Lorentzian Gradient Ricci Solitons

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Lorentzian Gradient Ricci Solitons

Miguel Brozos Vázquez

Departamento de Matemáticas Escola Politécnica Superior

Joint work with Eduardo García Río and Sandra Gavino Fernández

Lorentzian Gradient Ricci Solitons

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Index

1

Introduction

2

Einstein gradient Ricci solitons

3

Locally conformally at gradient Ricci solitons

4

Homogeneous gradient Ricci solitons

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Introduction

Context: Lorentzian manifolds

Context

(M,g)Lorentzian manifold of dimension n+2:

M dierentiable manifold of dimension n+2 g Lorentzian metric of signature(1,n+1)

Curvature

1 ∇denotes the Levi-Civita connection.

2 R(x,y) =∇_[x,y]−[∇_x,∇_y]is the curvature operator. For an orthonormal basis{e1, . . . ,en}:

Ricci tensor ρ(x,y) =P

iε_iR(x,ei,y,ei) =g(Ric(x),y)

Scalar curvature

τ =P

iε_iρ(ei,ei) Einstein manifolds

(M,g)is said to be Einstein if there exists a constantλsuch that ρ=λg.

Note thatλ= _n₊^τ₂.

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Introduction

Context: Lorentzian manifolds

Context

M dierentiable manifold of dimension n+2 g Lorentzian metric of signature(1,n+1) Curvature

2 R(x,y) =∇_[x,y]−[∇_x,∇_y]is the curvature operator.

For an orthonormal basis{e1, . . . ,en}:

Scalar curvature

τ =P

iε_iρ(ei,ei)

Einstein manifolds

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Introduction

Context: Lorentzian manifolds

Context

M dierentiable manifold of dimension n+2 g Lorentzian metric of signature(1,n+1) Curvature

2 R(x,y) =∇_[x,y]−[∇_x,∇_y]is the curvature operator.

For an orthonormal basis{e1, . . . ,en}:

Scalar curvature

τ =P

iε_iρ(ei,ei) Einstein manifolds

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Introduction

Denition of Ricci solitons

Ricci solitons

(M,g,X)is a Ricci soliton, where X is a smooth vector eld on M, if there exists a constantλsuch that

1

2L_Xg+ρ=λg

Motivation for the study Ricci solitons:

Ricci solitons generalize Einstein manifolds.

The main motivation to study Ricci solitons comes from the Ricci ow. Ricci solitons correspond to self-similar solutions of the Ricci ow.

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Introduction

Denition of Ricci solitons

Ricci solitons

1

2L_Xg+ρ=λg Motivation for the study Ricci solitons:

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Introduction

Denition of Ricci solitons

Ricci solitons

1

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Introduction

Denition of Ricci solitons

Ricci solitons

1

The main motivation to study Ricci solitons comes from the Ricci ow.

Ricci solitons correspond to self-similar solutions of the Ricci ow.

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Introduction

The Ricci ow

∂

∂tg(t) =−2ρ(t)

The xed points are Ricci at manifolds (ρ=0).

Ifρ₀=λg0forλ6=0, then g(t) = (1−2λt)g0 is a solution of the Ricci ow:

ifλ <0 then g(t)is dened for t>₂¹_λ and g(t)expands. ifλ >0 then g(t)is dened for t<₂¹_λ and g(t)shrinks. A solution of the Ricci ow is said to be self-similar if we allow the initial metric to change by homotheties and dieomorphisms:

g(t) =θ(t)φ(t)^∗g(0), whereθ:I→Randφ(t) :M→M.

Ricci solitons are the self-similar solutions of the Ricci ow.

Ricci solitons are, modulo dieomorphisms and scalings, xed points of the Ricci ow as a dynamical system.

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Introduction

The Ricci ow

∂

∂tg(t) =−2ρ(t) The xed points are Ricci at manifolds (ρ=0).

Ifρ₀=λg0forλ6=0, then g(t) = (1−2λt)g0 is a solution of the Ricci ow:

ifλ <0 then g(t)is dened for t>₂¹_λ and g(t)expands. ifλ >0 then g(t)is dened for t<₂¹_λ and g(t)shrinks. A solution of the Ricci ow is said to be self-similar if we allow the initial metric to change by homotheties and dieomorphisms:

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Introduction

The Ricci ow

∂

Ifρ₀=λg0 forλ6=0, then g(t) = (1−2λt)g0 is a solution of the Ricci ow:

ifλ <0 then g(t)is dened for t>₂¹_λ and g(t)expands.

ifλ >0 then g(t)is dened for t<₂¹_λ and g(t)shrinks.

A solution of the Ricci ow is said to be self-similar if we allow the initial metric to change by homotheties and dieomorphisms:

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Introduction

The Ricci ow

∂

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Introduction

The Ricci ow

∂

Ricci solitons are, modulo dieomorphisms and scalings, xed points of the

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Introduction

Denition of Ricci solitons

Ricci solitons

(M,g,X)is a Ricci soliton, where X is a smooth vector eld on M, if there exists a constantλsuch that 1

Ricci solitons are said to be: expanding ifλ <0, steady ifλ=0, shrinking ifλ >0.

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Introduction

Denition of Ricci solitons

Ricci solitons

(M,g,X)is a Ricci soliton, where X is a smooth vector eld on M, if there exists a constantλsuch that 1

Ricci solitons are said to be:

expanding ifλ <0, steady ifλ=0,

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Introduction

Gradient Ricci solitons

If X=∇f , then f satises the equation Hesf +ρ=λg

and(M,g,f)is called a gradient Ricci soliton. Note thatλ=^∆f_n+2^+τ

The Gaussian soliton

(Rⁿ,g0,f)where g0is the at pseudo-Euclidean metric and f(x) =^λ₂kxk² satises

Hes(f) +ρ=λg0

Rigid solitons

Let N be an Einstein metric withρ^N =λg^N. Then N×R^k with f(x) = ^λ₂kxk² on the Euclidean factor is a gradient Ricci soliton.

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Introduction

Gradient Ricci solitons

(Rⁿ,g0,f)where g0 is the at pseudo-Euclidean metric and f(x) =^λ₂kxk² satises

Hes(f) +ρ=λg0

Rigid solitons

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Introduction

Gradient Ricci solitons

(Rⁿ,g0,f)where g0 is the at pseudo-Euclidean metric and f(x) =^λ₂kxk² satises

Hes(f) +ρ=λg0

Rigid solitons

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Introduction

Notation and some formulas

If(M,g,f)is a gradient Ricci soliton,∇f may have dierent causal characters.

We say that a gradient Ricci soliton(M,g,f)is isotropic ifk∇fk=0,

non isotropic ifk∇fk 6=0.

Examples of isotropic gradient Ricci solitons

There exist nontrivial gradient Ricci solitons withk∇fk=0. For example, Cahen-Wallach symmetric spaces are isotropic steady gradient Ricci solitons.

W. Batat, , E. García-Río, S. Gavino-Fernández; Ricci solitons on Lorentzian manifolds with large isometry group, Bull. London Math. Soc. 43 (2011), 12191227.

The following formulas are satised by any gradient Ricci soliton:

∇τ=2 Ric(∇f),

τ+k∇fk²−2λf =const,

R(X,Y,Z,∇f) = (∇_Xρ)(Y,Z)−(∇_Yρ)(X,Z)

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Introduction

Notation and some formulas

If(M,g,f)is a gradient Ricci soliton,∇f may have dierent causal characters.

We say that a gradient Ricci soliton(M,g,f)is isotropic ifk∇fk=0,

non isotropic ifk∇fk 6=0.

Examples of isotropic gradient Ricci solitons

There exist nontrivial gradient Ricci solitons withk∇fk=0. For example, Cahen-Wallach symmetric spaces are isotropic steady gradient Ricci solitons.

W. Batat, , E. García-Río, S. Gavino-Fernández; Ricci solitons on Lorentzian manifolds with large isometry group, Bull. London Math. Soc. 43 (2011), 12191227.

The following formulas are satised by any gradient Ricci soliton:

∇τ=2 Ric(∇f),

τ+k∇fk²−2λf =const,

R(X,Y,Z,∇f) = (∇_Xρ)(Y,Z)−(∇_Yρ)(X,Z)

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Einstein gradient Ricci solitons

Index

1

Introduction

2

Einstein gradient Ricci solitons

3

Locally conformally at gradient Ricci solitons

4

Homogeneous gradient Ricci solitons General framework: arbitrary dimension

Three-dimensional homogeneous gradient Ricci solitons

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Einstein gradient Ricci solitons

Einstein equation Gradient Ricci soliton equation

ρ=λg Hesf +ρ=λg

Theorem

Let(M,g)be a Lorentzian Einstein manifold. If(M,g,f)is a gradient Ricci soliton with nonconstant f , then(M,g)is Ricci at. Moreover:

(i) Ifk∇fk 6=0, then(M,g)is locally a warped product of the form I×_f⁰N and the potential function f(t) = ^λ₂t²+at+b.

(ii) Ifk∇fk=0, then there exist coordinates(u,v,x1, . . . ,xn)in which the metric has the form g =2dudv+ ˜g, where the n-dimensional metricg˜ does not depend on v. Moreover, the potential function f is given by any function f(u)with f⁰⁰(u) =0 and the soliton is steady.

Ifk∇fk=0

τ+k∇fk²−2λf =const⇒ (

f =

×

constant

λ=0 ⇒Hesf =0⇒ ∇f is parallel isotropic ⇒g can be writen in Rosen coordinates(u,v,x1, . . . ,xn)as

g=dudv+ ˜g

whereg does not depend on v and is Ricci at for a xed u.˜

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Einstein gradient Ricci solitons

Einstein equation Gradient Ricci soliton equation

ρ=λg Hesf +ρ=λg

Theorem

Ifk∇fk=0

f =

×

constant

g=dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

0=∇τ =2 Ric(∇f)⇒ (

f =constant

f =

×

constant

Ric =0

⇒Hesf = _n^∆₊^f₂g Ifk∇fk=0

f =

×

constant

g=dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

0=∇τ =2 Ric(∇f)⇒ (

f =

×

constant Ric =0

⇒Hesf = _n^∆₊^f₂g Ifk∇fk=0

f =

×

constant

g=dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

0=∇τ =2 Ric(∇f)⇒ (

f =

×

constant

Ric =0 ⇒Hesf = _n^∆₊^f₂g

Ifk∇fk=0

f =

×

constant

g=dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

0=∇τ =2 Ric(∇f)⇒ (

f =

×

constant

Ric =0 ⇒Hesf = _n^∆₊^f₂g Ifk∇fk 6=0

The manifold decomposes locally as a warped product I×_f⁰N by a result by [Brinkmann, 1925].

Ifk∇fk=0

f =

×

constant

g=dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

0=∇τ =2 Ric(∇f)⇒ (

f =

×

constant

Ric =0 ⇒Hesf = _n^∆₊^f₂g

Ifk∇fk=0

f =

×

constant

g=dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

Ifk∇fk=0

f =

×

constant

λ=0 ⇒Hesf =0⇒ ∇f is

parallel isotropic ⇒g can be writen in Rosen coordinates(u,v,x1, . . . ,xn)as g=dudv+ ˜g

where˜g does not depend on v and is Ricci at for a xed u.

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Einstein gradient Ricci solitons

Theorem

Ifk∇fk=0

τ+k∇fk²−2λf =const

⇒ (

f =

×

constant

parallel isotropic ⇒g can be writen in Rosen coordinates(u,v,x1, . . . ,xn)as g=dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

Ifk∇fk=0

f =constant

f =

×

constant

λ=0

⇒Hesf =0

⇒ ∇f is parallel isotropic ⇒g can be writen in Rosen coordinates (u,v,x1, . . . ,xn)as

g=dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

Ifk∇fk=0

f =

×

constant λ=0

⇒Hesf =0⇒ ∇f is parallel isotropic ⇒g can be writen in Rosen coordinates(u,v,x1, . . . ,xn)as

g=dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

Ifk∇fk=0

f =

×

constant

λ=0 ⇒Hesf =0

⇒ ∇f is parallel isotropic ⇒g can be writen in Rosen coordinates(u,v,x1, . . . ,xn)as

g=dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

Ifk∇fk=0

f =

×

constant

parallel isotropic

⇒g can be writen in Rosen coordinates(u,v,x1, . . . ,xn)as g=dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

Ifk∇fk=0

f =

×

constant

parallel isotropic ⇒g can be writen in Rosen coordinates(u,v,x1, . . . ,xn)as g =dudv+ ˜g

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Einstein gradient Ricci solitons

Theorem

Ifk∇fk=0

f =

×

constant

parallel isotropic ⇒g can be writen in Rosen coordinates(u,v,x1, . . . ,xn)as g =dudv+ ˜g

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Riemannian vs. Lorentzian

Riemannian Lorentzian

Einstein Specic warped product Specic warped product Isotropic steady examples

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Locally conformally at gradient Ricci solitons

Index

1

Introduction

2

Einstein gradient Ricci solitons

3

Locally conformally at gradient Ricci solitons

4

Homogeneous gradient Ricci solitons General framework: arbitrary dimension

Three-dimensional homogeneous gradient Ricci solitons

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Locally conformally at gradient Ricci solitons

Theorem. Let ( M , g , f ) be a locally conformally at Lorentzian gradient Ricci soliton.

(i) If k∇ f k 6= 0, M is locally isometric to a Robertson-Walker warped product I ×

_ψ

N with metric ε dt

²

+ ψ

²

g

_N

, where I is a real interval and ( N , g

_N

) is a space of constant curvature c.

(ii) If k∇ f k = 0 on a non-empty open set, then ( M , g ) is locally isometric to a plane wave, i.e., M is locally dieomorphic to R

²

× R

ⁿ

with metric

g = 2dudv + H (u, x

₁

, . . . , x

_n

)du

²

+

n

X

i=1

dx

_i²

, where H (u, x

₁

, . . . , x

_n

) = a(u) P

_n

i=1

x

_i²

+ P

_n

i=1

b

_i

(u)x

_i

+ c (u) for some functions a ( u ) , b

i

( u ) , c ( u ) and the potential function is given by f ( u , x

₁

, . . . , x

_n

) = f

₀

( u ) , with

f

⁰⁰

(u) = −ρ

_uu

= n a(u), and the soliton is steady.

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Sketch of the proof. Non isotropic case.

1

∇ f is an eigenvector of the Ricci operator

2

Choose a local orthonormal frame { V =

_k∇^∇^f_f_k

, E

₁

, . . . , E

_n+1

}

3

The level sets of f are totally umbilical hypersurfaces. We prove this using the soliton equation and that the curvature expresses in terms of the Ricci tensor: Hes

_f

( E

_i

, E

_i

) = λε

_i

+

_n₊¹₁

(ρ( V , V )ε − τ ) ε

_i

.

4

( M , g ) is a twisted product I ×

_ω

N.

5

Since ρ( V , E

i

) = 0, the twisted product reduces to a warped product, so

( M , g ) = ( I × N , ε dt

²

+ ψ( t )

²

g

N

)

where (N , g

_N

) is a Riemannian or a Lorentzian manifold of constant sectional curvature c

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Sketch of the proof. Non isotropic case.

1

∇ f is an eigenvector of the Ricci operator

2

Choose a local orthonormal frame { V =

_k∇^∇^f_f_k

, E

₁

, . . . , E

_n+1

}

3

The level sets of f are totally umbilical hypersurfaces. We prove this using the soliton equation and that the curvature expresses in terms of the Ricci tensor: Hes

_f

( E

_i

, E

_i

) = λε

_i

+

_n₊¹₁

(ρ( V , V )ε − τ ) ε

_i

.

4

( M , g ) is a twisted product I ×

_ω

N.

5

Since ρ( V , E

i

) = 0, the twisted product reduces to a warped product, so

( M , g ) = ( I × N , ε dt

²

+ ψ( t )

²

g

N

)

where (N , g

_N

) is a Riemannian or a Lorentzian manifold of

constant sectional curvature c

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Sketch of the proof. Non isotropic case.

1

∇ f is an eigenvector of the Ricci operator

2

Choose a local orthonormal frame { V =

_k∇^∇^f_f_k

, E

₁

, . . . , E

_n+1

}

3

The level sets of f are totally umbilical hypersurfaces. We prove this using the soliton equation and that the curvature expresses in terms of the Ricci tensor: Hes

_f

( E

_i

, E

_i

) = λε

_i

+

_n₊¹₁

(ρ( V , V )ε − τ ) ε

_i

.

4

( M , g ) is a twisted product I ×

_ω

N.

5

Since ρ( V , E

i

) = 0, the twisted product reduces to a warped product, so

( M , g ) = ( I × N , ε dt

²

+ ψ( t )

²

g

N

)

where (N , g

_N

) is a Riemannian or a Lorentzian manifold of constant sectional curvature c

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Sketch of the proof. Non isotropic case.

1

∇ f is an eigenvector of the Ricci operator

2

Choose a local orthonormal frame { V =

_k∇^∇^f_f_k

, E

₁

, . . . , E

_n+1

}

3

The level sets of f are totally umbilical hypersurfaces.

We prove this using the soliton equation and that the curvature expresses in terms of the Ricci tensor:

Hes

_f

( E

_i

, E

_i

) = λε

_i

+

_n₊¹₁

(ρ( V , V )ε − τ ) ε

_i

.

4

( M , g ) is a twisted product I ×

_ω

N.

5

Since ρ( V , E

i

) = 0, the twisted product reduces to a warped product, so

( M , g ) = ( I × N , ε dt

²

+ ψ( t )

²

g

N

)

where (N , g

_N

) is a Riemannian or a Lorentzian manifold of

constant sectional curvature c

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Sketch of the proof. Non isotropic case.

1

∇ f is an eigenvector of the Ricci operator

2

Choose a local orthonormal frame { V =

_k∇^∇^f_f_k

, E

₁

, . . . , E

_n+1

}

3

The level sets of f are totally umbilical hypersurfaces.

We prove this using the soliton equation and that the curvature expresses in terms of the Ricci tensor:

Hes

_f

( E

_i

, E

_i

) = λε

_i

+

_n₊¹₁

(ρ( V , V )ε − τ ) ε

_i

.

4

( M , g ) is a twisted product I ×

_ω

N.

5

Since ρ( V , E

i

) = 0, the twisted product reduces to a warped product, so

( M , g ) = ( I × N , ε dt

²

+ ψ( t )

²

g

N

)

where (N , g

_N

) is a Riemannian or a Lorentzian manifold of constant sectional curvature c

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Sketch of the proof. Non isotropic case.

1

∇ f is an eigenvector of the Ricci operator

2

Choose a local orthonormal frame { V =

_k∇^∇^f_f_k

, E

₁

, . . . , E

_n+1

}

3

The level sets of f are totally umbilical hypersurfaces.

We prove this using the soliton equation and that the curvature expresses in terms of the Ricci tensor:

Hes

_f

( E

_i

, E

_i

) = λε

_i

+

_n₊¹₁

(ρ( V , V )ε − τ ) ε

_i

.

4

( M , g ) is a twisted product I ×

_ω

N.

5

Since ρ( V , E

i

) = 0, the twisted product reduces to a warped product, so

( M , g ) = ( I × N , ε dt

²

+ ψ( t )

²

g

_N

)

where (N , g

_N

) is a Riemannian or a Lorentzian manifold of

constant sectional curvature c

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Sketch of the proof. Non isotropic case.

Theorem. Let ( M , g , f ) be a locally conformally at Lorentzian gradient Ricci soliton.

(i) If k∇ f k 6= 0, M is locally isometric to a Robertson-Walker warped product I ×

_ψ

N with metric ε dt

²

+ ψ

²

g

_N

, where I is a real interval and ( N , g

_N

) is a space of constant curvature c.

(ii) If k∇ f k = 0 on a non-empty open set, then ( M , g ) is locally isometric to a plane wave, i.e., M is locally dieomorphic to R

²

× R

ⁿ

with metric

g = 2dudv + H ( u , x

₁

, . . . , x

_n

) du

²

+

n

X

i=1

dx

_i²

, where H (u, x

₁

, . . . , x

_n

) = a(u) P

_n

i=1

x

_i²

+ P

_n

i=1

b

_i

(u)x

_i

+ c (u) for some functions a ( u ) , b

i

( u ) , c ( u ) and the potential function is given by f ( u , x

₁

, . . . , x

_n

) = f

₀

( u ) , with

f

₀⁰⁰

(u) = −ρ

_uu

= n a(u), and the soliton is steady.

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Sketch of the proof. Non isotropic case.

Theorem. Let ( M , g , f ) be a locally conformally at Lorentzian gradient Ricci soliton.

(i) If k∇ f k 6= 0, M is locally isometric to a Robertson-Walker warped product I ×

_ψ

N with metric εdt

²

+ ψ

²

g

_N

, where I is a real interval and ( N , g

_N

) is a space of constant curvature c.

Remark

f is a radial function f ( t ) , hence the soliton equation gives:

f

⁰⁰

= ε λ+ ( n + 1 ) ψ

⁰⁰

ψ , ε ψ ψ

⁰

f

⁰

= λ ψ

²

− n c +ε(ψ ψ

⁰⁰

+ n (ψ

⁰

)

²

).

The warped product is not arbitrary.

(ii) If k∇ f k = 0 on a non-empty open set, then ( M , g ) is locally isometric to a plane wave, i.e., M is locally dieomorphic to R

²

× R

ⁿ

with metric

g = 2dudv + H ( u , x

₁

, . . . , x

_n

) du

²

+ X

n

i=1

dx

_i²

, where H ( u , x

1

, . . . , x

n

) = a ( u ) P

_n

i=1

x

_i²

+ P

_n

i=1

b

i

( u ) x

i

+ c ( u ) for some functions a ( u ) , b

_i

( u ) , c ( u ) and the potential function is given by f (u, x

₁

, . . . , x

_n

) = f

₀

(u), with

f

₀⁰⁰

( u ) = −ρ

_uu

= n a ( u ) , and the soliton is steady.

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Sketch of the proof. Non isotropic case.

Theorem. Let ( M , g , f ) be a locally conformally at Lorentzian gradient Ricci soliton.

(i) If k∇ f k 6= 0, M is locally isometric to a Robertson-Walker warped product I ×

_ψ

N with metric ε dt

²

+ ψ

²

g

_N

, where I is a real interval and ( N , g

_N

) is a space of constant curvature c.

(ii) If k∇ f k = 0 on a non-empty open set, then ( M , g ) is locally isometric to a plane wave, i.e., M is locally dieomorphic to R

²

× R

ⁿ

with metric

g = 2dudv + H (u, x

₁

, . . . , x

_n

)du

²

+

n

X

i=1

dx

_i²

, where H (u, x

₁

, . . . , x

_n

) = a(u) P

_n

i=1

x

_i²

+ P

_n

i=1

b

_i

(u)x

_i

+ c (u) for some functions a ( u ) , b

i

( u ) , c ( u ) and the potential function is given by f ( u , x

₁

, . . . , x

_n

) = f

₀

( u ) , with

f

₀⁰⁰

(u) = −ρ

_uu

= n a(u), and the soliton is steady.

(50)

Sketch of the proof. Isotropic case.

1

∇ f is an eigenvector of the Ricci operator for the eigenvalue λ

2

Choose a local pseudo-orthonormal frame { U , V = ∇ f , E

₁

, . . . , E

_n

}

3

Compute the Ricci tensor to see that the only nonzero component is ρ( U , U ) , so τ = 0.

4

The Ricci soliton is steady ( λ = 0), so Hes

_f

= −ρ . Hence

∇

_∇_f

∇ f = 0 and ∇ f is a geodesic vector eld.

5

Moreover, ∇ f is a recurrent vector eld: ∇

_X

∇ f = σ( X )∇ f with σ( U ) = −ρ( U , U ) , σ( V ) = 0 and σ( E

i

) = 0.

6

Also R (V

^⊥

, V

^⊥

, ·, ·) = 0 and the Ricci tensor is isotropic, so

( M , g ) is a pp-wave.

(51)

Sketch of the proof. Isotropic case.

1

∇ f is an eigenvector of the Ricci operator for the eigenvalue λ

2

Choose a local pseudo-orthonormal frame { U , V = ∇ f , E

₁

, . . . , E

_n

}

g =







0 1 0 . . . 0 1 0 0 . . . 0 0 0 1

... ... ...

0 0 1







3

Compute the Ricci tensor to see that the only nonzero component is ρ( U , U ) , so τ = 0.

4

The Ricci soliton is steady ( λ = 0), so Hes

_f

= −ρ . Hence

∇

_∇_f

∇ f = 0 and ∇ f is a geodesic vector eld.

5

Moreover, ∇ f is a recurrent vector eld: ∇

_X

∇ f = σ( X )∇ f with σ( U ) = −ρ( U , U ) , σ( V ) = 0 and σ( E

_i

) = 0.

6

Also R (V

^⊥

, V

^⊥

, ·, ·) = 0 and the Ricci tensor is isotropic, so ( M , g ) is a pp-wave.

(52)

Sketch of the proof. Isotropic case.

1

∇ f is an eigenvector of the Ricci operator for the eigenvalue λ

2

Choose a local pseudo-orthonormal frame { U , V = ∇ f , E

₁

, . . . , E

_n

}

3

Compute the Ricci tensor to see that the only nonzero component is ρ( U , U ) , so τ = 0.

ρ =







? 0 0 . . . 0

0 0 0 . . . 0 0 0 0

... ... ...

0 0 0







4

The Ricci soliton is steady ( λ = 0), so Hes

_f

= −ρ . Hence

∇

_∇_f

∇ f = 0 and ∇ f is a geodesic vector eld.

5

Moreover, ∇ f is a recurrent vector eld: ∇

_X

∇ f = σ( X )∇ f with σ( U ) = −ρ( U , U ) , σ( V ) = 0 and σ( E

_i

) = 0.

6

Also R ( V

^⊥

, V

^⊥

, ·, ·) = 0 and the Ricci tensor is isotropic, so

( M , g ) is a pp-wave.

(53)

Sketch of the proof. Isotropic case.

1

∇ f is an eigenvector of the Ricci operator for the eigenvalue λ

2

Choose a local pseudo-orthonormal frame { U , V = ∇ f , E

₁

, . . . , E

_n

}

3

Compute the Ricci tensor to see that the only nonzero component is ρ( U , U ) , so τ = 0.

4

The Ricci soliton is steady ( λ = 0), so Hes

_f

= −ρ . Hence

∇

_∇_f

∇ f = 0 and ∇ f is a geodesic vector eld.

5

Moreover, ∇ f is a recurrent vector eld: ∇

_X

∇ f = σ( X )∇ f with σ( U ) = −ρ( U , U ) , σ( V ) = 0 and σ( E

i

) = 0.

6

Also R (V

^⊥

, V

^⊥

, ·, ·) = 0 and the Ricci tensor is isotropic, so ( M , g ) is a pp-wave.

(54)

Sketch of the proof. Isotropic case.

1

∇ f is an eigenvector of the Ricci operator for the eigenvalue λ

2

Choose a local pseudo-orthonormal frame { U , V = ∇ f , E

₁

, . . . , E

_n

}

3

Compute the Ricci tensor to see that the only nonzero component is ρ( U , U ) , so τ = 0.

4

The Ricci soliton is steady ( λ = 0), so Hes

_f

= −ρ . Hence

∇

_∇_f

∇ f = 0 and ∇ f is a geodesic vector eld.

5

Moreover, ∇ f is a recurrent vector eld: ∇

_X

∇ f = σ( X )∇ f with σ( U ) = −ρ( U , U ) , σ( V ) = 0 and σ( E

i

) = 0.

6

Also R (V

^⊥

, V

^⊥

, ·, ·) = 0 and the Ricci tensor is isotropic, so

( M , g ) is a pp-wave.

(55)

Sketch of the proof. Isotropic case.

1

∇ f is an eigenvector of the Ricci operator for the eigenvalue λ

2

Choose a local pseudo-orthonormal frame { U , V = ∇ f , E

₁

, . . . , E

_n

}

3

Compute the Ricci tensor to see that the only nonzero component is ρ( U , U ) , so τ = 0.

4

The Ricci soliton is steady ( λ = 0), so Hes

_f

= −ρ . Hence

∇

_∇_f

∇ f = 0 and ∇ f is a geodesic vector eld.

5

Moreover, ∇ f is a recurrent vector eld: ∇

_X

∇ f = σ( X )∇ f with σ( U ) = −ρ( U , U ) , σ( V ) = 0 and σ( E

i

) = 0.

6

Also R(V

^⊥

, V

^⊥

, ·, ·) = 0 and the Ricci tensor is isotropic, so ( M , g ) is a pp-wave.

(56)

Isotropic case: Gradient Ricci solitons on pp-waves

pp-wave

( R

ⁿ⁺²

, g

_ppw

) . In coordinates ( u , v , x

₁

, . . . , x

_n

) g

_ppw

= 2dudv + H ( u , x

₁

, . . . , x

_n

) du

²

+

X

n

i=1

dx

_i²

, where H ( u , x

1

, .., x

n

) is an arbitrary smooth function.

Gradient Ricci solitons on pp-waves

(M, g

_ppw

, f ) is a nontrivial gradient Ricci soliton if and only if it is steady and the potential function f satises

f ( u , x

₁

, . . . , x

_n

) = f

₀

( u ) + P

ⁿ

i=1

κ

_i

x

_i

, where κ

_i

are arbitrary constants and

f

₀⁰⁰

( u ) = −ρ

_uu

− 1 2

X

n

i=1

κ

_i

∂

_i

H ( u , x

1

, . . . , x

n

).

(57)

Isotropic case: Gradient Ricci solitons on pp-waves pp-wave

( R

ⁿ⁺²

, g

_ppw

) . In coordinates ( u , v , x

₁

, . . . , x

_n

) g

ppw

= 2dudv + H ( u , x

1

, . . . , x

n

) du

²

+

n

X

i=1

dx

_i²

, where H ( u , x

₁

, .., x

_n

) is an arbitrary smooth function.

Gradient Ricci soliton equations



 



 



12

P

n

i=1

∂

_i

H ∂

_i

f + ∂

_uu²

f −

¹₂

∂

_u

H ∂

_v

f + ρ

_uu

= λ H ,

∂

_ui²

f −

¹₂

∂

_i

H ∂

_v

f = 0 , 1 ≤ i ≤ n ,

∂

_ii²

f = λ, 1 ≤ i ≤ n ,

∂

_uv²

f = λ,

∂

_ij²

f = ∂

_vi²

f = ∂

²_vv

f = 0 , 1 ≤ i 6= j ≤ n .

Gradient Ricci solitons on pp-waves

( M , g

_ppw

, f ) is a nontrivial gradient Ricci soliton if and only if it is steady and the potential function f satises

f (u, x

₁

, . . . , x

_n

) = f

₀

(u) + P

ⁿ

i=1

κ

_i

x

_i

, where κ

_i

are arbitrary constants and

f

₀⁰⁰

(u) = −ρ

_uu

− 1 2

n

X

i=1

κ

_i

∂

_i

H (u, x

₁

, . . . , x

_n

).

(58)

Isotropic case: Gradient Ricci solitons on pp-waves

pp-wave

( R

ⁿ⁺²

, g

_ppw

) . In coordinates ( u , v , x

₁

, . . . , x

_n

) g

_ppw

= 2dudv + H ( u , x

₁

, . . . , x

_n

) du

²

+

X

n

i=1

dx

_i²

, where H ( u , x

1

, .., x

n

) is an arbitrary smooth function.

Gradient Ricci solitons on pp-waves

(M, g

_ppw

, f ) is a nontrivial gradient Ricci soliton if and only if it is steady and the potential function f satises

f ( u , x

₁

, . . . , x

_n

) = f

₀

( u ) + P

ⁿ

i=1

κ

_i

x

_i

, where κ

_i

are arbitrary constants and

f

₀⁰⁰

( u ) = −ρ

_uu

− 1 2

X

n

i 1

κ

_i

∂

_i

H ( u , x

1

, . . . , x

n

).