

g = g − 2U∗ ⊗ U∗ is a Lorentz metric on M.

(By contrast with the Riemannian case) Not every smooth manifoldcan be made a Lorentz manifold.

There exists a Lorentz metric on M if and only if there is a nowherevanishing vector field on M .

1O’Neill B., Semi-Riemannian geometry, Academic Press, New York, 1983.Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 5 / 15

Solutions and conditions

O’Neill1: let (M, g) be a Riemannian manifold and U be a unit vectorfield. Then

g = g − 2U∗ ⊗ U∗ is a Lorentz metric on M.

(By contrast with the Riemannian case) Not every smooth manifoldcan be made a Lorentz manifold.

There exists a Lorentz metric on M if and only if there is a nowherevanishing vector field on M .

1O’Neill B., Semi-Riemannian geometry, Academic Press, New York, 1983.Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 5 / 15

Solutions and conditions

O’Neill1: let (M, g) be a Riemannian manifold and U be a unit vectorfield. Then

g = g − 2U∗ ⊗ U∗ is a Lorentz metric on M.

(By contrast with the Riemannian case)

Not every smooth manifoldcan be made a Lorentz manifold.

There exists a Lorentz metric on M if and only if there is a nowherevanishing vector field on M .

1O’Neill B., Semi-Riemannian geometry, Academic Press, New York, 1983.Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 5 / 15

Solutions and conditions

O’Neill1: let (M, g) be a Riemannian manifold and U be a unit vectorfield. Then

g = g − 2U∗ ⊗ U∗ is a Lorentz metric on M.

(By contrast with the Riemannian case) Not every smooth manifoldcan be made a Lorentz manifold.

There exists a Lorentz metric on M if and only if there is a nowherevanishing vector field on M .

1O’Neill B., Semi-Riemannian geometry, Academic Press, New York, 1983.Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 5 / 15

S-manifolds

In an S-manifold (M,ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, let p be an integer such

that 1 ≤ p ≤ s:

g = g − 2

p∑i=1

ηi ⊗ ηi.

g is an indefinite metric with index p,

ξ1, . . . , ξp are timelike and ξp+1, . . . , ξs are spacelike

ηi = ηi, for any i ∈ {1, . . . , s},(M,ϕ, ξi, η

i, g) is an indefinite S-manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 6 / 15

S-manifolds

In an S-manifold (M,ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, let p be an integer such

that 1 ≤ p ≤ s:

g = g − 2

p∑i=1

ηi ⊗ ηi.

g is an indefinite metric with index p,

ξ1, . . . , ξp are timelike and ξp+1, . . . , ξs are spacelike

ηi = ηi, for any i ∈ {1, . . . , s},(M,ϕ, ξi, η

i, g) is an indefinite S-manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 6 / 15

S-manifolds

In an S-manifold (M,ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, let p be an integer such

that 1 ≤ p ≤ s:

g = g − 2

p∑i=1

ηi ⊗ ηi.

g is an indefinite metric with index p,

ξ1, . . . , ξp are timelike and ξp+1, . . . , ξs are spacelike

ηi = ηi, for any i ∈ {1, . . . , s},(M,ϕ, ξi, η

i, g) is an indefinite S-manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 6 / 15

S-manifolds

In an S-manifold (M,ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, let p be an integer such

that 1 ≤ p ≤ s:

g = g − 2

p∑i=1

ηi ⊗ ηi.

g is an indefinite metric with index p,

ξ1, . . . , ξp are timelike and ξp+1, . . . , ξs are spacelike

ηi = ηi, for any i ∈ {1, . . . , s},(M,ϕ, ξi, η

i, g) is an indefinite S-manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 6 / 15

S-manifolds

In an S-manifold (M,ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, let p be an integer such

that 1 ≤ p ≤ s:

g = g − 2

p∑i=1

ηi ⊗ ηi.

g is an indefinite metric with index p,

ξ1, . . . , ξp are timelike and ξp+1, . . . , ξs are spacelike

ηi = ηi, for any i ∈ {1, . . . , s},

(M,ϕ, ξi, ηi, g) is an indefinite S-manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 6 / 15

S-manifolds

In an S-manifold (M,ϕ, ξi, ηi, g), i ∈ {1, . . . , s}, let p be an integer such

that 1 ≤ p ≤ s:

g = g − 2

p∑i=1

ηi ⊗ ηi.

g is an indefinite metric with index p,

ξ1, . . . , ξp are timelike and ξp+1, . . . , ξs are spacelike

ηi = ηi, for any i ∈ {1, . . . , s},(M,ϕ, ξi, η

i, g) is an indefinite S-manifold.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 6 / 15

Fundamental relations

The Levi-Civita connections: for any X,Y ∈ X(M)

∇XY = ∇XY + 2

p∑i=1

{ηi(Y )ϕX + ηi(X)ϕY }.

The ϕ-sectional curvatures2: for any X ∈ Imϕ

K(X,ϕX) = K(X,ϕX) + 6p,

since RXϕXϕX = RXϕXϕX + 6pX.

2For any X ∈ Imϕ the ϕ-sectional curvature is the sectional curvature of a

ϕ-plane π = span{X,ϕX}, i.e. K(X,ϕX) =g(X,RXϕXϕX)

g(X,X)2

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 7 / 15

Fundamental relations

The Levi-Civita connections: for any X,Y ∈ X(M)

∇XY = ∇XY + 2

p∑i=1

{ηi(Y )ϕX + ηi(X)ϕY }.

The ϕ-sectional curvatures2: for any X ∈ Imϕ

K(X,ϕX) = K(X,ϕX) + 6p,

since RXϕXϕX = RXϕXϕX + 6pX.

2For any X ∈ Imϕ the ϕ-sectional curvature is the sectional curvature of a

ϕ-plane π = span{X,ϕX}, i.e. K(X,ϕX) =g(X,RXϕXϕX)

g(X,X)2

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 7 / 15

Fundamental relations

The Levi-Civita connections: for any X,Y ∈ X(M)

∇XY = ∇XY + 2

p∑i=1

{ηi(Y )ϕX + ηi(X)ϕY }.

The ϕ-sectional curvatures2: for any X ∈ Imϕ

K(X,ϕX) = K(X,ϕX) + 6p,

since RXϕXϕX = RXϕXϕX + 6pX.

2For any X ∈ Imϕ the ϕ-sectional curvature is the sectional curvature of a

ϕ-plane π = span{X,ϕX}, i.e. K(X,ϕX) =g(X,RXϕXϕX)

g(X,X)2

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 7 / 15

Fundamental relations

The Levi-Civita connections: for any X,Y ∈ X(M)

∇XY = ∇XY + 2

p∑i=1

{ηi(Y )ϕX + ηi(X)ϕY }.

The ϕ-sectional curvatures2: for any X ∈ Imϕ

K(X,ϕX) = K(X,ϕX) + 6p,

since RXϕXϕX = RXϕXϕX + 6pX.

2For any X ∈ Imϕ the ϕ-sectional curvature is the sectional curvature of a

ϕ-plane π = span{X,ϕX}, i.e. K(X,ϕX) =g(X,RXϕXϕX)

g(X,X)2

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 7 / 15

Fundamental relations

The Levi-Civita connections: for any X,Y ∈ X(M)

∇XY = ∇XY + 2

p∑i=1

{ηi(Y )ϕX + ηi(X)ϕY }.

The ϕ-sectional curvatures2: for any X ∈ Imϕ

K(X,ϕX) = K(X,ϕX) + 6p,

since RXϕXϕX = RXϕXϕX + 6pX.

2For any X ∈ Imϕ the ϕ-sectional curvature is the sectional curvature of a

ϕ-plane π = span{X,ϕX}, i.e. K(X,ϕX) =g(X,RXϕXϕX)

g(X,X)2

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 7 / 15

The purpose of the above construction

How can we use this construction?

We have to apply this construction to obtain a model for compact LorentzSasaki space form3

A (Lorentz) Sasaki manifold is a (Lorentz) S-manifold with onecharacteristic vector field (i.e. s = 1).

Let (M,ϕ, ξ, η, g) be a Sasaki manifold, considering

g = g − 2η ⊗ η,

we have the Lorentz Sasaki manifold (M,ϕ, ξ, η, g), ξ timelike, such that,for any X ∈ Imϕ, K(X,ϕX) = K(X,ϕX) + 6.

3A Sasaki space form is a Sasaki manifold with constant ϕ-sectionalcurvatureLetizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 8 / 15

The purpose of the above construction

How can we use this construction?We have to apply this construction to obtain a model for compact Lorentz

Sasaki space form3

A (Lorentz) Sasaki manifold is a (Lorentz) S-manifold with onecharacteristic vector field (i.e. s = 1).

Let (M,ϕ, ξ, η, g) be a Sasaki manifold, considering

g = g − 2η ⊗ η,

we have the Lorentz Sasaki manifold (M,ϕ, ξ, η, g), ξ timelike, such that,for any X ∈ Imϕ, K(X,ϕX) = K(X,ϕX) + 6.

3A Sasaki space form is a Sasaki manifold with constant ϕ-sectionalcurvatureLetizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 8 / 15

The purpose of the above construction

How can we use this construction?We have to apply this construction to obtain a model for compact Lorentz

Sasaki space form3

A (Lorentz) Sasaki manifold is a (Lorentz) S-manifold with onecharacteristic vector field (i.e. s = 1).

Let (M,ϕ, ξ, η, g) be a Sasaki manifold, considering

g = g − 2η ⊗ η,

we have the Lorentz Sasaki manifold (M,ϕ, ξ, η, g), ξ timelike, such that,for any X ∈ Imϕ, K(X,ϕX) = K(X,ϕX) + 6.

3A Sasaki space form is a Sasaki manifold with constant ϕ-sectionalcurvatureLetizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 8 / 15

The purpose of the above construction

How can we use this construction?We have to apply this construction to obtain a model for compact Lorentz

Sasaki space form3

A (Lorentz) Sasaki manifold is a (Lorentz) S-manifold with onecharacteristic vector field (i.e. s = 1).

Let (M,ϕ, ξ, η, g) be a Sasaki manifold, considering

g = g − 2η ⊗ η,

we have the Lorentz Sasaki manifold (M,ϕ, ξ, η, g), ξ timelike, such that,for any X ∈ Imϕ, K(X,ϕX) = K(X,ϕX) + 6.

3A Sasaki space form is a Sasaki manifold with constant ϕ-sectionalcurvatureLetizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 8 / 15

The sphere S2n+1 and Sasakian structures on S2n+1[c]

Standard one: (S2n+1, g∗) as an hypersurface of the complex manifold(R2n+2, J,<,>):

ξ∗ = −JN, JX = ϕ∗X + η∗(X)N, ∀X ∈ X(S2n+1).

Another one: considering the D-homothetic transformation with α ∈ R∗+

ϕ = ϕ∗, ξ = α−1ξ∗, η = αη∗ g = αg∗ + (α2 − α)η∗ ⊗ η∗.

S2n+1[c] = (S2n+1, ϕ, ξ, η, g) is a Sasakian manifold with constantϕ-sectional curvature c > −3, (c = 4

α − 3).

Lorentz structure: considering the Lorentz metric g:S2n+11 [c] = (S2n+1, ϕ, ξ, η, g) is a Lorentz Sasaki manifold with constantϕ-sectional curvature c > 3, (c = c+ 6).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 9 / 15

The sphere S2n+1 and Sasakian structures on S2n+1[c]

Standard one: (S2n+1, g∗) as an hypersurface of the complex manifold(R2n+2, J,<,>):

ξ∗ = −JN, JX = ϕ∗X + η∗(X)N, ∀X ∈ X(S2n+1).

Another one: considering the D-homothetic transformation with α ∈ R∗+

ϕ = ϕ∗, ξ = α−1ξ∗, η = αη∗ g = αg∗ + (α2 − α)η∗ ⊗ η∗.

S2n+1[c] = (S2n+1, ϕ, ξ, η, g) is a Sasakian manifold with constantϕ-sectional curvature c > −3, (c = 4

α − 3).

Lorentz structure: considering the Lorentz metric g:S2n+11 [c] = (S2n+1, ϕ, ξ, η, g) is a Lorentz Sasaki manifold with constantϕ-sectional curvature c > 3, (c = c+ 6).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 9 / 15

The sphere S2n+1 and Sasakian structures on S2n+1[c]

Standard one: (S2n+1, g∗) as an hypersurface of the complex manifold(R2n+2, J,<,>):

ξ∗ = −JN, JX = ϕ∗X + η∗(X)N, ∀X ∈ X(S2n+1).

Another one: considering the D-homothetic transformation with α ∈ R∗+

ϕ = ϕ∗, ξ = α−1ξ∗, η = αη∗ g = αg∗ + (α2 − α)η∗ ⊗ η∗.

S2n+1[c] = (S2n+1, ϕ, ξ, η, g) is a Sasakian manifold with constantϕ-sectional curvature c > −3, (c = 4

α − 3).

Lorentz structure: considering the Lorentz metric g:S2n+11 [c] = (S2n+1, ϕ, ξ, η, g) is a Lorentz Sasaki manifold with constantϕ-sectional curvature c > 3, (c = c+ 6).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 9 / 15

The sphere S2n+1 and Sasakian structures on S2n+1[c]

Standard one: (S2n+1, g∗) as an hypersurface of the complex manifold(R2n+2, J,<,>):

ξ∗ = −JN, JX = ϕ∗X + η∗(X)N, ∀X ∈ X(S2n+1).

Another one: considering the D-homothetic transformation with α ∈ R∗+

ϕ = ϕ∗, ξ = α−1ξ∗, η = αη∗ g = αg∗ + (α2 − α)η∗ ⊗ η∗.

S2n+1[c] = (S2n+1, ϕ, ξ, η, g) is a Sasakian manifold with constantϕ-sectional curvature c > −3, (c = 4

α − 3).

Lorentz structure: considering the Lorentz metric g:S2n+11 [c] = (S2n+1, ϕ, ξ, η, g) is a Lorentz Sasaki manifold with constantϕ-sectional curvature c > 3, (c = c+ 6).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 9 / 15

The sphere S2n+1 and Sasakian structures on S2n+1[c]

Standard one: (S2n+1, g∗) as an hypersurface of the complex manifold(R2n+2, J,<,>):

ξ∗ = −JN, JX = ϕ∗X + η∗(X)N, ∀X ∈ X(S2n+1).

Another one: considering the D-homothetic transformation with α ∈ R∗+

ϕ = ϕ∗, ξ = α−1ξ∗, η = αη∗ g = αg∗ + (α2 − α)η∗ ⊗ η∗.

S2n+1[c] = (S2n+1, ϕ, ξ, η, g) is a Sasakian manifold with constantϕ-sectional curvature c > −3, (c = 4

α − 3).

Lorentz structure: considering the Lorentz metric g:S2n+11 [c] = (S2n+1, ϕ, ξ, η, g) is a Lorentz Sasaki manifold with constantϕ-sectional curvature c > 3, (c = c+ 6).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 9 / 15

The sphere S2n+1 and Sasakian structures on S2n+1[c]

Standard one: (S2n+1, g∗) as an hypersurface of the complex manifold(R2n+2, J,<,>):

ξ∗ = −JN, JX = ϕ∗X + η∗(X)N, ∀X ∈ X(S2n+1).

Another one: considering the D-homothetic transformation with α ∈ R∗+

ϕ = ϕ∗, ξ = α−1ξ∗, η = αη∗ g = αg∗ + (α2 − α)η∗ ⊗ η∗.

S2n+1[c] = (S2n+1, ϕ, ξ, η, g) is a Sasakian manifold with constantϕ-sectional curvature c > −3, (c = 4

α − 3).

Lorentz structure: considering the Lorentz metric g:S2n+11 [c] = (S2n+1, ϕ, ξ, η, g) is a Lorentz Sasaki manifold with constantϕ-sectional curvature c > 3, (c = c+ 6).

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 9 / 15

Theorem

Let (M2n+1, ϕ, ξ, η, g) be a Sasakian manifold and (M2n+1, ϕ, ξ, η, g) theassociated Lorentz Sasaki manifold. Then the Sasaki manifold hasconstant ϕ-sectional curvature c ∈ R if and only if the Lorentz Sasakimanifold has constant ϕ-sectional curvature c = c+ 6 and we have

c > −3⇔ c > 3,

c = −3⇔ c = 3,

c < −3⇔ c < 3.

It is well known the Tanno’s classification4 for complete and simplyconnected Sasakian manifolds with constant ϕ-sectional curvature.

4S. Tanno, Sasakian manifolds with constant φ-holomorphic sectionalcurvature, Tohoku Math. J. (2) 21 (1969), 501–507.Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 10 / 15

Theorem

Let (M2n+1, ϕ, ξ, η, g) be a Sasakian manifold and (M2n+1, ϕ, ξ, η, g) theassociated Lorentz Sasaki manifold. Then the Sasaki manifold hasconstant ϕ-sectional curvature c ∈ R if and only if the Lorentz Sasakimanifold has constant ϕ-sectional curvature c = c+ 6 and we have

c > −3⇔ c > 3,

c = −3⇔ c = 3,

c < −3⇔ c < 3.

It is well known the Tanno’s classification4 for complete and simplyconnected Sasakian manifolds with constant ϕ-sectional curvature.

4S. Tanno, Sasakian manifolds with constant φ-holomorphic sectionalcurvature, Tohoku Math. J. (2) 21 (1969), 501–507.Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 10 / 15

Theorem

Let (M2n+1, ϕ, ξ, η, g) be a Sasakian manifold and (M2n+1, ϕ, ξ, η, g) theassociated Lorentz Sasaki manifold. Then the Sasaki manifold hasconstant ϕ-sectional curvature c ∈ R if and only if the Lorentz Sasakimanifold has constant ϕ-sectional curvature c = c+ 6 and we have

c > −3⇔ c > 3,

c = −3⇔ c = 3,

c < −3⇔ c < 3.

It is well known the Tanno’s classification4 for complete and simplyconnected Sasakian manifolds with constant ϕ-sectional curvature.

4S. Tanno, Sasakian manifolds with constant φ-holomorphic sectionalcurvature, Tohoku Math. J. (2) 21 (1969), 501–507.Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 10 / 15

Problem to obtain models for Lorentz Sasaki space forms

Theorem

Let (M2n+1, ϕ, ξ, η, g) be a compact simply connected Lorentz Sasakianmanifold with ξ timelike and constant ϕ-sectional curvature c > 3. ThenM is isomorphic to S2n+1

1 [c].

Sketch proof.

g is Lorentz metric, hence g = g + 2η ⊗ η is Riemannian and(M2n+1, ϕ, ξ, η, g) turns out to be a compact simply connected Sasakianmanifold with constant ϕ-sectional curvature c = c− 6 > −3. It isS2n+1[c] up to isomorphisms.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 11 / 15

Problem to obtain models for Lorentz Sasaki space forms

Theorem

Let (M2n+1, ϕ, ξ, η, g) be a compact simply connected Lorentz Sasakianmanifold with ξ timelike and constant ϕ-sectional curvature c > 3. ThenM is isomorphic to S2n+1

1 [c].

Sketch proof.

g is Lorentz metric, hence g = g + 2η ⊗ η is Riemannian and(M2n+1, ϕ, ξ, η, g) turns out to be a compact simply connected Sasakianmanifold with constant ϕ-sectional curvature c = c− 6 > −3. It isS2n+1[c] up to isomorphisms.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 11 / 15

Problem to obtain models for Lorentz Sasaki space forms

Theorem

Let (M2n+1, ϕ, ξ, η, g) be a compact simply connected Lorentz Sasakianmanifold with ξ timelike and constant ϕ-sectional curvature c > 3. ThenM is isomorphic to S2n+1

1 [c].

Sketch proof.

g is Lorentz metric, hence g = g + 2η ⊗ η is Riemannian and(M2n+1, ϕ, ξ, η, g) turns out to be a compact simply connected Sasakianmanifold with constant ϕ-sectional curvature c = c− 6 > −3. It isS2n+1[c] up to isomorphisms.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 11 / 15

At least in the compact case we have a model space, unique up toisomorphisms preserving the Lorentz Sasaki structures.

Theorem

Let (M2n+1, ϕ, ξ, η, g) and (M ′2n+1, ϕ′, ξ′, η′, g′) be compact simplyconnected Lorentz Sasakian manifolds with ξ and ξ′ timelike and havingthe same constant ϕ-sectional curvature c > 3. Then they are isomorphic.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 12 / 15

At least in the compact case we have a model space, unique up toisomorphisms preserving the Lorentz Sasaki structures.

Theorem

Let (M2n+1, ϕ, ξ, η, g) and (M ′2n+1, ϕ′, ξ′, η′, g′) be compact simplyconnected Lorentz Sasakian manifolds with ξ and ξ′ timelike and havingthe same constant ϕ-sectional curvature c > 3. Then they are isomorphic.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 12 / 15

Final remarks

We want to note that

A compact indefinite S-manifold (M2n+s, ϕ, ξi, ηi, g) such that

ind g = p, where p is an integer such that 2p ≤ 2n+ s and having ptimelike characteristic vector fields is complete by Theorem 45

Every compact Lorentz Sasaki manifold is complete.

We will try to obtain a full classification for the Lorentz Sasaki spaceforms.

5M. Guediri and J. Lafontaine, Sur la completude des varietespseudo-riemanniennes, J. Geom and Physics 15 (1995), 150–158.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 13 / 15

Final remarks

We want to note that

A compact indefinite S-manifold (M2n+s, ϕ, ξi, ηi, g) such that

ind g = p, where p is an integer such that 2p ≤ 2n+ s and having ptimelike characteristic vector fields is complete by Theorem 45

Every compact Lorentz Sasaki manifold is complete.

We will try to obtain a full classification for the Lorentz Sasaki spaceforms.

5M. Guediri and J. Lafontaine, Sur la completude des varietespseudo-riemanniennes, J. Geom and Physics 15 (1995), 150–158.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 13 / 15

Final remarks

We want to note that

A compact indefinite S-manifold (M2n+s, ϕ, ξi, ηi, g) such that

ind g = p, where p is an integer such that 2p ≤ 2n+ s and having ptimelike characteristic vector fields is complete by Theorem 45

Every compact Lorentz Sasaki manifold is complete.

We will try to obtain a full classification for the Lorentz Sasaki spaceforms.

5M. Guediri and J. Lafontaine, Sur la completude des varietespseudo-riemanniennes, J. Geom and Physics 15 (1995), 150–158.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 13 / 15

Final remarks

We want to note that

A compact indefinite S-manifold (M2n+s, ϕ, ξi, ηi, g) such that

ind g = p, where p is an integer such that 2p ≤ 2n+ s and having ptimelike characteristic vector fields is complete by Theorem 45

Every compact Lorentz Sasaki manifold is complete.

We will try to obtain a full classification for the Lorentz Sasaki spaceforms.

5M. Guediri and J. Lafontaine, Sur la completude des varietespseudo-riemanniennes, J. Geom and Physics 15 (1995), 150–158.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 13 / 15

Thank you for the attention!

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 14 / 15

Guediri and Lafontaine’s Theorem

Theorem (Guediri – Lafontaine)

Let (M, g) be a compact pseudo-Riemannian manifold with signature(n− s, s) such that 2s ≤ n. We suppose that there exist s linearindependent timelike Killing vector fields on M . Then (M, g) isgeodesically complete.

Letizia Brunetti (University of Bari) Lorentzian Sasaki space forms September 7, 2012 15 / 15