Proponemos un nuevo enfoque al estudio de las subvariedades luz, cuyo estudio hasta el momento dependía de una elección arbitraria de una distribución pantalla que hacía completamente inadecuado el estudio de la geometría intrínseca.  Las técnicas utilizarán geometrías de Cartan.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Palomo-Ruiz, Francisco Jose

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Palomo-Ruiz, Francisco José

Repositorio Institucional Universidad de Málaga

IntroductionKlein and Cartan GeometriesLightlike manifoldsA new approach to the study of lightlike manifoldsFrancisco J. PalomoDepartment of Applied MathematicsFirst Joint Meeting Brazil-Spain in MathematicsFortaleza, December 2015Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsINTRODUCTIONLet (M, g) be a (n + 2)-dimensional Lorentz manifold and a hypersurfaceψ : L → MLightlike hypersurfaceψ : L → M is said to be a lightlike hypersurface wheneverψ∗(g) is degenerate at every point, i.e., Rad(TpL) 6= 0⇓Rad(TpL) = {v ∈ TpL : g(v , u) = 0 for all u ∈ TpL} 6= 0 ⇒ dim(Rad(TpL)) = 1.D(p) = Rad(TpL)=(TpL)⊥ ⊂ Tψ(p)M, p ∈ MD defines a 1-dimensional distribution (and the characteristic foliation) on L.The normal fiber bundle satisfies(TL)⊥= D ⊂ TL and therefore is nottransverse to L.The classical submanifold theory cannot be applied!!Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsMotivations...1 Cauchy Horizons.2 Characteristic hypersurfaces of the wave equation.3 Degenerate orbits of Lorentz isometric actions.4 Lightlike cones on Lorentz manifolds.Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsAn ad hoc technique for lightlike hypersurfaces ψ : L → Mn+2The following technique was introduced by Duggal and Bejancu in 1996.Screen distribution S(L)Fix an arbitrary n-distribution S(L) on L such that TL = D ⊕ S(L). Then,1 S(L) inherits a Riemannian metric.2 There exists a unique lightlike transverse fiber bundle tr(L) such thatTM |L= TL ⊕ tr(L).Let ∇ be the Levi-Civita connection of M and X ,Y ∈ X(L),N ∈ Γ(tr(L)).The following decompositions strongly depend on S(L).∇XY = ∇XY + II(X ,Y ), Gauss equation∇XN = −ANY +∇⊥X N, Weingarten equationand the usual terminology is introduced: induced connection, secondfundamental form, Weingarten operator...Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsSeveral issues to be mentionedLet (L, g , S(L)) be a lightlike hypersurface of (M, g).The notion of totally geodesic lightlike hypersurface (II = 0) does notdepend on the screen distribution.There exists also a notion of totally umbilical lightlike hypersurface whichis independent on the screen distribution. But as far as we know, thisnotion has no clear geometric meaning.The intrinsic geometry of L strongly depends on S(L).Intrinsic geometry of L ⊂ MThe induced connection ∇ is independent of S(L) if and only if II = 0.∇g = 0 if and only if II = 0.Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsIt will be useful an intrinsic definition of lightlike manifold L:Lightlike manifoldsA lightlike manifold is a pair (Ln+1, g) whereg ∈ T0,2(L) is a symmetric tensor (to simplify we assume g is positivesemidefinite)Rad(g) is a 1-dimensional distribution on L.For a lightlike manifold (L, g), the distribution Rad(g) is Killing when everyvector field ξ ∈ Rad(g) is Killing. That is, the Lie derivative of g satisfiesLξg = 0.Duggal-Jin, 2007A torsion free linear connection on a lightlike manifold Ln+1 compatible with gexists if and only if Rad(g) is a Killing distribution. In this case, there is aninfinitude of connections with none distinguished.Rad(g) is a Killing distribution.mThere are coordinates systems (r , x1, ..., xn) with∂∂rspans Rad(g) and∂gij∂r= 0.Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsSumming up... from my point of view:1 The Screen distribution construction could be a good approach to studyextrinsic properties but not for intrinsic geometry.2 The lightlike hypersurfaces are conformal invariants. It would be desirablecertain conformal invariance.Natural questionsIs it possible to construct an intrinsic geometric structure on L ?This intrinsic geometric structure should be independent of any arbitraryelection...... and should provide local invariants which permit to distinguish locallytwo lightlike manifolds.Where can we look for such geometric structures ?Although I do not have a definitive answer to the above questions, let meintroduce you the tangle of ideas I have developed hoping to find thesegeometric structures.The main ideas come from the notion of Cartan geometry.Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesKLEIN AND CARTAN GEOMETRIESIn the early 1920s, Elie Cartan found a common generalization for the Klein’sErlangen program and Riemann geometry. He called Espaces ge´ne´ralize´s andnow we call Cartan Geometries.Elie Cartan (Wikimedia Commons)From a Klein Geometry G/H⇓A Cartan Geometry with model (G ,H) on M is a differential geometricstructure on M whose objects may be thought as curved analogs of thehomogeneous model.Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesAround 1950, Charles Ehresmann gave for the first time a rigorous globaldefinition of a Cartan connection as a particular case of a more general notionnow called Ehresmann connection.Charles EhresmannThis is the point of view of the influential book Foundations of DifferentialGeometry, Volumes I and II by S. Kobayashi and K. Nomizu.Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesThe models of Cartan geometriesKlein GeometryA pair (G ,H), where G is a Lie group and H a closed subgroup of G , is said tobe a Klein Geometry when M = G/H is connected and is considered with ageometric structure such that1 The left translations `g for g ∈ G .`g : M → M, x 7→ g · x .are automorphisms of the geometric structure.2 For every automorphism f : M → M of the geometric structure, there isg ∈ G such that f = `g .Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesExamples of Klein geometriesSemi-Euclidean Geometry Rn(p,q)= (Rn, 〈 , 〉(p,q)), p + q = nG ={( 1 0x A): x ∈ Rn, A ∈ O(p, q)}Consider the transitive actionG × Rn → Rn,(1 0x A)·(1p)=(1x + Ap)Thus, the isotropy group at the origin is H ∼= O(p, q) and thereforeRn = G/O(p, q), Iso(Rn(p,q)) = GFor q = 0 ⇒ En, G = Euc(n), H = O(n) ⇒ Riemann Geometry.For q = 1 ⇒ Ln, G = Lor(n), H = O(n, 1) ⇒ Lorentz Geometry.Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesThe Mo¨bius sphere (S(n), [g ]), n ≥ 3Let C n+1 ⊂ Ln+2 be the cone of the (nonzero) lightlike vectors andconsider the space of lines in C n+1 (the Mo¨bius sphere)S(n) ⊂ RPn+1.From the obvious projectionΠ : C n+1 → S(n), v 7→ R · v ,S(n) carries a natural Riemannian conformal structure [g ]As a manifold S(n) = Sn and [g ] corresponds to the conformal class of theround metric of radius one.How can we see (S(n), [g ]) as a Klein geometry?Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesConsider the transitive (and natural) action,O(n + 1, 1)× S(n) → S(n)This action is not effective and it is better to use the groupG := PO(n + 1, 1) = O(n + 1, 1)/{±Id}Thus, we can identifyS(n) = G/Pwhere P ⊂ G is the isotropy group of a lightlike line.P is called the Poincare´ conformal group.Conf(S(n)) = O(n + 1, 1)/{±Id}(G ,P) ⇒ Riemannian Conformal Geometry.Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesSeveral types of Klein GeometriesLet (G ,H) be a Klein Geometry and M = G/HThe Klein geometry (G ,H) is said to be of first order when therepresentation of H given byAd : H → Gl(g/h), h 7→ Ad(h)(X + h) = Ad(h)(X ) + h.is injective.Otherwise, (G ,H) is a higher-order geometry.For a Klein geometry (G ,H) of first order, we can see G ⊂ L(M) (as afiber bundle of frames over M).Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesLet g and h be the Lie algebras of G and H, respectively.A Klein geometry (G ,H) is said to be...Reductive (with complement fixed m)g = h⊕m as vector spaces andAd(H)(m) ⊂ m.Under the assumption H is connected this condition is equivalent to[h,m] ⊂ m.Semi-Euclidean geometries are reductive Klein geometries of first order| k |-graded (k ≥ 1 and the grading is assumed to be fixed)g = g−k ⊕ ...⊕ g−1 ⊕ g0 ⊕ g1 ⊕ ...⊕ gk ,[gi , gj ] ⊂ gi+j , h = g0 ⊕ g1 ⊕ ...⊕ gk .where g−k 6= {0}, gk 6= {0} and gi = {0} for | i |> k.Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesThe Mo¨bius sphere (S(n), [g ]) is a | 1 |-graded Klein geometryIt is better to considerLn+2 = En ⊕⊥ L2This yields to write the usual scalar Lorentzian product as the corresponding tothe matrix(0 0 10 In 01 0 0). RecallG = O(n + 1, 1)/{±Id}P = λ −λw tC −λ2 〈w ,w〉0 C w0 0 λ−1 : λ ∈ R∗,w ∈ Rn,C ∈ O(n)Thus,g = a z 0x A −z t0 −x t −a : a ∈ R, x ∈ Rn, z ∈ (Rn)∗,A ∈ o(n)g = g−1 ⊕ g0 ⊕ g1 and p = g0 ⊕ g1Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesLooking for the model of lightlike manifolds L = G/H1 The isometries of the degenerate tensor g on L should be exactly the leftmultiplications by elements of G .2 (Liouville Theorem) Suppose that L = G/H is connected. Then anyisometry between two connected open subset of L uniquely globalizes toan isometry of LWhy not L = Rn+1?Why don’t we use (Rn+1 = R⊕ En, 0⊕ 〈 , 〉) as the model for lightlikemanifolds?The group of isometric tranformations is infinite dimensional in this case!!!What could it be the model for lightlike manifolds?Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesL: The upper lightlike cone Qn+1 ⊂ Ln+2Recall that Ln+2 = En ⊕⊥ L2Fix `, η ∈ L2 lightlike vectors such that g(`, η) = 1 and take T = 1√2(`− η).Qn+1 ={v ∈ Ln+2 : g(v , v) = 0, g(v ,T ) < 0}≈ C n+1/Z2is a lightlike hypersurface (g is the Lorentz metric).The natural action, O(n + 1, 1)× C n+1 → C n+1 induces a transitive actionPO(n + 1, 1)× Qn+1 → Qn+1Thus, we can identify Qn+1 = PO(n + 1, 1)/H where H ⊂ PO(n + 1, 1) isthe isotropy group of the first basis vector `.H = 1 −w tC − 12 〈w ,w〉0 C w0 0 1 : w ∈ Rn,C ∈ O(n) ≡ Euc(n).(here 〈 , 〉 is the Euclidean scalar product)Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesWe have the above required properties 1,1 For n ≥ 3Iso(Qn+1) = PO(n + 1, 1) ≈ O+(n + 1, 1)2 This is true even locally: any isometry between two connected open subsetof Qn+1 is the restriction of an element in PO(n + 1, 1).1Bekkara, Frances and Zeghib, 2009Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesTaking a look at the model Qn+1 = G/H at an algebraical levelThe Klein geometry (G ,H) is of first order.In particular observe that G ⊂ L(Qn+1) as follows(v | e1 | e2 | ... | en | w)7→ (e1, ..., en,w) ⊂ TvQn+1.The Klein geometry Qn+1 = G/H is not reductive.Summing-up:g = g−1 ⊕ g0 ⊕ g1, p = g0 ⊕ g1 conformal modelg = g−1 ⊕ g0 ⊕ g1, h = [g0, g0]⊕ g1 ≤ p lightlike model?h = 0 z 00 A −z t0 0 0 : z ∈ (Rn)∗,A ∈ o(n) = [g0, g0]⊕ g1 ≤ p ≤ gNow we would like to introduce the meaning on the sentence:The Klein geometry (G ,H) is a model for a curved geometric structure.Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesCartan Geometry of type (G ,H) on MA principal fiber bundle p : P → M with structure group H.A g-valuated one form ω ∈ Ω1(P, g), called the Cartan connection suchthat:1 ω(u) : TuP → g is a linear isomorphism for all u ∈ P.2 For ξX , the fundamental vector field corresponding to X ∈ h,ω(ξX ) = X(ξX (u) :=ddt|0 (u · exp(tX )))3 For every h ∈ H, let rh be the corresponding right multiplication on P. Then(rh)∗ω = Ad(h−1) ◦ ωThat is, the following diagram commutes.TuP ω(u)−−−−−→ gTu rhy yAd(h−1)TuhP −−−−−→ω(uh)gdim(P) = dim(G), dim(M) = dim(G/H)Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesThe curvature of a Cartan connectionLet (p : P → M, ω) be a Cartan geometry of type (G ,H) on M.The curvature form K ∈ Ω2(P, g) is given, for every ξ, η ∈ X(P), by:K(ξ, η) := dω(ξ, η) + [ω(ξ), ω(η)].The Klein Geometry (G ,H) as a Cartan geometryThe principal fiber bundle: pi : G → M ∼= G/H, g 7→ gH;The Cartan connection: the Maurer-Cartan form ωG ∈ Ω1(G , g):ωG (g) : TgG → g, ξ ∈ TgG 7→ ωG (g)(ξ) := Tgλg−1 · ξ ∈ TeG ∼= g,where λg : G → G , σ 7→ gσ.This is called the homogeneous model of a Cartan geometry of type (G ,H).The homogeneous models are always flat (Maurer-Cartan equation gives K = 0)and the converse is essentially true.Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsKlein GeometriesCartan geometriesThe tangent bundle of a Cartan geometry of type (G ,H)Consider the representation of H given byAd : H → Gl(g/h), h 7→ Ad(h)(X + h) = Ad(h)(X ) + h.For each u ∈ P with p(u) = x ∈ M, there is a canonical linear isomorphismφu : TxM → g/hsuch that the following diagram conmutesTuP ω(u)−−−−−→ gTupy yρTxM −−−−−→φu∼=g/hwith φuh = Ad(h−1)φu for all h ∈ H.TM ∼= P ×H (g/h), (x , v) ∈ TM 7→ [u, φu(v)], where p(u) = x .Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsLightlike manifolds as Cartan geometriesCorrespondence spacesLIGHTLIKE MANIFOLDSFrom now on, we return to the upper lightlike cone Qn+1 ⊂ Ln+2 and considerour chain of Lie groups.H ⊂ P ⊂ G ⇒ Qn+1 = G/H −→ S(n) = G/PG = Iso(Qn+1), G = Conf(S(n)).PropositionLet (pi : P → L, ω) be a Cartan geometry with model (G ,H).Then, L is a lightlike manifold with tensor g and there is a global vector fieldξ ∈ X(L) which spans Rad(g).For every u ∈ P with pi(u) = x , consider φu : TxL → g/h ∼= Rn ⊕ R.Ad : H → Gl(g/h), h 7→ Ad(h)(X , a) = (CX , a− 〈C−1w ,X 〉),where h↔ (C ,w) ∈ O(n)× Rn.Note that the natural hyperplane of Rn ⊕ R is not invariant by Ad(H).In order to get a first approach to the converse...Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsLightlike manifolds as Cartan geometriesCorrespondence spacesCorrespondence spacesLet (p : P → M, ω) be a Cartan geometry with arbitrary model (G ,P).We define the correspondence space C(M) of M for H ⊂ P to be the quotientspace P/H:P↓ ↘P/H = C(M) Π−→ M = P/PThe projection Π : C(M)→ M is a fiber bundle with fiber the homogeneousspace P/H and(pi : P → C(M), ω) is a Cartan geometry of type (G ,H).Return to ours fixed Lie groups H ⊂ P ⊂ G ...Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsLightlike manifolds as Cartan geometriesCorrespondence spacesWhich are the correspondence spaces for H ⊂ P ⊂ G?Conformal Riemannian structures on Mm1-1Cartan connections on M with model (G ,P)(satisfying certain curvature properties)PropositionLet (M, [g ]) be a conformal Riemannian manifold and (p : P → M, ω) thecorresponding Cartan geometry with model (G ,P).Then the correspondence space C(M) for H ⊂ P isE(M) = ∪x∈M{c · gx : c > 0}the bundle of scales associated to (M, [g ]).In particular, E(M) is a lightlike manifold and admits a distinguished Cartanconnection of type (G ,H).Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsLightlike manifolds as Cartan geometriesCorrespondence spacesWhich are the lightlike manifolds which can be seenas bundles of scales of Riemann conformal manifolds?(these will admit a distinguished Cartan connection of type (G ,H))Assume (L, g) is a lightlike manifold with Rad(g) = Span(ξ) for ξ ∈ X(L).The vector field ξ is said to be conformal when there is a smooth function ρsuch that the Lie derivative Lξg = 2ρ g .LemmaFor (L, g) a lightlike manifold with Rad(g) = Span(ξ) are equivalent:1 ξ is conformal.2 For every point p ∈ L there is a coordinate system (r , x1, ..., xn) around psuch that, for all i , j = 1, .., n,ξ =∂∂rand gij(r , x1, ..., xn) = qij(x1, ...xn)eΦ(r,x1,...,xn).Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsLightlike manifolds as Cartan geometriesCorrespondence spacesLet Π : L → L/Rad(g) be the natural projection on the space of orbits.TheoremLet (L, g) be a lightlike manifold with Rad(g) = Span(ξ) and ξ a conformalvector field.Assume that around every point p ∈ L there is a regular coordinate system(U, ψ = (r , x1, ..., xn)) respect to the distribution Rad(g). That is,1 ψ(U) is an open cube.2 ξ |U= ∂∂r .3 Every integral curve of ξ intersects U in at most one slicex1 = c1, ..., xn = cn.Then, the local orbits space Π(U) can be endowed with a conformal class ofRiemannian metrics [σ]. This conformal class does not change if we replace gby any e f g .... moreover, the correspondence space C(Π(U)) for H ⊂ P is an extension of UU ↪→ C(Π(U))?Thus, (L, g) could be covered by open subsets U ′s which are base of adistinguished Cartan geometry with model (G ,H).Francisco J. Palomo Lightlike manifoldsIntroductionKlein and Cartan GeometriesLightlike manifoldsLightlike manifolds as Cartan geometriesCorrespondence spacesTheoremLet ψ : L → M be lightlike hypersurface in a strongly causal Lorentzianmanifold (M, g). Assume the radical of g|L is spanned by a conformal vectorfield.Then every point p ∈ L is in an open subset U ⊂ L such that Π(U) admits aconformal class of Riemannian metric [σ] which depends only on the conformalclass of g... in a such way that the correspondence space C(Π(U)) for H ⊂ P is anextension of U?Thus, (L, g) could be covered by open subsets U ′s which are base of adistinguished Cartan geometry with model (G ,H).EXAMPLES1 Every integral lightlike hypersurface of ξ⊥ in a Brinkmann space(∇ξ = 0).Cahen-Wallach spaces.2-symmetric Lorentzian spaces (which are not 1-symmetric).Plane-fronted waves (as introduced by Candela, Flores and Sa´nchez).2 Every integral lightlike hypersurface of ξ⊥ in a Lorentz manifold with∇ξ = ω ⊗ ξ.Francisco J. Palomo Lightlike manifolds

A new approach to the study of lightlike manifolds

https://riuma.uma.es/xmlui/bitstream/10630/10838/1/PalomoFortaleza2015.pdf

A new approach to the study of lightlike manifolds

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Repositorio Institucional Universidad de Málaga