Quasihomogeneous three-dimensional real analytic Lorentz metrics do not exist
aa r X i v : . [ m a t h . DG ] M a y QUASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICSDO NOT EXIST
SORIN DUMITRESCU AND KARIN MELNICKA
BSTRACT . We show that a germ of a real analytic Lorentz metric on R which is locally homogeneous on anopen set containing the origin in its closure is necessarily locally homogeneous. We classifiy Lie algebras that canact quasihomogeneously —meaning they act transitively on an open set admitting the origin in its closure, but notat the origin—and isometrically for such a metric. In the case that the isotropy at the origin of a quasihomogeneousaction is semi-simple, we provide a complete set of normal forms of the metric and the action.
1. I
NTRODUCTION
A Riemannian or pseudo-Riemannian metric is called locally homogeneous if any two points can be con-nected by flowing along a finite sequence of local Killing fields. The study of such metrics is a traditional fieldin differential geometry. In dimension two, they are exactly the semi-Riemannian metrics of constant sec-tional curvature. Locally homogeneous Riemannian metrics of dimension three are the subject of Thurston’s3-dimensional geometrization program [Thu97]. The classification of compact locally homogeneous Lorentz3-manifolds was given in [DZ10].The most symmetric geometric structures after the locally homogeneous ones are those which are quasi-homogeneous , meaning they are locally homogeneous on an open set containing the origin in its closure,but not locally homogeneous in the neighborhood of the origin. In particular, all the scalar invariants of aquasihomogeneous geometric structure are constant. Recall that, for Riemannian metrics, constant scalarinvariants implies local homogeneity (see [PTV96] for an effective result).In a recent joint work with A. Guillot, the first author obtained the classification of germs of quasihomo-geneous, real analytic, torsion free, affine connections on surfaces [DG13]. The article [DG13] also classifiesthe quasihomogeneous germs of real analytic, torsion free, affine connections which extend to compact sur-faces. In particular, such germs of quasihomogeneous connections do exist.The first author proved in [Dum08] that a real analytic Lorentz metric on a compact -manifold which islocally homogeneous on a nontrivial open set is locally homogeneous on all of the manifold . In other words,quasihomogeneous real analytic Lorentz metrics do not extend to compact threefolds. The same is knownto be true, by work of the second author, for real analytic Lorentz metrics on compact manifolds of higherdimension, under the assumptions that the Killing algebra is semisimple, the metric is geodesically complete,and the universal cover is acyclic [Mel09]. In the smooth category, A. Zeghib proved in [Zeg96] that compactLorentz 3-manifolds which admit essential Killing fields are necessarily locally homogeneous. Key words and phrases. real analytic Lorentz metrics, transitive Killing Lie algebras, local differential invariants.The authors acknowledge support from U.S. National Science Foundation grants DMS-1107452, 1107263, 1107367, "RNMS: Geo-metric Structures and Representation Varieties (the GEAR Network)." Melnick was also supported during work on this project by aCentennial Fellowship from the American Mathematical Society and by NSF grants DMS-1007136 and 1255462.MSC 2010: 53A55, 53B30, 53C50.
Here we simplify arguments of [Dum08] and introduce new ideas in order to dispense with the compact-ness assumption and prove the following local result:
Theorem 1.
Let g be a real-analytic Lorentz metric in a connected open neighborhood U of the origin in R .If g is locally homogeneous on a nontrivial open subset in U, then g is locally homogeneous on all of U. As a by-product of this new proof, we classifiy Lie algebras that can act isometrically for a three-dimensionalLorentz metric and quasihomogeneously , meaning they act transitively on an open set admitting the origin inits closure, but not at the origin. In the case that the isotropy at the origin of such a quasihomogeneous actionis semisimple, we provide a complete set of normal forms of the metric and the action, which, by Theorem 1above, are all locally homogeneous (see Proposition 10 and Proposition 11).We also present a new approach to the problem in Section 5, relying on the Cartan connection associatedto a Lorentzian metric. This approach yields a nice alternate proof of our results.Our work is motivated by Gromov’s
Open-Dense Orbit Theorem [DG91, Gro88] (see also [Ben97, Fer02]).Gromov’s result asserts that, if the pseudogroup of local automorphisms of a rigid geometric structure —suchas a Lorentz metric or a connection—acts with a dense orbit, then this orbit is open. In this case, the rigidgeometric structure is locally homogeneous on an open dense set. Gromov’s theorem says little about thismaximal open and dense set of local homogeneity, which appears to by mysterious (see [DG91, 7.3.C]). Inmany interesting geometric situations, it can be shown to be all of the connected manifold. This was proved,for instance, for Anosov flows preserving a pseudo-Riemannian metric arising from differentiable stable andunstable foliations and a transverse contact structure [BFL92]. In [BF05], the authors deal with this question;their results indicate ways in which some rigid geometric structures cannot degenerate off the open dense set.The composition of this article is the following. In Section 2 we use the geometry of Killing fields andgeometric invariant theory to prove that the Killing Lie algebra of a three-dimensional quasihomogeneousLorentz metric g is a three-dimensional, solvable, nonunimodular Lie algebra. We also show that g is locallyhomogeneous away from a totally geodesic surface S , on which the isotropy is a one parameter semisim-ple group or a one parameter unipotent group. In the case of semisimple isotropy, Theorem 1 is proved inSection 3. The proof of this case relies on the classification of normal forms of the metrics admitting quasi-homogeneous isometric actions (see Proposition 10 and Proposition 11). In the case of unipotent isotropy,Theorem 1 is proved in Section 4. Section 5 provides an alternative proof of Theorem 1 using the formalismof Cartan connections.Our result raises the following question: Question 1.
Let g be a smooth Lorentz metric on a connected three-dimensional manifold M. If g is locallyhomogeneous on an open, dense subset of M, then must g be locally homogeneous on all of M?
We are aware of noncompact quasihomogeneous examples of lower regularity C , recently discovered byC. Frances. We would like to thank C. Frances for interesting conversations on the topic of this paper. Wethank the referee for her/his careful reading of our manuscript and many useful remarks.2. K ILLING L IE A LGEBRA . I
NVARIANT T HEORY
Let g be a real analytic Lorentz metric defined in a connected open neighborhood U of the origin in R ,which we assume is also simply connected. In this section we recall the definition and several properties ofthe Killing algebra of ( U , g ) . These were proved in [Dum08] without use of the compactness assumption.For completeness, we briefly explain their derivation again here.Classically, (see, for instance [Gro88, DG91]) one considers the k -jet of g by taking at each point u ∈ U the expression of g up to order k in exponential coordinates. In these coordinates, the 0-jet of g is the UASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICS DO NOT EXIST 3 standard flat Lorentz metric dx + dydz . At each point u ∈ U , the space of exponential coordinates is actedon simply transitively by O ( , ) , the identity component of which is isomorphic to PSL ( , R ) . The spaceof all exponential coordinates in U compatible with a fixed orientation and time orientation is a principal PSL ( , R ) -bundle over U , which we will call the orthonormal frame bundle and denote by R ( U ) .Geometrically, the k -jets of g form an analytic PSL ( , R ) -equivariant map g ( k ) : R ( U ) → V ( k ) , where V ( k ) is the finite-dimensional vector space of k -jets at 0 of Lorentz metrics on R with fixed 0-jet dx + dydz . Thegroup O ( , ) ≃ PSL ( , R ) acts linearly on this space, in which the origin corresponds to the k -jet of the flatmetric. One can find the details of this classical construction in [DG91].Recall also that a local vector field is a Killing field for a Lorentz metric g if its flow preserves g whereverit is defined. Note that local Killing fields preserve orientation and time orientation, so they act on R ( U ) . Thecollection of all germs of local Killing fields at a point u has the structure of a finite dimensional Lie algebra g called the local Killing algebra of g at u . At a given point u ∈ U , the subalgebra i of the local Killing algebraconsisting of the local Killing fields X with X ( u ) = isotropy algebra at u .The proof of Theorem 1 will use analyticity in an essential way. We will make use of an extendabilityresult for local Killing fields proved first by Nomizu in the real-analytic Riemannian setting [Nom60] andgeneralized then for any C w rigid geometric structure by Amores and Gromov [Amo79, Gro88, DG91]. Thisphenomenon states that a local Killing field of g can be extended analytically along any curve g in U , and theresulting Killing field germ at the endpoint only depends on the homotopy type of g . Because U is assumedconnected and simply connected, local Killing fields extend to all of U. Therefore, the local Killing algebraat any u ∈ U equals the algebra of globally defined Killing fields on U , which we will denote by g . Definition . The Lorentz metric g is locally homogeneous on an open subset W ⊂ U , if for any w ∈ W andany tangent vector V ∈ T w W , there exists X ∈ g such that X ( w ) = V . In this case, we will say that the Killingalgebra g is transitive on W .Any two points in a connected open subset W on which g is locally homogeneous can be related by flowingalong a finite sequence of local Killing fields of g .Notice that Nomizu’s extension phenomenon does not imply that the extension of a family of pointwiselinearly independent Killing fields stays linearly independent. The assumption of Theorem 1 is that g istransitive on a nonempty open subset W ⊂ U . Choose three elements X , Y , Z ∈ g that are linearly independentat a point u ∈ W . The function vol g ( X ( u ) , Y ( u ) , Z ( u )) is analytic on U and nonzero in a neighborhood of u .The vanishing set of this function is a closed analytic proper subset S ′ of U containing the points where g isnot transitive. Its complement is an open dense set of U on which g is transitive.From now on we will assume that g is a quasihomogeneous Lorentz metric in the neighborhood U of theorigin in R , with Killing algebra g . Let S be the complement of the maximal open subset of U on which g actstransitively—that is, of a maximal locally homogeneous subset of U . It is an intersection of closed, analyticproper subsets, so S is a nontrivial closed and analytic subset of positive codimension passing through theorigin . The aim of this article is to prove that this is impossible.We will next derive some basic properties of g that follow from quasihomogeneity. Lemma 3 ([Dum08] Lemme 3.2(i)) . The Killing algebra g cannot be both three-dimensional and unimodular.Proof. Let ( K , K , K ) be a basis of the Killing algebra. Again consider the analytic function v = vol g ( K , K , K ) .Since g is unimodular and preserves the volume form of g , the function v is nonzero and constant on eachopen set where g is transitive. On the other hand, v vanishes on S : a contradiction. (cid:3) Lemma 4 ([Dum08] Lemme 2.1, Proposition 3.1, Lemme 3.2(i)) . SORIN DUMITRESCU AND KARIN MELNICK (i) The dimension of the isotropy at a point u ∈ U differs from two.(ii) The Killing algebra g is of dimension three.(iii) The Killing algebra g is solvable.Proof. (i) Assume for a contradiction that the isotropy algebra i at a point u ∈ U has dimension two. Elementsof i act linearly in exponential coordinates at u . Since elements of i preserve g , they preserve, in particular,the k -jet of g at u , for all k ∈ N . This gives an embedding of i in the Lie algebra of PSL ( , R ) such that thecorresponding two-dimensional connected subgroup of PSL ( , R ) preserves the k -jet of g at u , for all k ∈ N .But stabilizers in a finite-dimensional linear algebraic PSL ( , R ) -action never have dimension two . Indeed, itsuffices to check this statement for irreducible linear representations of PSL ( , R ) , for which it is well-knownthat the stabilizer in PSL ( , R ) of a nonzero element is zero- or one-dimensional [Kir74].It follows that the stabilizer in PSL ( , R ) of the k -jet of g at u is of dimension three and hence equals PSL ( , R ) . Consequently, in exponential coordinates at u , each element of sl ( , R ) gives rise to a local linearvector field which preserves g , because it preserves all k -jets of the analytic metric g at u . The isotropy algebra i thus contains a copy of sl ( , R ) : a contradiction, since i was assumed of dimension two.(ii) Since g is quasihomogeneous, the Killing algebra is of dimension at least 3. For a three-dimensionalLorentz metric, the maximal dimension of the Killing algebra is 6. This characterizes Lorentz metrics ofconstant sectional curvature. Indeed, in this case, the isotropy is, at each point, of dimension three (see, forinstance, [Wol67]). These Lorentz metrics are locally homogeneous.Suppose that the Killing algebra of g is of dimension 5. Then, on any open set of local homogeneity theisotropy is two-dimensional. This is in contradiction with point (i).Last, suppose that the Killing algebra of g is of dimension 4. Then, at a point s ∈ S , the isotropy hasdimension ≥
2. Hence, point (i) implies that the isotropy at s has dimension three and thus is isomorphic to sl ( , R ) . Moreover, the standard linear action of the isotropy on T s U preserves the image of the evaluationmorphism ev ( s ) : g → T s U , which is a line. But the standard 3-dimensional PSL ( , R ) -representation doesnot admit invariant lines: a contradiction.Therefore, the Killing algebra is three-dimensional.(iii) A Lie algebra of dimension three is semisimple or solvable [Kir74]. Since semisimple Lie algebrasare unimodular, Lemma 3 implies that g is solvable. (cid:3) Let us recall Singer’s result [Sin60, DG91, Gro88] which asserts that g is locally homogeneous if and onlyif the image of g ( k ) is exactly one PSL ( , R ) -orbit in V ( k ) , for a certain k (big enough). This theorem is thekey ingredient in the proof of the following fact.
Proposition 5 ([Dum08] Lemme 2.2) . If g is quasihomogeneous, then the Killing algebra g does not preserveany nontrivial vector field of constant norm ≤ .Proof. Let k ∈ N be given by Singer’s Theorem. First suppose, for a contradiction, that there exists anisotropic vector field X in U preserved by g . Then the g -action on R ( U ) , lifted from the action on U , preservesthe subbundle R ′ ( U ) , where R ′ ( U ) is a reduction of the structural group PSL ( , R ) ∼ = O o ( , ) to the stabilizerof an isotropic vector in the standard linear representation on R : H = (cid:26)(cid:18) T (cid:19) ∈ PSL ( , R ) : T ∈ R (cid:27) . Restricting to exponential coordinates with respect to frames the first vector of which is X gives an H -equivariant map g ( k ) : R ′ ( U ) → V ( k ) . On each open set W on which g is locally homogeneous, the image g ( k ) ( R ′ ( W )) is exactly one H -orbit O ⊂ V ( k ) . Let s ∈ S be a point in the closure of W . Then the image under UASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICS DO NOT EXIST 5 g ( k ) of the fiber R ′ ( W ) s lies in the closure of O . But H is unipotent, and a classical result due to Kostant andRosenlicht [Ros61] asserts that for algebraic representations of unipotent groups, the orbits are closed. Thisimplies that the image g ( k ) ( R ′ ( W ) s ) is also O . Moreover, this holds for all s ∈ S . Indeed, the restriction of g to S being transitive (as will be proved independently in point (i) of Lemma 6), this holds for all s ∈ S .Any open set of local homogeneity in U admits points of S in its closure. It follows that the image of R ′ ( U ) under g ( k ) is exactly the orbit O . Singer’s theorem implies that g is locally homogeneous, a contradiction toquasihomogeneity.If there exists a g -invariant vector field X in U of constant strictly negative g -norm, then the g -action on R ( U ) preserves a subbundle R ′ ( U ) with structural group H ′ , where H ′ is the stabilizer of a strictly negativevector in the standard linear representation of PSL ( , R ) on R . In this case, H ′ is a compact one param-eter group in PSL ( , R ) . The previous argument again yields a contradiction, after replacing the Kostant-Rosenlicht Theorem by the obvious fact that orbits of smooth compact group actions are closed. (cid:3) Lemma 6 (compare [Dum08], Proposition 3.3) . After possibly shrinking U, we have(i) S is a connected, real analytic submanifold of codimension one, on which g acts transitively.(ii) The isotropy at a point of S is unipotent or R -semisimple.(iii) The restriction of g to S is degenerate.Proof. (i) The fact that S is a real analytic set was already established above: it coincides with the vanishingof the analytic function v = vol g ( K , K , K ) , where ( K , K , K ) is a basis of the Killing algebra. If needed,one can shrink the open set U in order that S be connected. By point (i) in Lemma 4, the isotropy algebra atpoints in S has dimension one or three. We prove that this dimension must be equal to one.Assume, for a contradiction, that there exists s ∈ S such that the isotropy at s has dimension three. Then,the isotropy algebra at s is isomorphic to sl ( , R ) . On the other hand, since both are 3-dimensional, theisotropy algebra at s is isomorphic to g . Hence, g is semisimple, which contradicts Lemma 4 (iii).It follows that the isotropy algebra at each point s ∈ S is of dimension one. Equivalently, the evaluationmorphism ev ( s ) : g → T s U has rank two. Since the g -action preserves S , this implies that S is a smoothsubmanifold of codimension one in U and T s S coincides with the image of ev ( s ) . The restriction of g to S satisfies Definition 2, so is transitive.(ii) Let i be the isotropy Lie algebra at s ∈ S . It corresponds to a 1-parameter subgroup of PSL ( , R ) ,which is elliptic, R -semisimple, or unipotent. In any case, there is a tangent vector V ∈ T s U annihilated by i .Then i also vanishes along the curve exp s ( tV ) , where defined. Because points of U \ S have trivial isotropy,this curve must be contained in S . Thus the fixed vector V of the flow of i is tangent to S .If i is elliptic, it preserves a tangent direction at s transverse to the invariant subspace T s S ⊂ T s U . Within T s S , there must also be an invariant line independent from V . But now an elliptic flow with three invariantlines must be trivial. We conclude that i is semisimple or unipotent.(iii) If the isotropy is unipotent, the vector V annihilated by i must be isotropic, and the invariant subspace T s S must equal V ⊥ . So S is degenerate in this case.If i is semisimple over R , then V is spacelike. The other two eigenvectors of i have nontrivial eigenvaluesand must be isotropic. On the other hand, i preserves the plane T s S , so it preserves a line of T s U transverseto S and a line independent from V in T s S . These lines must be the eigenspaces of i . If the plane T s S ⊂ T s U contains an isotropic line and is transverse to an isotropic line, then it is degenerate. (cid:3) According to Lemma 6 we have two different geometric situations, which will be treated separately inSections 3 and 4. The case of R -semisimple isotropy will be referred to as just “semisimple” below. SORIN DUMITRESCU AND KARIN MELNICK
3. N
O QUASIHOMOGENEOUS L ORENTZ METRICS WITH SEMISIMPLE ISOTROPY
If the isotropy at s ∈ S is semisimple, then it fixes a vector V ∈ T s S of positive g -norm. Using the transitive g -action on S , we can extend V to a g -invariant vector field X on S with constant positive g -norm. In thissection we assume that the isotropy is semisimple . We can suppose that X is of constant norm equal to 1.Recall that the affine group of the real line Aff is the group of transformations of R given by x ax + b ,with a ∈ R ∗ and b ∈ R . If Y is the infinitesimal generator of the one-parameter group of homotheties and H the infinitesimal generator of the one parameter group of translations, then [ Y , H ] = H . Lemma 7 (compare [Dum08], Proposition 3.6) . (i) The Killing algebra g is isomorphic to R ⊕ aff . Thestabilizer of a point of S corresponds to a one-parameter group of homotheties in Aff .(ii) The vector field X is the restriction to S of a central element X ′ in g .(iii) The restriction of the Killing algebra to S has, in adapted analytic coordinates ( x , h ) , a basis ( − h ¶¶ h , ¶¶ h , ¶¶ x ) .(iv) In the above coordinates, the restriction of g to S is dx .Proof. (i) We show first that the derived Lie algebra g ′ = [ g , g ] is 1-dimensional. It is a general fact that thederived algebra of a solvable Lie algebra is nilpotent [Kir74]. Remark first that [ g , g ] =
0. Indeed, otherwise g is abelian and the action of the isotropy i ⊂ g at a point s ∈ S is trivial on g and hence on T s S , which isidentified with g / i . The isotropy action on the tangent space T s S being trivial implies that the isotropy actionis trivial on T s U (An element of O ( , ) which acts trivially on a plane in R is trivial). This implies that theisotropy is trivial at s ∈ S : a contradiction. As g is 3-dimensional, g ′ is a nilpotent Lie algebra of dimension1 or 2, hence g ′ ≃ R , or g ′ ≃ R .Assume, for a contradiction, that g ′ ≃ R . We first prove that the isotropy i lies in [ g , g ] . Suppose this isnot the case. Then [ g , g ] ≃ R acts freely and transitively on S , preserving the vector field X . Then X is therestriction to S of a Killing vector field X ′ ∈ [ g , g ] .Let Y be a generator of the isotropy at s ∈ S . Since X is fixed by the isotropy, one gets, in restrictionto S , the following Lie bracket relation: [ Y , X ′ ] = [ Y , X ] = aY , for some a ∈ R . On the other hand, by ourassumption, Y / ∈ [ g , g ] , meaning that a =
0. This implies that X ′ is a central element in g . In particular, g ′ isat most one-dimensional: a contradiction. Hence i ⊂ [ g , g ] .Now let Y be a generator of i , { Y , X ′ } be generators of [ g , g ] , and ( Y , X ′ , Z ) be a basis of g . The tangentspace of S at a point s ∈ S is identified with g / i . Denote ¯ X ′ , ¯ Z the projections of X ′ and Z to this quotient. Theinfinitesimal action of Y on this tangent space is given in the basis { ¯ X ′ , ¯ Z } by the matrix ad ( Y ) = (cid:18) ∗ (cid:19) because g ′ ≃ R and ad ( Y )( g ) ⊂ g ′ . Moreover, ad ( Y ) =
0, since the restriction of the isotropy action to T s S is injective. From this form of ad ( Y ) , we see that the isotropy is unipotent with fixed direction R X ′ : acontradiction.We have proved that [ g , g ] is 1-dimensional. Notice that i = [ g , g ] . Indeed, if they are equal, then the actionof the isotropy on the tangent space T s U at s ∈ S is trivial: a contradiction.Let H be a generator of [ g , g ] , and Y the generator of i . Then [ Y , H ] = aH , with a ∈ R . If a =
0, then theimage of ad ( Y ) , which lies in [ g , g ] , belongs to the kernel of ad ( Y ) , which contradicts semisimplicity of theisotropy. Therefore a = Y of the isotropy, that a =
1, so [ Y , H ] = H .Let X ′ ∈ g be such that { X ′ , H } span the kernel of ad ( H ) . Then ( Y , X ′ , H ) is a basis of g . There is b ∈ R such that [ X ′ , Y ] = bH . After replacing X ′ by X ′ + bH , we can assume [ X ′ , Y ] =
0. It follows that g is the Lie UASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICS DO NOT EXIST 7 algebra R ⊕ aff ( R ) . The Killing field X ′ spans the center, the isotropy Y spans the one-parameter group ofhomotheties, and H spans the one-parameter group of translations.(ii) This comes from the fact that X is the unique vector field tangent to S invariant by g .(iii) The commuting Killing vector fields X ′ and H are nonsingular on S . This implies that, in adaptedcoordinates ( x , h ) on S , H = ¶¶ h and X = ¶¶ x . Because [ Y , X ] =
0, the restriction of Y to S has the expression f ( h ) ¶¶ h , with f an analytic function vanishing at the origin. The Lie bracket relation [ Y , H ] = H reads (cid:20) f ( h ) ¶¶ h , ¶¶ h (cid:21) = ¶¶ h , and leads to f ( h ) = − h .(iv) Since H = ¶¶ h and X = ¶¶ x are Killing fields, the restriction of g to S admits constant coefficients withrespect to the coordinates ( x , h ) . Since H is expanded by the isotropy, it follows that H is of constant g -normequal to 0. On the other hand, X is of constant g -norm equal to one. It follows that the expression of g on S is dx . (cid:3) Lemma 8.
Assume g as in Lemma 7 acts quasihomogeneously on ( U , g ) . In adapted analytic coordinates ( x , h , z ) on U, g = dx + dhdz + Cz dh + Dzdxdh for some C , D ∈ R . Moreover, in these coordinates, ¶¶ x , ¶¶ h , and − h ¶¶ h + z ¶¶ z are Killing fields.Proof. Consider the commuting Killing vector fields X ′ and H constructed in Lemma 7. Their restrictions to S have the expressions H = ¶ / ¶ h and X = ¶ / ¶ x . Recall that on S , the vector field H is of constant g -normequal to 0 and X is of constant g -norm equal to one. Point (iv) in Lemma 7 also shows that g ( X , H ) = S .Moreover, being central, X ′ is of constant g -norm on U \ S , hence of constant g -norm one on all of U .Define a geodesic vector field Z as follows. At each point s ∈ S , there exists a unique tangent vector Z s , transverse to T s S , such that g ( Z s , Z s ) = , g ( X s , Z s ) = , and g ( H s , Z s ) =
1. In fact, Z s spans the secondisotropic line (other than that generated by H s ) in X ⊥ s . In this line Z s is uniquely determined by the relation g ( H s , Z s ) =
1. Now X ′ and H are Killing and, in restriction to S , commute. So along S , the vector field Z isstable by the flow of X and H . Now extend Z via the geodesic flow: Z ( exp s ( tZ s )) : = ( exp s ) ∗ tZ s ( Z s ) = dd t exp s ( tZ s ) The resulting geodesic vector field is well defined on a sufficiently small open neighborhood of S in U . Since X ′ and H are Killing, their flows commute with the exponential map, so Z commutes with X ′ and H .The image of S through the flow of Z defines a foliation by surfaces. Each leaf is given by exp S ( zZ ) , forsome z small enough. The leaf S corresponds to z = ( x , h , z ) be analytic coordinates in the neighborhood of the origin such that X ′ = ¶ / ¶ x , H = ¶ / ¶ h , Z = ¶ / ¶ z . The scalar product g ( Z , X ′ ) is constant along the orbits of Z . This comes from the following classicalcomputation : Z · g ( X ′ , Z ) = g ( (cid:209) Z X ′ , Z ) + g ( X ′ , (cid:209) Z Z ) = (cid:209) Z Z = (cid:209) · X ′ is skew-symmetric with respect to g . The same is true for g ( Z , H ) . In particular, thecoefficients in g of dxdz and dhdz are constant on the orbits of Z .Moreover, the invariance of the metric by the commutative Killing algebra generated by X ′ and H impliesthat dxdz and dhdz are also constant along the orbits of X ′ and of H . This implies that the coefficients of dxdz and dhdz are 0 and 1, respectively, not only on S , but over all of U . SORIN DUMITRESCU AND KARIN MELNICK
The coefficients of dh and dxdh depend only on z . Then g = dx + dhdz + c ( z ) dh + d ( z ) dxdh with c and d analytic functions which both vanish at z = g by Y . Recall that [ Y , X ′ ] = [ Y , H ] = H . Note that Y preserves the twoisotropic directions of X ′⊥ , which are spanned by Z and H − d ( z ) X ′ . From g ( X ′ , H − d ( z ) X ′ ) ≡
1, compute0 = Y . ( g ( X ′ , H − d ( z ) X ′ )) = g ([ Y , X ′ ] , H − d ( z ) X ′ ) + g ( X ′ , [ Y , H − d ( z ) X ′ ])= g ( X ′ , H ) − ( Y . d ) g ( X ′ , X ′ ) = d ( z ) − ( Y . d )( z ) , so Y . d = d . Then [ Y , H − d ( z ) X ′ ] = H − d ( z ) X ′ . Next, from g ( H − d ( z ) X ′ , Z ) ≡ = g ([ Y , H − d ( z ) X ′ ] , Z ) + g ( H − d ( z ) X ′ , [ Y , Z ]) = + g ( H − d ( z ) X ′ , [ Y , Z ]) , so [ Y , Z ] = − Z . Now, since Y and X ′ commute, the general expression for Y is Y = u ( h , z ) ¶¶ h + v ( h , z ) ¶¶ z + t ( h , z ) ¶¶ x with u , v , and t analytic functions, where u ( h , ) = − h , and v and t vanish on { z = } .The other Lie bracket relations read [ u ( h , z ) ¶¶ h + v ( h , z ) ¶¶ z + t ( h , z ) ¶¶ x , ¶¶ h ] = ¶¶ h and [ u ( h , z ) ¶¶ h + v ( h , z ) ¶¶ z + t ( h , z ) ¶¶ x , ¶¶ z ] = − ¶¶ z . The first relation gives ¶ u ¶ h = − ¶ v ¶ h = ¶ t ¶ h = . The second one leads to ¶ u ¶ z = ¶ v ¶ z = ¶ t ¶ z = . We get u ( h , z ) = − h v ( h , z ) = z t ( h , z ) = . Hence, in our coordinates, Y = − h ¶ / ¶ h + z ¶ / ¶ z . The invariance of g under the action of this linear vectorfield implies c ( e − t z ) e t = c ( z ) and d ( e − t z ) e t = d ( z ) , for all t ∈ R . This implies then that c ( z ) = Cz and d ( z ) = Dz , with C , D real constants. (cid:3) Computation of the Killing algebra.
We need to understand now whether the metrics g C , D = dx + dhdz + Cz dh + Dzdxdh constructed in Lemma 8 really are quasihomogeneous. In other words, do the metrics in this family admitother Killing fields than ¶ / ¶ x , ¶ / ¶ h and − h ¶ / ¶ h + z ¶ / ¶ z ? In this section we compute the full Killingalgebra g of g C , D . In particular, we obtain that the metrics g C , D = dx + dhdz + Cz dh + Dzdxdh alwaysadmit additional Killing fields and, by Lemma 4 (ii) are locally homogeneous.The formula for the Lie derivative of g (see, eg, [KN96]) gives ( L T g C , D ) (cid:18) ¶¶ x i , ¶¶ x j (cid:19) = T · g C , D (cid:18) ¶¶ x i , ¶¶ x j (cid:19) + g C , D (cid:18)(cid:20) ¶¶ x i , T (cid:21) , ¶¶ x j (cid:19) + g C , D (cid:18) ¶¶ x i , (cid:20) ¶¶ x j , T (cid:21)(cid:19) . UASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICS DO NOT EXIST 9
Let T = a¶ / ¶ x + b¶ / ¶ h + g¶ / ¶ z . The pairs (cid:18) ¶¶ x i , ¶¶ x j (cid:19) = ( ) (cid:18) ¶¶ z , ¶¶ z (cid:19) ( ) (cid:18) ¶¶ x , ¶¶ x (cid:19) ( ) (cid:18) ¶¶ x , ¶¶ z (cid:19) ( ) (cid:18) ¶¶ x , ¶¶ h (cid:19) ( ) (cid:18) ¶¶ h , ¶¶ z (cid:19) ( ) (cid:18) ¶¶ h , ¶¶ h (cid:19) give the following system of PDEs on a , b and g in order for T to be a Killing field:0 = b z , (1) 0 = a x + Dz b x , (2) 0 = b x + Dz b z + a z , (3) 0 = g D + Dz a x + Cz b x + g x + a h + Dz b h , (4) 0 = b h + Cz b z + Dz a z + g z , (5) 0 = zC g + Cz b h + Dz a h + g h . (6)The following proposition finishes the proof of Theorem 1 in the case of semisimple isotropy on S : Proposition 9.
The Lorentz metrics g C , D are locally homogeneous for all C , D ∈ R .Proof. It is straightforward to verify that T = Dh ¶¶ x + ( D − C ) h ¶¶ h + (( C − D ) zh − ) ¶¶ z satisfies equations (1)–(6). Note that T ( ) = − ¶ / ¶ z , so T / ∈ g , and ( U , g ) is locally homogeneous. (cid:3) We explain now our method to find the extra Killing field T in Proposition 9, and we compute the fullKilling algebra, g , of g C , D . Recall the n -dimensional Lorentzian manifolds AdS n , Min n , and dS n , of constantsectional curvature − ,
0, and 1, respectively (see, eg, [Wol67]). Recall also that AdS is isometric to SL ( , R ) with the bi-invariant Cartan-Killing metric. Proposition 10. (i) If D = and C / ∈ { , D } , then ( U , g C , D ) is locally isometric to a left-invariant metric on SL ( , R ) with g ∼ = R ⊕ sl ( , R ) . The isotropy is the graph of a Lie algebra homomorphism of the R factor to thesubalgebra spanned by a R -semisimple element of sl ( , R ) .(ii) If D = and C = D , then ( U , g C , D ) is locally isometric to a left-invariant metric on the Heisenberggroup with g ∼ = R ⋉ heis . The isotropy is the R factor, which acts by a semisimple automorphism of heis .(iii) If C = and D = , then ( U , g C , D ) is locally isometric to AdS , so g ∼ = sl ( , R ) ⊕ sl ( , R ) .(iv) If C = and D = , then ( U , g C , D ) is locally isometric to R × dS , for which g ∼ = R ⊕ sl ( , R ) . Theisotropy is generated by a semisimple element of sl ( , R ) .(v) If C = and D = , then ( U , g C , D ) is locally isometric to Min , so g ∼ = sl ( , R ) ⋉ R .Proof. Recall that ( x , h , z ) are analytic coordinates on U , with S = z − ( ) , such that all Lorentz metrics g C , D admit the Killing fields X ′ = ¶¶ x , Y = − h ¶¶ h + z ¶¶ z and H = ¶¶ h , for which the Lie bracket relations are [ Y , X ′ ] = [ H , X ′ ] = [ Y , H ] = H . Moreover, Proposition 9 shows that all Lorentz metrics g C , D are locallyhomogeneous and that their full Killing algebra g is of dimension at least four. In particular, the Killingalgebra g strictly contains the previous three-dimensional Lie algebra as a subalgebra l acting quasihomoge-neously in the neighborhood of the origin.Assuming g C , D is not of constant sectional curvature, then Lemma 4 (i) implies dim g =
4. We first derivesome information on the algebraic structure of g in this case. If dim g =
4, then it is generated by X ′ , Y , H , and an additional Killing field T . Since the isotropy R Y atthe origin fixes the spacelike vector X ( ) and expands H , we can choose a fourth generator T of g evaluatingat the origin to a generator of the second isotropic direction of the Lorentz plane X ( ) ⊥ . As the action of Ad ( Y ) on g is g -skew symmetric, we get at the origin : [ Y , T ]( ) = − T ( ) . Hence [ Y , T ] = − T + aY forsome constant a ∈ R , and we can replace T with T − aY in order that [ Y , T ] = − T . Since X ′ and Y commute, [ X ′ , T ] is also an eigenvector of ad ( Y ) with eigenvalue −
1. This eigenspace of ad ( Y ) is one-dimensional, so [ T , X ′ ] = cT , for some c ∈ R .The Jacobi relation [ Y , [ T , H ]] = [[ Y , T ] , H ] + [ T , [ Y , H ]] = [ − T , H ] + [ T , H ] = [ T , H ] commutes with Y . The centralizer of Y in g is R Y ⊕ R X ′ . We conclude that [ H , T ] = aX ′ − bY ,for some a , b ∈ R .(i) Assume D = C / ∈ { , D } . A straightforward computation shows that g C , D is not of constantsectional curvature. We will construct a Killing field T = a¶ / ¶ x + b¶ / ¶ h + g¶ / ¶ z , meaning the functions a , b and g solve the PDE system (1)–(6). We will moreover construct it so that c = a = T and l . Remark that, since T and X ′ commute,the coefficients a , b and g of T do not depend on the coordinate x ; in particular, equation (2) is satisfied. Therelation [ H , T ] = aX ′ − bY reads, when a = (cid:20) ¶¶ h , T (cid:21) = ¶¶ x + b (cid:18) h ¶¶ h − z ¶¶ z (cid:19) . This leads to a h = , b h = bh , and g h = − bz . Using equation (1), we obtain b = bh + b . We can take theadditive constant b = ¶¶ h ∈ l . Now equation (4) gives g = − bzh − / D .Equation (6) now reads0 = zC ( − D − zbh ) + Cz bh + Dz − bz = − CzD + Dz − bz which yields b = D − C / D . Now g can be written − / D − zh ( D − C / D ) .Equation (3) says a z =
0, so we conclude a = h . The resulting vector field is T = h ¶¶ x + ( D − CD ) h ¶¶ h + (cid:18) zh ( CD − D ) − D (cid:19) ¶¶ z . (7)Note that the coefficients of T also satisfy equation (5), so T is indeed a Killing field.We obtained this solution setting c =
0, so the Lie algebra g generated by { T , X ′ , Y , H } contains X ′ as acentral element. We also set a =
1, and found b = D − C / D , so [ H , T ] = X ′ + ( C / D − D ) Y , which we willcall Y ′ . It is straightforward to verify that for T as above, [ Y , T ] = − T . Under the hypothesis C = D , the Liesubalgebra generated by { Y ′ , H , T } is isomorphic to sl ( , R ) , with Y ′ R -semisimple, and it acts transitively on U . Consequently, g C , D is locally isomorphic to a left-invariant Lorentz metric on SL ( , R ) . The full Killingalgebra is g ∼ = R ⊕ sl ( , R ) , with center generated by X ′ , and isotropy R Y = R ( X ′ + Y ′ ) . This terminates theproof of point (i).(ii) When D = C = D , then (7) still solves the Killing equations. The bracket relations are the same,but now [ H , T ] = X ′ . Then g ∼ = R ⋉ heis , where the heis factor is generated by { H , T , X ′ } and acts transitively,and the R factor is generated by the isotropy Y , which acts by a semisimple automorphism on heis . Up tohomothety, there is a unique left-invariant Lorentz metric on Heis in which X ′ is spacelike, by Proposition1.1 of [DZ10], where it is called the Lorentz-Heisenberg geometry . UASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICS DO NOT EXIST 11 (iii) When C = D =
0, then (7) again solves the Killing equations. It now simplifies to T = h ¶¶ x + Dh ¶¶ h + (cid:18) − zhD − D (cid:19) ¶¶ z . The bracket relation is [ H , T ] = X ′ − DY , and g still contains a copy of R ⊕ sl ( , R ) , with center generatedby X ′ and sl ( , R ) generated by { X ′ − DY , H , T ′ } . The sl ( , R ) factor still acts simply transitively. On theother hand, one directly checks that a = b = g = e − Dx is a solution of the PDE system, meaning that e − Dx ¶ / ¶ z is also a Killing field. From (cid:20) X ′ , e − Dx ¶¶ z (cid:21) = − De − Dx ¶¶ z = { T , X ′ , Y , H } , inwhich X ′ is central. It follows that the Killing algebra is of dimension at least five, hence six by Lemma 4 (i),which implies that g , D is of constant sectional curvature. Since g , D is locally isomorphic to a left-invariantLorentz metric on SL ( , R ) , the sectional curvature is negative. Up to normalization, g , D is locally isometricto AdS .(iv) The Killing field T in (7) multiplied by D gives T D = Dh ¶¶ x + ( D − C ) h ¶¶ h + (cid:0) zh ( C − D ) − (cid:1) ¶¶ z . Setting C = D = T = − Ch ¶¶ h + ( zhC − ) ¶¶ z which is indeed a Killing field of g C , . The brackets are [ X ′ , T ] = , [ H , T ] = CY , and [ Y , T ] = − T . As incase (i), the Killing Lie algebra contains a copy of R ⊕ sl ( , R ) , with center generated by X ′ , and sl ( , R ) gen-erated by { Y , H , T } . Here the isotropy generator Y lies in the sl ( , R ) -factor, which acts with two-dimensionalorbits. This local sl ( , R ) -action defines a two-dimensional foliation tangent to X ′⊥ . Recall that X ′ is of con-stant g -norm equal to one, so X ′⊥ has Lorentzian signature. The metric is, up to homotheties on the twofactors, locally isomorphic to the product R × dS .(v) If C = D =
0, then g C , D is flat and g ∼ = sl ( , R ) ⋉ R . (cid:3) As a by-product of the proof of Theorem 1 in the case of semisimple isotropy, we have obtained thefollowing more technical result:
Proposition 11.
Let g be a real-analytic Lorentz metric in a neighborhood of the origin in R . Suppose thatthere exists a three-dimensional subalgebra l of the Killing Lie algebra acting transitively on an open setadmitting the origin in its closure, but not in the neighborhood of the origin. If the isotropy at the origin is aone-parameter R -semisimple subgroup in O ( , ) , then(i) There exist local analytic coordinates ( x , h , z ) in the neighborhood of the origin and real constants C , Dsuch that g = g C , D = dx + dhdz + Cz dh + Dzdxdh . (ii) The algebra l is solvable, and equals, in these coordinates, l = h ¶¶ x , ¶¶ h , − h ¶¶ h + z ¶¶ z i . In particular, l ∼ = R ⊕ aff ( R ) , where aff ( R ) is the Lie algebra of the affine group of the real line. (iii) All the metrics g C , D are locally homogeneous. They admit a Killing field T / ∈ l of the formT = Dh ¶¶ x + ( D − C ) h ¶¶ h + (( C − D ) zh − ) ¶¶ z . The possible geometries on ( U , g C , D ) are given by (i) - (v) of Proposition 10.
4. N
O QUASIHOMOGENEOUS L ORENTZ METRICS WITH UNIPOTENT ISOTROPY
We next treat the unipotent case of Lemma 6. The following results can be found in [Dum08] Propositions3.4 and 3.5 in Section 3.1, where they are proved without making use of compactness. See also [Zeg96,Proposition 9.2] for point (iii).
Proposition 12. (i) The surface S is totally geodesic.(ii) The Levi-Civita connection (cid:209) restricted to S is either flat, or locally isomorphic to the canonical bi-invariant connection on the affine group of the real line
Aff .(iii) The restriction of the Killing algebra g to S is isomorphic either to the Lie algebra of the Heisenberggroup in the flat case, or otherwise to a solvable subalgebra sol ( , a ) of Aff × Aff , spanned by theelements ( t , ) , ( , t ) and ( w , aw ) , where t is the infinitesimal generator of the one-parameter group oftranslations, w the infinitesimal generator of the one-parameter group of homotheties, and a ∈ R . Recall that, as S has codimension one, the restriction to S of the Killing Lie algebra g of g is an isomor-phism. The Heisenberg group and sol ( , − ) are unimodular, so by Lemma 3, g is isomorphic to sol ( , a ) ,with a = − , and S is non flat .Recall that in dimension three, the curvature is completely determined by its Ricci tensor, which is asymmetric bilinear form. The Ricci tensor is determined by the Ricci operator, which is a field of g -symmetricendomorphisms A : TU → TU such that Ricci ( u , v ) = g ( Au , v ) , for any tangent vectors u , v . Definition . The metric g is said to be curvature homogeneous if for any pair of points u , u ′ ∈ U , thereexists a linear isomorphism from T u U to T u ′ U preserving both g and the curvature tensor.In dimension three, it is equivalent to assume in the previous definition that these linear maps preserveboth g and the Ricci operator A . Proposition 14. (i) The only eigenvalue of the Ricci operator is , everywhere on U.(ii) The metric g is curvature homogeneous; more precisely, in an adapted framing on U, the Ricci operatorreads A = a , a ∈ R ∗ . Proof. (i) Pick a point s in S . The Ricci operator A ( s ) must be invariant by the unipotent isotropy (whichidentifies with the stabilizer in the orthogonal group of g ( s ) of an isotropic vector X ( s ) ∈ T s U ).The action of the isotropy on T s U fixes an isotropic vector e = X ( s ) tangent to S and so preserves thedegenerate plane e ⊥ = T s S . In order to define an adapted basis, consider two vectors e , e ∈ T s U such that g ( e , e ) = g ( e , e ) = g ( e , e ) = g ( e , e ) = g ( e , e ) = UASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICS DO NOT EXIST 13
The action on T s U of the one-parameter group of isotropy is given in the basis ( e , e , e ) by the matrix L t = t − t − t , t ∈ R . First we show that A ( s ) : T s U → T s U has, in our adapted basis, the following form: l b a l − b l , a , b , l ∈ R . Since A ( s ) is invariant by the isotropy, it commutes with L t for all t . Each eigenspace of A ( s ) is preserved by L t , and eigenspaces of L t are preserved by A ( s ) . As L t does not preserve any non trivial splitting of T s U , itfollows that all eigenvalues of A ( s ) are equal to some l ∈ R . Moreover, the unique line and plane invariantby L t must also be invariant by A ( s ) , so A ( s ) is upper-triangular in the basis ( e , e , e ) . A straightforwardcalculation of the top corner entry of A ( s ) L t = L t A ( s ) leads to the relation on the b entries and thus to ourclaimed form for A ( s ) .Now the g -symmetry of A ( s ) means g ( A ( s ) e , e ) = g ( e , A ( s ) e ) , which gives b =
0. Since the symmetricfunctions of the eigenvalues of A are scalar invariants, they must be constant on all of U . This implies thatthe only eigenvalue of A is l , on all of U . It remains only to prove that l =
0. Consider an open set in U onwhich the Killing algebra sol ( , a ) is transitive, so g is locally isomorphic to a left-invariant Lorentz metricon SOL ( , a ) .The sectional and Ricci curvatures and Ricci operator of a left-invariant Lorentz metric on a given Liegroup can be calculated, starting from the Koszul formula, in terms of the brackets between left-invariantvector fields forming an adapted framing of the metric. In [CK09] Calvaruso and Kowalski calculate Riccioperators for left-invariant Lorentz metrics on three-dimensional Lie groups, assuming they are not sym-metric (see also previous curvature calculations in [Nom79], [CP97], [Cal07]). If the metric on U \ S weresymmetric, then the covariant derivative of the curvature would vanish on all of U , which would imply U lo-cally symmetric, hence locally homogeneous; therefore, we need consider only nonsymmetric left-invariantmetrics here. A consequence of their Theorems 3.5, 3.6, and 3.7 is that the Ricci operator of a left-invariant,nonsymmetric Lorentz metric on a nonunimodular three-dimensional Lie group admits a triple eigenvalue l if and only if l =
0, and the Ricci operator is nilpotent of order two. We conclude l =
0, so A ( s ) has theform claimed. Moreover, A is nilpotent of order two on U \ S .(ii) Because g acts transitively on S , there is an adapted framing along S in which A ≡ A ( s ) . The parameter a in A ( s ) cannot vanish; otherwise the curvature of g vanishes on S and ( S , (cid:209) ) is flat, which was provedto be impossible in Proposition 12. Now the Ricci operator on S is nontrivial and lies in the closure of the PSL ( , R ) -orbit O of the Ricci operator on U \ S . But we know from (i) that on U \ S , the Ricci operatoris g -symmetric and nilpotent of order 2, so it has the same form as A ( s ) , meaning it also belongs to the PSL ( , R ) -orbit of . (cid:3) Now Ricc ( u , u ) is a quadratic form of rank one equal to g ( W , u ) , for some nonvanishing isotropic vectorfield W on U , which coincides with X on S . Invariance of Ricci by g implies invariance of W . Proposition 5implies that g is locally homogeneous.5. A LTERNATE PROOFS USING THE C ARTAN CONNECTION
The aim of this section is to give a second proof of Theorem 1 using the Cartan connection associatedto a Lorentz metric. The reader can find more details about the geometry of Cartan connections in thebook [Sha97]. We still consider g a Lorentz metric defined in a connected open neighborhood U of the originin R .5.1. Introduction to the Cartan connection.
Let h = o ( , ) ⋉ R , . Let P = O ( , ) < O ( , ) ⋉ R , , so p = o ( , ) ⊂ h . Let p : B → U be the principal P -bundle of normalized frames on U , in which the Lorentzmetric g has the matrix form I = . (Note that B is nearly the same as the bundle R ( U ) from Section 2, though it has been enlarged to allow allpossible orientations and time orientations.)The Cartan connection associated to ( U , g ) is the 1-form w ∈ W ( B , h ) formed by the sum of the Levi-Civita connection of the metric n ∈ W ( B , p ) and the tautological 1-form q ∈ W ( B , R , ) , defined by q b ( v ) = b − ( p ∗ v ) . The form w satisfies the following axioms for a Cartan connection:(1) It gives a parallelization of B —that is, for all b ∈ B , the restriction w b : T b B → h is an isomorphism.(2) It is P -equivariant: for all p ∈ P , the pullback R ∗ p w = Ad p − ◦ w .(3) It recognizes fundamental vertical vector fields: for all X ∈ p , if X ‡ is the vertical vector field on B generated by X , then w ( X ‡ ) ≡ X .The Cartan curvature of w is K ( X , Y ) = d w ( X , Y ) + [ w ( X ) , w ( Y )] . This 2-form is always semibasic, meaning K b ( X , Y ) only depends on the projections of X and Y to T p ( b ) U ;in particular, K vanishes when either input is a vertical vector. We will therefore express the inputs to K b astangent vectors at p ( b ) . Torsion-freeness of the Levi-Civita connection implies that K has values in p . Thus K is related to the usual Riemannian curvature tensor R ∈ W ( U ) ⊗ End ( T M ) by b ◦ R p ( b ) ( u , v ) ◦ b − = K b ( u , v ) . The benefit here of working with the Cartan curvature is that, when applied to Killing vector fields, it gives aprecise relation between the brackets on the manifold U and the brackets in the Killing algebra g .The P -equivariance of w leads to P -equivariance of K : ( R ∗ p K )( X , Y ) = ( Ad p − )( K ( X , Y )) . The infinites-imal version of this statement is, for A ∈ p , K ([ A ‡ , X ] , Y ) + K ( X , [ A ‡ , Y ]) = [ K ( X , Y ) , A ] . A Killing field Y on U lifts to a vector field on B , which we will also denote Y , with L Y w =
0. Note thatalso L Y K = X and Y are Killing fields, then X . ( w ( Y )) = w [ X , Y ] and Y . ( w ( X )) = w [ Y , X ] . UASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICS DO NOT EXIST 15
In this case, K ( X , Y ) = X . ( w ( Y )) − Y . ( w ( X )) − w [ X , Y ] + [ w ( X ) , w ( Y )]= w [ X , Y ] − w [ Y , X ] − w [ X , Y ] + [ w ( X ) , w ( Y )]= w [ X , Y ] + [ w ( X ) , w ( Y )] so, when X and Y are Killing, then w [ X , Y ] = [ w ( Y ) , w ( X )] + K ( X , Y ) . (8)Via the parallelization given by w , the semibasic, p -valued 2-form K corresponds to a P -equivariant,automorphism-invariant function k : B → ∧ R , ∗ ⊗ p . The P -representation on the target vector space is associated naturally to the adjoint representation of G restricted to P , and will be denoted g · k ( b ) , for g ∈ P and b ∈ B . We will use the same notation below forother P -represenations associated to the adjoint, and also for the corresponding Lie algebra representations—for example, X · k ( b ) for X ∈ p .5.2. Curvature representation.
Denote ( e , h , f ) a basis of R , in which the inner product is given by I . Let E , H , F be generators of p with matrix expression in the basis ( e , h , f ) E = −
10 10 H = − F = − . Therefore this representation of p is equivalent to ad p via the isomorphism sending ( e , h , f ) to ( E , H , F ) .Denote by ∗ the isomorphism R , → R , ∗ with w ∗ ( u ) = h w , u i . Note that for p ∈ O ( , ) and x ∈ R , ,we have ( px ) ∗ = p ∗ x ∗ for the dual represention p ∗ x ∗ = x ∗ ◦ p − .Next we define an O ( , ) -equivariant homomorphism j : ∧ R , ∗ ⊗ o ( , ) → R ∗ ⊗ R , where the rep-resentation on End R is by conjugation. Define j on simple tensors by j ( v ∗ ∧ w ∗ ⊗ X ) = ( Xv ) ∗ ⊗ w − ( Xw ) ∗ ⊗ v = ( w ∗ ◦ X ) ⊗ v − ( v ∗ ◦ X ) ⊗ w . Equivariance is easy to check. When the input lies in the submodule W satisfying the Bianchi identity,then the output is I -symmetric (see [Sha97], Section 6, Proposition 1.4 (ii)(c)). The Ricci endomorphism A ,defined in terms of the curvature tensor by h A x v , w i = tr R x ( v , · ) w = Ricci x ( v , w ) , ∀ v , w ∈ T x M corresponds via w to the function j ◦ k . Recall that in dimension 3, the curvature tensor is determined by theRicci curvature, so j restricted to W is actually an isomorphism onto its image.This image is the sum E ⊕ E of two irreducible components of the O ( , ) -representation on End R .The first, denoted E , is the one-dimensional trivial representation, generated by the identity on R , which wewill denote m d . Another irreducible component E corresponds to endomorphisms in o ( , ) , which satisfy X I = − I X t . The O ( , ) -invariant complementary subspace, consisting of the I -symmetric endomorphisms,splits into E and the last irreducible component, E , which is five-dimensional. The component E capturesthe scalar curvature, while E corresponds to the tracefree Ricci endomorphism.In the second column of the following table, we list a basis for E ⊕ E , with notation for each element inthe first column, and the elements of W ⊂ ∧ R , ∗ ⊗ o ( , ) mapping to them under j in the third column.Note that the elements in the last column span the space of all possible values of k . R × W ⊂ ∧ R , ∗ ⊗ o ( , ) m d ( f ∗ ⊗ e + h ∗ ⊗ h + e ∗ ⊗ f ) h ∗ ∧ e ∗ ⊗ F + e ∗ ∧ f ∗ ⊗ H + f ∗ ∧ h ∗ ⊗ Em e e ∗ ⊗ e e ∗ ∧ h ∗ ⊗ Em eh h ∗ ⊗ e + e ∗ ⊗ h f ∗ ∧ e ∗ ⊗ E + f ∗ ∧ h ∗ ⊗ Hm h − e f h ∗ ⊗ h − f ∗ ⊗ e − e ∗ ⊗ f f ∗ ∧ e ∗ ⊗ H + f ∗ ∧ h ∗ ⊗ E + h ∗ ∧ e ∗ ⊗ Fm h f f ∗ ⊗ h + h ∗ ⊗ f h ∗ ∧ f ∗ ⊗ H + f ∗ ∧ e ∗ ⊗ Fm f f ∗ ⊗ f h ∗ ∧ f ∗ ⊗ F Assume now that g is quasihomogeneous. Recall that, by the results in Section 2, the Killing algebra g is three-dimensional. It acts transitively on U , away from a two-dimensional, degenerate submanifold S passing through the origin. Moreover, g acts transitively on S and the isotropy at points of S is conjugatedto a one-parameter semisimple group or to a one-parameter unipotent group in PSL ( , R ) . We will study theinteraction of g , w ( g ) , and k , both on and off S .5.3. Semisimple isotropy.
Let b be a point of B lying over the origin and assume that the isotropy action of g at 0 is semisimple, as in Section 3. A semisimple element of p is conjugate in P into R H , so up to changingthe choice of b ∈ p − ( ) , we may assume that w b ( g ) ∩ p is spanned by H . Proposition 15. (compare Lemma 7 (i)) If the isotropy of g at the origin is semisimple, then g ∼ = R ⊕ aff ( R ) .Proof. Let Y ∈ g have w b ( Y ) = H , so the corresponding Killing field vanishes at the origin. The projection w b ( g ) of w b ( g ) to R , is 2-dimensional, degenerate, and H -invariant. Again, by changing the point b in the fiber above the origin, we may conjugate by an element normalizing R H so that this projection isspan { e , h } . Therefore, there is a basis ( X , Y , Z ) of g such that w b ( X ) = h + a E + b F and w b ( Z ) = e + g E + d F for some a , b , g , d ∈ R . Because K b ( Y , · ) =
0, equation (8) gives w b [ Y , X ] = [ h + a E + b F , H ] = − a E + b F ∈ w b ( g ) so a = b = [ Y , X ] =
0. A similar computation gives w b [ Y , Z ] = [ e + g E + d F , H ] = − e − g E + d F so d =
0, and [ Y , Z ] = − Z .Infinitesimal invariance of K by Y gives K b ([ Y , X ] , Z ) + K b ( X , [ Y , Z ]) = Y . ( K ( X , Z )) b = H ‡ . ( K ( X , Z )) b = [ − H , K b ( X , Z )] , which reduces to K b ( X , Z ) = [ H , K b ( X , Z )] . Since K takes values in p , where ker ( ad H − Id ) = R E , we get K b ( X , Z ) = k b ( h , e ) = rE for some r ∈ R . Now equation (8) gives for X and Z , w b [ X , Z ] = [ e + g E , h ] + rE = − g e + rE . In order that this element belong to w b ( g ) = span { H , h , e + g E } , we must have r = − g , and [ X , Z ] = − g Z . UASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICS DO NOT EXIST 17
The structure of the algebra g in the basis ( X , Y , Z ) isad Y = − ad X = − g ad Z = g . This g is isomorphic to aff ( R ) ⊕ R , with center generated by g Y − X . (cid:3) Let W = X − g Y . Note that W ( ) has norm 1 because w b ( W ) = h . As in Section 3, where the centralelement of g is called X ′ , the norm of W is constant 1 on U because it is g -invariant and equals 1 at a point of S . Existence of a Killing field of constant norm 1 has the following consequences for the geometry of U : Proposition 16. (i) The local g -action on U preserves a splitting of TU into three line bundles, L − ⊕ R W ⊕ L + , with L − and L + isotropic.(ii) The distributions L − ⊕ R W and L + ⊕ R W are each tangent to g -invariant, degenerate, totally geodesicfoliations P − and P + , respectively; moreover, the surface S is a leaf of one of these foliations, whichwe may assume is P + .Proof. (i) Because g preserves W , it preserves W ⊥ , which is a 2-dimensional Lorentz distribution. A 2-dimensional Lorentz vector space splits into two isotropic lines preserved by all linear isometries. Therefore W ⊥ = L − ⊕ L + , with both line bundles isotropic and g -invariant.(ii) Because the flow along W preserves L − and L + , the distributions L − ⊕ R W and L + ⊕ R W are involutive,and thus they each integrate to foliations P − and P + by degenerate surfaces.Let x ∈ U . Let V − ∈ G ( L − ) and V + ∈ G ( L + ) be vector fields with V ± ( x ) = [ W , V ± ]( x ) =
0. It iswell known that a Killing field of constant norm is geodesic: (cid:209) W W =
0. Moreover, because g ( V ± , V ± ) isconstant zero, W . ( g ( V ± , V ± )) = V ± . ( g ( V ± , V ± )) =
0, from which g x ( (cid:209) W V ± , V ± ) = g x ( (cid:209) V ± W , V ± ) = g x ( (cid:209) V ± V ± , V ± ) = . The tangent distributions T P ± equal ( V ± ) ⊥ , and it is now straightforward to verify from the axioms for (cid:209) that P − and P + are totally geodesic through x .The Killing field W is tangent to the surface S . Because S is degenerate, T S ⊥ is an isotropic line of W ⊥ and therefore coincides with L + or L − . We can assume it is L + , so S is a leaf of P + ; in particular, we haveshown S is totally geodesic. (cid:3) Proposition 17. (i) For x ∈ U and u , v ∈ T P ± x , the curvature R x ( u , v ) annihilates ( P ± x ) ⊥ .(ii) The Ricci endomorphism at x preserves each of the line bundles L + , R W , and L − .Proof. (i) The argument is the same for P + and P − , so we write it for P − . Let x ∈ U \ S . Because g actstransitively on a neighborhood of x , there is a Killing field A − evaluating at x to a nonzero element of L − ( x ) .Note that [ A − , W ] =
0. The orbit of x under A − and W coincides near x with an open subset of P − x . Because L − is g -invariant, the values of A − in this relatively open set belong to L − .Now A − . ( g ( A − , A − )) = g ( (cid:209) A − A − , A − ) =
0, and A − . ( g ( A − , W )) = = g x ( (cid:209) A − A − , W ) + g x ( A − , (cid:209) A − W ) = g x ( (cid:209) A − A − , W ) , using that P − x is totally geodesic. Therefore ( (cid:209) A − A − ) x = aA − for some a ∈ R . The flows along A − and W act locally transitively on P − x preserving the connection (cid:209) and commuting with A − . Thus (cid:209) A − A − ≡ aA − ona neighborhood of x in P − x . Next, W . ( g ( A − , W )) = = g ( (cid:209) W A − , W ) + g ( A − , (cid:209) W W ) = g ( (cid:209) W A − , W ) , using that W is geodesic. Therefore ( (cid:209) W A − ) x = bA − for some b ∈ R . Again invariance of (cid:209) , A − , and W implies that (cid:209) W A − ≡ bA − on a neighborhood of x in P − x . Now we compute R x ( A − , W ) A − = ( (cid:209) A − (cid:209) W − (cid:209) W (cid:209) A − − (cid:209) [ A − , W ] ) A − = (cid:209) A − ( bA − ) − (cid:209) W ( aA − ) = abA − − baA − = . This property of the curvature we have proved on U \ S remains true on S because it is a closed condition.(ii) It suffices to show that the Ricci endomorphism preserves L − ⊕ R W = T P − and L + ⊕ R W = T P + .Then invariance of L + and L − will follow from symmetry of A with respect to g . Again, we just write theproof for P − . The Ricci endomorphism preserves T P − if and only if Ricci x ( u , v ) = Ricci x ( v , u ) = u ∈ L − x , v ∈ T P − x . Assume u = ( u , w , z ) of T x U with w = W ( x ) , z ∈ L + x ,and g x ( u , z ) =
1. Then, by part (i),Ricci x ( v , u ) = g x ( R ( v , u ) u , z ) + g x ( R ( v , w ) u , w ) + g x ( R ( v , z ) u , u ) = + + = . (cid:3) Let R be the g -invariant reduction of B to the subbundle comprising frames ( x , ( v − , W ( x ) , v + )) with v − ∈ L − x and v + ∈ L + x . Now R is a principal A -bundle, where R ∗ ∼ = A < P is the subgroup with matrix form A = l l − : l ∈ R ∗ . Note that, at any b ∈ R , the projection w b ( W ) = h . Proposition 17 translates to the following statementon R . Proposition 18.
For any b ∈ R , the component ¯ k b in the representation E ⊕ E , corresponding to the Ricciendomorphism, is diagonal, so has the form ¯ k b = ym d + zm h − e f y , z ∈ R . Note that H · ¯ k b =
0, so by P -equivariance of ¯ k , the derivative in the vertical direction H ‡ . ¯ k b =
0. Becausethis curvature function is also g -invariant, it is constant on R U \ S . By continuity, we conclude that on all R ,¯ k ≡ ym d + zm h − e f y , z ∈ R . Since g acts transitively on U \ S and preserves R , for any b ∈ R | U \ S there exists a sequence a n in A such that j n ba − n → b , where each j n is in the pseudo-group generated by flows along local Killing fieldsin g ; then ( Ad a n )( w b ( g )) → w b ( g ) in the Grassmannian Gr ( , h ) . Let us consider such a sequence a n corresponding to a point b ∈ B lying above U \ S . Then we prove the following Lemma 19.
Write a n = l n l − n , l n ∈ R ∗ . Then l n → ¥ . UASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICS DO NOT EXIST 19
Proof.
First note that l n cannot converge to a nonzero number, because in this case lim n ( Ad a n )( w b ( g )) = w b ( g ) would still project onto R , modulo p , contradicting that the g -orbit of 0 is two-dimensional. Thisalso shows that a n cannot admit a convergent subsequence, meaning that a n goes to infinity in A .The space w b ( g ) can be written as span { e + r ( e ) , h + r ( h ) , f + r ( f ) } for r : R , → p a linear map. Thespace ( Ad a n )( w b ( g )) contains l − n f + a n · r ( f ) , so it contains f + l n a n · r ( f ) . If l n →
0, then this lastterm converges to f + x ∈ w b ( g ) , for some x ∈ p (because the adjoint action of a n on p is diagonal witheigenvalues l n , 1 and l − n ). But w b ( g ) is spanned by e and h , so this is a contradiction. (cid:3) Differentiating the function ¯ k : B → V ( ) = E ⊕ E gives, via the parallelization of B arising from w ,a P -equivariant, automorphism-invariant function D ( ) ¯ k : B → V ( ) = h ∗ ⊗ V , and similarly, by iteration,functions D ( i ) ¯ k : B → V ( i ) = h ∗ ⊗ V ( i − ) ; automorphism-invariant here means D ( i ) ¯ k ( f ( b )) = D ( i ) ¯ k ( b ) for all b ∈ B and all automorphisms f . For vertical directions X ∈ p , the derivative is determined by equivariance: X ‡ . ¯ k = − X · ¯ k . Our goal, in order to show local homogeneity of U , is to show that D ( i ) ¯ k has values on B in a single P -orbit. Because ¯ k determines k for 3-dimensional metrics, it will follow that D ( i ) k has valueson B in a single P -orbit, which suffices by Singer’s theorem to conclude local homogeneity (see Proposition3.8 in [Mel11] for a version of Singer’s theorem for real analytic Cartan connections and also [Pec14] for thesmooth case). By P -equivariance of these functions, it suffices to show that the values on R lie in a single A -orbit. We will prove the following slightly stronger result: Proposition 20.
The curvature ¯ k and all of its derivatives D ( i ) ¯ k are constant on R .Proof. Recall that ¯ k ≡ ym d + zm h − e f on all of R , for some fixed y , z ∈ R . The proof proceeds by induction on i . Suppose that for i ≥
0, thederivative D ( i ) ¯ k is constant on R , so that in particular, the value D ( i ) ¯ k is annihilated by H . As in the proof for i = D ( i + ) ¯ k is constant on R , it suffices to show that H ‡ . D ( i + ) ¯ k b = − H · D ( i + ) ¯ k b = b ∈ R | U \ S .To complete the induction step, we will need the following information on w b ( g ) . Lemma 21.
At b ∈ R lying over x ∈ U \ S, the Killing algebra evaluates to w b ( g ) = span { e + g E + b H , h − g H , f + a H + d F } , g , b , a , d ∈ R . Proof.
Write w b ( g ) = span { e + r ( e ) , h + r ( h ) , f + r ( f ) } . From Proposition 15, we know that ( Ad a n )( w b ( g )) → w b ( g ) = span { e + g E , h , H } . Now Lemma 19 implies that r ( h ) and r ( f ) both have zero component on E . Indeed, since this componentis dilated by l n , it must vanish in order that E / ∈ w b ( g ) .At the point b , let A − be a Killing field with p ∗ b A − ∈ L − p ( b ) , so we can assume w b ( A − ) = e . We have w b ( A − ) = e + r ( e ) and w b ( W ) = h + r ( h ) . Recall from Proposition 15 that k b ( h , e ) = rE . The fact that¯ k b = ¯ k b implies that the full curvature k b = k b , so also k b ( e , h ) = K b ( A − , W ) = rE . On the other hand, equation (8) gives0 = w b [ A − , W ] = [ h + r ( h ) , e + r ( e )] + rE , so r ( h ) e = r ( e ) h and [ r ( h ) , r ( e )] = − rE . Writing r ( e ) = g E + b H + d F and r ( h ) = b ′ H + d ′ F gives b ′ = − g and d = d ′ = g = − r , which is consistent with Proposition 15. (cid:3) We now use g -invariance of D ( i ) ¯ k . For abitrary X ∈ h , write X ‡ for the coresponding w -constant vectorfield on B . Lemma 21 gives(1) ( e + g E + b H ) ‡ ( b ) . D ( i ) ¯ k ≡ ( h − g H ) ‡ ( b ) . D ( i ) ¯ k ≡ ( f + a H + d F ) ‡ ( b ) . D ( i ) ¯ k ≡ D ( i + ) ¯ k b ( e ) = − ( g E + b H ) ‡ ( b ) . D ( i ) ¯ k = ( g E + b H ) · D ( i ) ¯ k b = g E · D ( i ) ¯ k b . The last equality above follows from the induction hypothesis. Then ( H · D ( i + ) ¯ k b )( e ) = H · ( D ( i + ) ¯ k b ( e )) − D ( i + ) ¯ k b ([ H , e ])= H · ( g E ) · D ( i ) ¯ k b − D ( i + ) ¯ k b ( e )= g ([ H , E ] + EH ) · D ( i ) ¯ k b − g E · ( D ( i ) ¯ k b )= g E · D ( i ) ¯ k b − g E · D ( i ) ¯ k b = D ( i + ) ¯ k b ( h ) = − g H · D ( i ) ¯ k b = ( H · D ( i + ) ¯ k b )( h ) = . Finally, (3) gives D ( i + ) ¯ k b ( f ) = d F · D ( i ) ¯ k b and again ( H · D ( i + ) ¯ k b )( f ) = . We have thus shown vanishing of H · D ( i + ) ¯ k b on R , . The remainder of h is obtained by taking linearcombinations with p . The H -invariance of D ( i ) ¯ k and P -equivariance of D ( i + ) ¯ k give, for X ∈ p , ( H · D ( i + ) ¯ k b )( X ) = H · ( D ( i + ) ¯ k b ( X )) − D ( i + ) ¯ k b ([ H , X ])= − H · X · D ( i ) ¯ k b + [ H , X ] · D ( i ) ¯ k b = − X · H · D ( i ) ¯ k b = . The conclusion is H · D ( i + ) ¯ k b =
0, as desired. (cid:3)
Now if ¯ k and all its derivatives are constant on R , then U is curvature homogeneous to all orders, andtherefore, U is locally homogeneous by Singer’s theorem for Cartan connections [Mel11, Pec14].Let us consider now the remaining case where the isotropy at the origin is unipotent. UASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICS DO NOT EXIST 21
Unipotent isotropy.Proposition 22.
If the isotropy at ∈ S is unipotent, then g is isomorphic to sol ( a , b ) , for b = − a.Proof. Let Y ∈ g generate the isotropy at 0. There is b ∈ p − ( ) for which w b ( Y ) = E . The projection w b ( g ) of w b ( g ) to R , is 2-dimensional and E -invariant, so it must be span { e , h } . Therefore, there is abasis ( X , Y , Z ) of g such that w b ( X ) = e + a H + b F and w b ( Z ) = h + g H + d F for some a , b , g , d ∈ R . Because K b ( Y , · ) =
0, equation (8) gives w b [ Y , X ] = [ e + a H + b F , E ] = a E − b H ∈ w b ( g ) so b = [ Y , X ] = a Y . A similar computation gives w b [ Y , Z ] = [ h + g H + d F , E ] = e + g E − d H so d = − a , and [ Y , Z ] = X + g Y .Infinitesimal invariance of K by Y gives K b ([ Y , X ] , Z ) + K b ( X , [ Y , Z ]) = [ − E , K b ( X , Z )] . But the left side is 0 because [ Y , X ]( ) = [ Y , Z ]( ) = X ( ) . Therefore E commutes with K b ( X , Z ) ∈ p ,which means K b ( X , Z ) = rE for some r ∈ R . Now equation (8) gives for X and Z , w b [ X , Z ] = [ h + g H − a F , e + a H ] + rE = g e + a h − a F + rE . In order that this element belongs to w b ( g ) , one must have a = g =
0. First consider g =
0. Thestructure of the algebra g in the basis ( X , Y , Z ) isad Y = a ad X = − a r a ad Z = − − r − a . This algebra is unimodular, so this case does not arise, by Lemma 3.Next consider a =
0. Then the Lie algebra isad Y = g ad X = g r ad Z = − g − − r − g . In order that g not be unimodular, g must be nonzero (notice also that for g = r =
0, we would get a Heisenbergalgebra). We obtain a solvable Lie algebra g ∼ = R ⋉ j R , where j = (cid:18) − g − − r − g (cid:19) . If r >
0, then g ∼ = sol ( a , b ) , where a = − g + √ r , b = − g − √ r . Conversely, j is R -diagonalizable only if r > (cid:3) Proposition 23. (compare Proposition 14 (i))(i) At points of S, there is only one eigenvalue of the Ricci operator.(ii) This triple eigenvalue is positive if and only if the Killing algebra sol ( a , b ) is R -diagonalizable.Proof. (i) The invariance of the Ricci endomorphism ¯ k b by E means (see the table in Subsection 5.2):¯ k b ∈ span { m d , m e } . The triple eigenvalue is the coefficient of m d .(ii) The full curvature k b ∈ W is E -invariant, so it is in the span of the elements of W corresponding to m d and m e . Referring to the column labeled ∧ R , ∗ ⊗ p in the table reveals that m d is the only of these twocomponents of k b possibly assigning a nonzero value to the input pair ( e , h ) . Therefore the parameter r inthe proof of Proposition 22 coincides with the coefficient of the element corresponding to m d in k b and withhalf the triple eigenvalue of the Ricci endomorphism at 0. (cid:3) But, by the point (iii) in Proposition 12, we know that the Killing algebra sol ( a , b ) is R -diagonalizable.This implies that r > g on three-dimensional Lie groups. In particular, they proved (see their Theorems3.5, 3.6 and 3.7) that a Ricci operator of a left-invariant Lorentz metric on a nonunimodular three-dimensionalLie group admits a triple eigenvalue r = g is of constant sectional curvature. Since on U \ S ,our Lorentz metric g is locally isomorphic to a left-invariant Lorentz metric on the nonunimodular Lie group SOL ( a , b ) corresponding to the Killing algebra, this implies that g is of constant sectional curvature. Inparticular, g is locally homogeneous. R EFERENCES[Amo79] A. M. Amores. Vector fields of a finite type G -structure. J. Differential Geom. , 14(1):1–6 (1980), 1979.[Ben97] Yves Benoist. Orbites des structures rigides (d’après M. Gromov). In
Integrable systems and foliations/Feuilletages et systèmesintégrables (Montpellier, 1995) , volume 145 of
Progr. Math. , pages 1–17. Birkhäuser Boston, Boston, MA, 1997.[BF05] E. Jerome Benveniste and David Fisher. Nonexistence of invariant rigid structures and invariant almost rigid structures.
Comm.Anal. Geom. , 13(1):89–111, 2005.[BFL92] Yves Benoist, Patrick Foulon, and François Labourie. Flots d’Anosov à distributions stable et instable différentiables.
J. Amer.Math. Soc. , 5(1):33–74, 1992.[Cal07] Giovanni Calvaruso. Einstein-like metrics on 3-dimensional homogeneous Lorentzian manifolds.
Geom. Ded. , 127:99–119,2007.[CK09] Giovanni Calvaruso and Oldˇrich Kowalski. On the Ricci operator of locally homogeneous lorentzian 3-manifolds.
CEJM ,7(1):124–139, 2009.[CP97] Luis A. Cordero and Phillip E. Parker. Left invariant Lorentzian metrics on 3-dimensional Lie groups.
Rend. Math. Appl.(7) ,17(1):129–155, 1997.[DG91] G. D’Ambra and M. Gromov. Lectures on transformation groups: geometry and dynamics. In
Surveys in differential geometry(Cambridge, MA, 1990) , pages 19–111. Lehigh Univ., Bethlehem, PA, 1991.[DG13] Sorin Dumitrescu and Adolfo Guillot. Quasihomogeneous real analytic connections on surfaces.
J. Topol Anal. , 5(4):491–532,2013.[Dum08] Sorin Dumitrescu. Dynamique du pseudo-groupe des isométries locales sur une variété lorentzienne analytique de dimension3.
Ergodic Theory Dynam. Systems , 28(4):1091–1116, 2008.[DZ10] Sorin Dumitrescu and Abdelghani Zeghib. Géométries lorentziennes de dimension trois : classification et complétude.
GeomDedicata , 149:243–273, 2010.[Fer02] Renato Feres. Rigid geometric structures and actions of semisimple Lie groups. In
Rigidité, groupe fondamental et dynamique ,volume 13 of
Panor. Synthèses , pages 121–167. Soc. Math. France, Paris, 2002.
UASIHOMOGENEOUS THREE-DIMENSIONAL REAL ANALYTIC LORENTZ METRICS DO NOT EXIST 23 [Gro88] Michael Gromov. Rigid transformations groups. In
Géométrie différentielle (Paris, 1986) , volume 33 of
Travaux en Cours ,pages 65–139. Hermann, Paris, 1988.[Kir74] A. Kirilov.
Eléments de la théorie des représentations . M.I.R., Moscou, 1974.[KN96] Shoshichi Kobayashi and Katsumi Nomizu.
Foundations of differential geometry. Vol. I . Wiley Classics Library. John Wiley& Sons Inc., New York, 1996. Reprint of the 1963 original.[Mel09] Karin Melnick. Compact Lorentz manifolds with local symmetry.
J. Differential Geom. , 81(2):355–390, 2009.[Mel11] Karin Melnick. A Frobenius theorem for Cartan geometries, with applications.
Enseign. Math. (2) , 57(1-2):57–89, 2011.[Nom60] Katsumi Nomizu. On local and global existence of Killing vector fields.
Ann. of Math. (2) , 72:105–120, 1960.[Nom79] Katsumi Nomizu. Left invariant Lorentz metrics on Lie groups.
Osaka J. Math. , 16:143–150, 1979.[Pec14] Vincent Pecastaing. On two theorems about local automorphisms of geometric structures.
Arxiv 1402.5048 , 2014.[PTV96] Friedbert Prüfer, Franco Tricerri, and Lieven Vanhecke. Curvature invariants, differential operators and local homogeneity.
Trans. Amer. Math. Soc. , 348(11):4643–4652, 1996.[Ros61] M. Rosenlicht. On quotient varieties and the affine embedding of certain homogeneous spaces.
Trans. Amer. Math. Soc. ,101:211–223, 1961.[Sha97] R. W. Sharpe.
Differential geometry , volume 166 of
Graduate Texts in Mathematics . Springer-Verlag, New York, 1997. Car-tan’s generalization of Klein’s Erlangen program, With a foreword by S. S. Chern.[Sin60] I. Singer. Infinitesimally homogeneous spaces.
Comm. Pure Appl. Math. , 13:685–697, 1960.[Thu97] William P. Thurston.
Three-dimensional geometry and topology. Vol. 1 , volume 35 of
Princeton Mathematical Series . Prince-ton University Press, Princeton, NJ, 1997. Edited by Silvio Levy.[Wol67] Joseph A. Wolf.
Spaces of constant curvature . McGraw-Hill Book Co., New York, 1967.[Zeg96] Abdelghani Zeghib. Killing fields in compact Lorentz 3-manifolds.
J. Differential Geom. , 43(4):859–894, 1996.U
NIVERSITÉ C ÔTE D ’A ZUR , U
NIVERSITÉ N ICE S OPHIA A NTIPOLIS , CNRS, L
ABORATOIRE
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IEUDONNÉ , UMR 7351,P
ARC V ALROSE , 06108 N
ICE C EDEX
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