Homogeneity of Lorentzian three-manifolds withreccurent curvature

E. Garcıa-Rıo, P. Gilkey, S. Nikcevic

Korea, August 2014.

Abstract

k-Curvature homogeneous three-dimensional Walker metrics aredescribed for k ≤ 2. This allows a complete description of locallyhomogeneous three-dimensional Walker metrics, showing thatthere exists exactly three isometry classes of such manifolds. As anapplication one obtains a complete description of all locallyhomogeneous Lorentzian manifolds with recurrent curvature.Moreover, potential functions are constructed in all the locallyhomogeneous manifolds resulting in steady gradient Ricci andCotton solitons.

Walker Lorentzian 3 dimensional manifolds

Let M = (M, gM ) be a 3-dimensional Lorentzian manifold whichadmits a parallel null vector field, i.e. M is a 3-dimensional Walkermanifold. Such a manifold admits local adapted coordinates(x, y, x) so that the (possibly) non-zero components of the metricare given by

g(∂x, ∂x) = −2f(x, y), g(∂x, ∂x) = g(∂y, ∂y) = 1 .

define a signature (2, 1) metric on R3. We examine curvaturehomogeneity in this setting; this extends previous work where weassumed f = f(y) was a function of only one variable.We shall denote this manifold by Mf .

Introduction

The study of Lorentzian three-manifolds admitting a parallel nullvector field is central both in geometry and physics. Physicallythey represent the simplest non-trivial pp-waves and, from ageometrical point of view, they are the underlying structure ofmany Lorentzian situations without Riemannian counterpart. ALorentzian manifold is said to be irreducible if the holonomy groupdoes not leave invariant any non-trivial subspace. Moreover, theaction of the holonomy is said to be indecomposable it leavesinvariant only non-trivial subspaces for which the restriction of themetric degenerates. Then the de Rham-Wu’s Theorem shows thatany complete and simply connected Lorentzian manifold is aproduct of indecomposable ones. Thus Walker three-manifoldsconstitute the basic material to build many Lorentzian metrics.

When trying to analyze the curvature of a given manifold, onemust deal not only with the curvature tensor itself, but also withits covariant derivatives. In the locally symmetric case, thecurvature tensor is parallel and hence the study can be reduced tothe purely algebraic level. Generalizing this condition, one naturallyconsiders the case when the curvature tensor is recurrent (i.e.,∇R = ω ⊗R for some 1-form ω) but not parallel (∇R 6= 0). Theclass of recurrent Lorentzian manifolds reduces to the study ofplane waves and the three-dimensional Walker manifolds(A.S.Galaev-Lorentzian manifolds with recurrent curvature tensor,arXiv; A.G.Walker 1950 -On Ruse’s spaces of recurrent curvature).Homogeneous plane waves were discussed in (M.Blau andM.O’Loughlin 2003) and hence one of the purposes of this paper isto give a complete description of all locally homogeneous Walkerthree-manifolds.

Our main result shows that there exists exactly three isometryclasses of such manifolds , which allows a complete description ofall locally homogeneous Lorentzian manifolds with recurrentcurvature. Recall that, in any locally homogeneouspseudo-Riemannian manifold, the curvature tensor and all itscovariant derivatives are the same at each point . Generalizing thiscondition, a manifold (M, g) is said to be k-curvaturehomogeneous if for any two points there exists a linear isometrybetween the corresponding tangent spaces which preserves thecurvature tensor and its covariant derivatives up to order k. Clearlyany locally homogeneous space is curvature homogeneous of anyorder; conversely, given m, there is k = k(m) so that anyk-curvature homogeneous manifold is in fact homogeneous(I.M.Singer-1960, F.Podesta and A.Spiro-1996, B.Opozda-1997).

Moreover it is shown (Bueken,Djoric-2000) that athree-dimensional Lorentzian manifold which is curvaturehomogeneous up order two is also locally homogeneous. Ourapproach is based on a careful analysis of the curvaturehomogeneity of a Walker three-manifold.A purpose of this work to give a complete description of all Walkerthree-manifolds . It is our second purpose in this paper to analyzetheir geometry, thus showing that any 1-curvature homogeneousWalker three-manifold is either a gradient Ricci soliton or agradient Cotton soliton.

Recall that, generalizing the Einstein condition, a Lorentzianmanifold (M, g) is said to be a Ricci soliton if and only if it is aself-similar solution of the Ricci flow, i.e., the one-parameter familyof metrics g(t) = σ(t)ψ∗t g is a solution of the Ricci flow∂∂tg(t) = −2Ricg(t), for some smooth function σ(t) and someone-parameter family of diffeomorphisms ψt of M , where Ricg(t)denotes the Ricci tensor of (M, g(t)). From a physical viewpoint,Ricci solitons can be interpreted as special solutions of the Einsteinfield equations, where the stress-energy tensor essentiallycorresponds to the Lie derivative of the metric. We analyze indetail the structure of gradient Ricci solitons on Walkerthree-manifolds , showing that any 1-curvature homogeneousLorentzian three-manifold with recurrent curvature is indeed asteady gradient Ricci soliton. While one of the possible 1-curvaturehomogeneous Walker three-manifold is indeed a plane wave andthus a expanding, steady and shrinking Ricci soliton, it is shownthat the non-homogeneous family does not support any non-steadyRicci soliton.

The Cotton tensor, C, measures the failure of the Schouten tensorto be Codazzi. The existence of self-similar solutions to the Cottonflow ∂

∂tg(t) = −λCg(t) also provide a family of three-dimensionalmetrics which generalize the locally conformally flat manifolds.Locally homogeneous Walker three-manifolds split essentially intotwo families, one of which is locally conformally flat. The existenceof gradient Cotton solitons is also studied, showing that any locallyhomogeneous Walker three-manifold is a steady gradient Cottonsoliton. Moreover, a locally homogeneous Walker three-manifoldadmits a non-steady Cotton soliton if and only if it is locallyconformally flat.

Preliminaries

A Lorentzian manifold admitting a parallel null vector field will berefereed in what follows as a Walker manifold. Let (x, y, x) becoordinates on R3, let O be a connected open subset of R2, letM := O × R, let f ∈ C∞(O), and let Mf := (M, gf ) whereg = gf is the Lorentz metric on M given by:

g(∂x, ∂x) = −2f(x, y), g(∂x, ∂x) = g(∂y, ∂y) = 1 .

We shall assume that fyy > 0 and thus the curvature does notvanish identically; the case fyy < 0 is similar. This implies thatMf is 0-curvature homogeneity.

We note for future reference that the non-zero covariantderivatives are given by:

∇∂x∂x = −fx∂x + fy∂y and ∇∂x∂y = ∇∂y∂x = −fy∂x.

Since ∇x∂x = −fx∂x + fy∂y involves a ∂y component which candepend on y, these manifolds are not generalized plane wavemanifolds and in fact can be geodesically incomplete; they canexhibit Ricci blowup.The only (potentially) non-zero components of the (1, 3) curvaturetensor R are given by

R(∂x, ∂y)∂y = fyy∂x and R(∂x, ∂y)∂x = −fyy∂y.

Thus the (0, 4) curvature tensor is determined byR(∂x, ∂y, ∂y, ∂x) = fyy. Similarly, when considering ∇R, the onlypossible contributions from the Christoffel symbols arise whenplugging ∂x or ∂y in the last entry ∇R(∂x, ∂y, ∂y, ∂x; · ).

Consequently the (possibly) non-zero entries in ∇R are given by:

∇R(∂x, ∂y, ∂y, ∂x; ∂x) = fxyy and ∇R(∂x, ∂y, ∂y, ∂x; ∂y) = fyyy.

It follows from previous equations that Mf is locally symmetricif and only if fyy is a constant. Whenever fyy = const. 6= 0, theresulting manifold is a Cahen-Wallach symmetric space(CLPTV-1990). In dimension 3, all Walker metrics haverecurrent curvature in a neighborhood of any point of non-zerocurvature, i.e., ∇R = ω ⊗R, for a 1-form

ω = (ln fyy)x dx+ (ln fyy)y dy.

We refer to (V,R,G,N,L-2009) and the references therein formore information on Walker three-manifolds.

We compute similarly that the (possibly) non-zero entries in ∇2Rare:

∇2R(∂x, ∂y, ∂y, ∂x; ∂x, ∂x) = fxxyy − fyfyyy,

∇2R(∂x, ∂y, ∂y, ∂x; ∂x, ∂y) = fxyyy,

∇2R(∂x, ∂y, ∂y, ∂x; ∂y, ∂x) = fxyyy,

∇2R(∂x, ∂y, ∂y, ∂x; ∂y, ∂y) = fyyyy .

Definition

Let Mf be a three-dimensional Walker manifold given byWalker-metric.

1. Let Nb be defined by taking f(x, y) = b−2eby for b 6= 0.

2. Let Pc be defined by taking f(x, y) = 12y

2α(x) where

αx = cα3/2 and α > 0.

3. Let CW be the three-dimensional Cahen-Wallach symmetricspace defined by taking f(x, y) = εy2.

TheoremThere is a geodesic γ(t) in Pc which defined for t ∈ [0, 1) andthere exists a parallel vector field Y (t) along γ(t) with

gf (γ, γ) = 1, gf (γ, Y ) = 0, gf (Y, Y ) = 1, limt→1−

R(γ, Y, Y, γ) =∞ .

Consequently, Pc is geodesically incomplete and can not beembedded in a geodesically complete manifold.

Theorem

1. The manifolds CWε are locally symmetric.

2. The manifolds Nb and Pc are locally homogeneous.

3. The manifolds {CWε,Nb,Pc} have non-isomorphic1-curvature models and represent different local isometrytypes.

The manifolds Pc are pp-waves one supposes since the functionf(x, y) is quadratic in y; they are not generalized plane. Themanifolds Nb are not pp-waves since fyy is not quadratic in y.Note that the Cahen-Wallach symmetric spaces CWε aregeodesically complete. The geodesic equations corresponding tothe manifolds Nb can be integrated explicitly.

Geometric Solitons

The objective of the different geometric evolution equations is toimprove a given initial metric by considering a flow associated tothe geometric object under consideration. The Ricci, Yamabe, andmean curvature flows are examples extensively studied in theliterature. Under suitable conditions, the Ricci flow evolves aninitial metric to an Einstein metric while the Yamabe flow evolvesan initial metric to a new one with constant scalar curvature withinthe same conformal class. There are however certain metrics which,instead of evolving by the flow, remain invariant up to scaling anddiffeomorphisms, i.e., they are self-similar solutions of the flow.For any solution of the form g(t) = σ(t)ψ∗t g(0), where σ(t) is asmooth function and {ψt} a one-parameter family ofdiffeomorphisms of M , there exists a vector field X (the solitonvector field) which relates the Lie derivative of the metric LXgwith the geometric object defining the flow under consideration.

Ricci solitons

A Ricci soliton is a pseudo-Riemannian manifold (M, g) whichadmits a smooth vector field X (which is called a soliton vectorfield) on M such that

LXg +Ric = λg, (1)

where LX denotes the Lie derivative in the direction of X, Ric isthe Ricci tensor, and λ is a real number (λ = 1

n(2divX + Sc),where n = dim M and Sc denotes the scalar curvature of (M, g)).A Ricci soliton is said to be shrinking, steady or expanding, ifλ > 0, λ = 0 or λ < 0, respectively. Moreover we say that a Riccisoliton (M, g) is a gradient Ricci soliton if the vector field Xsatisfies X = gradh, for some potential function h. In such a caseEquation (1) can be written in terms of h as 2Hesh +Ric = λg.

Three-dimensional Walker metrics admitting a non-trivial (i.e., notEinstein) gradient Ricci soliton were completely described inBVGRGF. where it is shown that one of the following twopossibilities must occur:[(R.1)] There exist coordinates (x, y, x) for some function fsatisfying f−1yy fyyy = b. Hence

f(x, y) =1

κ2eκyα(x) + yβ(x) + γ(x)

for some arbitrary functions α(x), β(x) and γ(x). The potentialfunction of the soliton is given by

h(x, y, x) =κ

2y + h(x), where hxx =

κ

2β(x),

and the soliton vector field is spacelike and given bygradh = κ

2∂y + hx(x)∂x.

[(R.2)] There exist coordinates (x, y, x) for some function fsatisfying fyyy = 0. Hence

f(x, y) = y2α(x) + yβ(x) + γ(x)

for some arbitrary functions α(x), β(x) and γ(x).The potential function of the soliton is given by

h(x, y, x) = h(x), where hxx = −α(y),

and the soliton vector field is lightlike and given bygradh = hx(x)∂x.Moreover, in both cases the Ricci soliton is steady. As we shall seeall gradient Ricci solitons above are 1-curvature homogeneous,provided that fyy has constant sign.

Further, note that all metrics corresponding to the second caseabove are plane waves (since the function f is quadratic on y), andhence they admit non-gradient vector fields X resulting inexpanding and shrinking Ricci solitons BBGG-2011. Howevermetrics corresponding to the first case above only admit steadyRicci solitons

TheoremA gradient Ricci soliton Mf admits a vector field X resulting in anon-steady Ricci soliton if and only if it is a locally conformally flatmetric.

Recall here that two Ricci soliton vector fields differ in ahomothetic vector field . Hence, a gradient Ricci soliton Walkermetric admits a vector field resulting in a non-steady Ricci solitonif and only if it admits a non-Killing homothetic vector field.Locally conformally flat Walker metrics are plane wavesCGRVA-2005, and thus they admit homothetic vector fieldsresulting in expanding, steady and shrinking Ricci solitons ....

Cotton solitons

The Schouten tensor of any pseudo-Riemannian manifold is given

by Sij = Ricij −Sc

4gij . Then the Cotton tensor,

Cijk = (∇iS)jk − (∇jS)ik measures the failure of the Schoutentensor to be a Codazzi tensor. The Cotton tensor is the uniqueconformal invariant in dimension three and it vanishes if and only ifthe manifold is locally conformally flat. Using the Hodge?-operator, the (0, 2)-Cotton tensor is given by

Cij =1

2√gCnmiε

nm`g`j , where ε123 = 1. Moreover, the

(0, 2)-Cotton tensor is trace-free and divergence-free York-1971and it appears naturally in many physical contexts (see, forexample Chow-Pope-Sezgin 2010, Garcia-Hehl-Heinicke 2004 andthe references therein).

The only non-zero component of the (0, 2)-Cotton tensor of aWalker manifold Mf is given by C(∂x, ∂x) = −1

2fyyy (and hencethe manifold is locally conformally flat if and only if fyyy = 0).A geometric flow associated to the Cotton tensor was introducedin Kisisel, Sarioglu,Tekin-2008 as

∂

∂tg(t) = −λCg(t),

where Cg(t) is the (0, 2)-Cotton tensor corresponding to (M, g(t)).Then one naturally considers soliton solutions of the Cotton flow.Following K,S,T -2008, a Cotton soliton is a triple (M, g,X) of athree-dimensional manifold and a vector field X satisfying

LXg + C = λ g, (2)

where λ is a real number. The Cotton soliton is said to beshrinking, steady or expanding if λ > 0, λ = 0 or λ < 0,respectively.

The necessary and sufficient conditions for a Walker manifold to bea gradient Cotton soliton were discussed in (CLGRVL-2012), whereit is shown that Mf is a non-trivial (i.e., not locally conformallyflat) gradient Cotton soliton if and only if it is steady andfyyyy = κfyy for some non-zero constant κ. Now one of thefollowing three possibilities must occur:(C.1)There exist coordinates (x, y, x) for some function fsatisfying fyyyy = κ2fyy .

f(x, y) =1

κ2(eκyα1(x) + e−κyα2(x)) + yβ(x) + γ(x)

where α1(x), α2(x), β(x) and γ(x) are arbitrary functions.Moreover, the potential function of the soliton is given byh(x, y, x) = κ

2y + h(x), where

hxx(x) =κ

2(eκyα1(x)− e−κyα2(x) + 2κβ(x)).

The soliton vector field is spacelike and given bygradh = κ2∂y + hx(x)∂x.

[(C.2)] There exist coordinates (x, y, x) for some function fsatisfying fyyyy = −κ2fyy . Hence

f(x, y) = − 1

κ2(cos(κy)α1(x) + sin(κy)α2(x)) + yβ(x) + γ(x))

where α1(x), α2(x), β(x) and γ(x) are arbitrary functions.Moreover, the potential function of the soliton is given byh(x, y, x) = −κ

2y + h(x), where

hxx(x) =κ

2(cos(κy)− sin(κy)− 2κβ(y)).

The soliton vector field is spacelike and given bygradh = −κ2∂y + hx(x)∂x.

[(C.3)] There exist coordinates (x, y, x) for some function fsatisfying fyyyy = 0 . Hence

f(x, y) = y3α1(x) + y2α2(x) + yβ(x) + γ(x)

where α1(x), α2(x), β(x) and γ(x) are arbitrary functions.Moreover, the potential function of the soliton is given byh(x, y, x) = h(x), where

hxx(x) = −3α1(x).

The soliton vector field is spacelike and given bygradh = −κ2∂y + hx(x)∂x.

Moreover, in all cases above the gradient Cotton soliton is steady.Note that two Cotton soliton vector fields(LXig+C = λig, i = 1, 2) differ by a homothetic vector field since

LX1−X2g = (λ1 − λ2)g.

Hence, as well as for Ricci solitons, no Walker metriccorresponding to (C.1) and (C.2) supports any non-trivial Cottonsoliton of non-steady type as a consequence of the following.

TheoremLet Mf be a non-flat Walker manifold satisfying fyyyy = bfyy(b 6= 0). Then any homothetic vector field on Mf is necessarily aKilling vector field.

Remark

Let Mf be given by f(x, y) = y3e−λx + y2 + yeλx + γ(x). Thenthe vector field X = 1

2∂x + λ2y∂y + {λx+ θ(x)} ∂x is a Cotton

soliton for any function θ(x) which satisfies the identityθx = 3e−λx + (12 − λ)γ(x). Moreover the Cotton soliton isexpanding or shrinking depending on the sign of λ. Hence thereare gradient Cotton solitons (C.3) which also admit non-Killinghomothetic vector fields.

Curvature homogeneity

A pseudo-Riemannian manifold (M, g) is said to be k-curvaturehomogeneous if for each pair of points p, q ∈M there is a linearisometry Φpq : TpM → TqM such that

Φ∗pqR(q) = R(p), Φ∗pq∇R(q) = ∇R(p), . . . , Φ∗pq∇kR(q) = ∇kR(p)

where R, ∇R, . . . , ∇kR stands for the curvature tensor and itscovariant derivatives up to order k.Clearly any locally homogeneous manifold is curvaturehomogeneous and the converse holds true if k is sufficiently large.An open question in the study of curvature homogeneity is todecide the minimum level of curvature homogeneity needed toshow that a space is locally homogeneous. A general estimate ofthe form kM + 1 ≤ n(n− 1)/2 (where n is the dimension of themanifold) was obtained by Singer.

However, there are sharper bounds in low dimensions. ARiemannian manifold which is 1-curvature homogeneous is locallyhomogeneous in dimension ≤ 4. However one needs 2-curvaturehomogenenity to ensure local homogeneity in the three-dimensionalLorentzian setting (see PG-2007) for more information andreferences).A 0-curvature homogeneous manifold is said to be modeled on asymmetric space if its curvature tensor at each point is that of asymmetric space. A complete and simply connectedindecomposable Lorentzian symmetric space is either irreducible,and hence of constant sectional curvature, or otherwise it is aCahen-Wallach symmetric space (CLPTV-1990). Now, animmediate application of Schur’s lemma shows that a curvaturehomogeneous Lorentz manifold modeled on an irreduciblesymmetric space has constant sectional curvature.

On the other hand, curvature homogeneous Lorentzian manifoldsmodeled on indecomposable symmetric spaces need not to besymmetric, but they are Walker manifolds (CLPTV-1990).The purpose of this work is to analyze the class ofthree-dimensional manifolds with recurrent curvature underdifferent curvature homogeneity assumptions. The question of1-curvature homogeneity is dealt with by the following result :

TheoremAssume that fyy > 0 and that fyy is non-constant. Then Mf is1-curvature homogeneous if and only if exactly one of the followingtwo possibilities holds:

1. fyy(x, y) = α(x)eby where 0 6= b ∈ R and where α(x) is arbitrary.This manifold is 1-curvature modeled on Nb.

2. fyy(x, y) = α(x) where α = c · α1.5x for some 0 6= c ∈ R. This

manifold is locally homogeneous and is locally isometric to Pc.

TheoremAny 1-curvature homogeneous Lorentzian three-manifold withrecurrent curvature is a steady gradient Ricci soliton.

TheoremThe manifold Mf is 2-curvature homogeneous if and only if it fallsinto one of the three families:

1. f = b−2α(x)eby + β(x)y+ γ(x) for β(x) = b−1α−1{αxx−α2xα−1}

where b 6= 0 and α > 0. The manifold Mf is locally isometric tothe manifold Nb and consequently is locally homogeneous.

2. fyy = α(x) > 0 where αx = cα3/2 for c > 0; thus α = c(x− x0)−2for some (c, x0). The manifold Mf is locally isometric to themanifold Pc and consequently locally homogeneous.

3. f = εy2 + β(x)y + γ(x) where 0 < ε ∈ R. The manifold Mf islocally isometric to the manifold CWε and consequently is locallyhomogeneous.

TheoremAny 2-curvature homogeneous Lorentzian three-manifold withrecurrent curvature is a steady gradient Cotton soliton.

Finally note that the model manifolds Pc and CWε admit Ricciand Cotton solitons of any kind (expanding, steady and shrinking),but Nb only admits steady Ricci and Cotton solitons.

There is a slightly different version of curvature homogeneity thatis due to Kowalski and Vanzurova 2011. A manifold (M, g) is saidto be k-curvature homogeneous of type (1, 3)(homothety curvaturehomogeneity) if for any two points there exists a linear homothetybetween the corresponding tangent spaces which preserves the(1, 3)-curvature tensor and its covariant derivatives up to order k.This concept lies between the notion of affine k-curvaturehomogeneity and k-curvature homogeneity since the conformalgroup lies between the orthogonal group and the general lineargroup. This is genuinely a different concept.

TheoremSuppose fyy is never zero and non-constant.

1. If fyyy never vanishes, then Mf is homothety 1-curvaturehomogeneous.

2. If fyy = α(x) with αx never zero, then Mf is homothety1-curvature homogeneous if and only iff = a(x− x0)−2y2 + β(x)y + γ(x) where 0 6= a ∈ R. Thismanifold is locally homogeneous.

3. Assume that Mf is homothety 2-curvature homogeneous, and thatfyy and fyyy never vanish. Then Mf is locally isometric to one ofthe examples:

3.1 M±eay for some a 6= 0 and M = R3. This manifold ishomogeneous.

3.2 M± ln(y) for some a 6= 0 and M = R× (0,∞)× R. This manifoldis homothety homogeneous but not locally homogeneous.

3.3 M±yε for ε 6= 0, 1, 2 and M = R× (0,∞)× R. This manifold ishomothety homogeneous but not locally homogeneous.

It will follow from our analysis that the manifolds M± ln(y) andM±yc are homothety homogeneous VSI manifolds which arecohomogeneity one, thereby exhibiting non-trivial examples in theVSI setting. We also refer to recent work of Dunn and McDonald(2013) for related work on homothetical curvature homogeneousmanifolds.

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