A new approach to the study of lightlike manifolds

Introduction Klein and Cartan Geometries Lightlike manifolds A new approach to the study of lightlike manifolds Francisco J. Palomo Department of Applied Mathematics First Joint Meeting Brazil-Spain in Mathematics Fortaleza, December 2015 Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds INTRODUCTION Let (M,g) be a (n+ 2)-dimensional Lorentz manifold and a hypersurface ψ:L → M Lightlike hypersurface ψ:L → Mis said to be a lightlike hypersurface whenever ψ∗(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∈M Ddefines a 1-dimensional distribution (and the characteristic foliation) on L. The normal fiber bundle satisfies TL⊥=D ⊂ TLand therefore is not transverse to L. The classical submanifold theory cannot be applied!! Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Motivations... 1Cauchy Horizons. 2Characteristic hypersurfaces of the wave equation. 3Degenerate orbits of Lorentz isometric actions. 4Lightlike cones on Lorentz manifolds. Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds An ad hoc technique for lightlike hypersurfaces ψ:L → Mn+2 The following technique was introduced by Duggal and Bejancu in 1996. Screen distribution S(L) Fix an arbitrary n-distribution S(L) on Lsuch that TL=D ⊕ S(L).Then, 1S(L) inherits a Riemannian metric. 2There exists a unique lightlike transverse fiber bundle tr(L) such that TM |L=TL ⊕ tr(L). Let ∇be the Levi-Civita connection of Mand 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+∇⊥ XN,Weingarten equation and the usual terminology is introduced: induced connection, second fundamental form, Weingarten operator... Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Several issues to be mentioned Let (L,g,S(L)) be a lightlike hypersurface of (M,g). The notion of totally geodesic lightlike hypersurface (II = 0) does not depend on the screen distribution. There exists also a notion of totally umbilical lightlike hypersurface which is independent on the screen distribution. But as far as we know, this notion has no clear geometric meaning. The intrinsic geometry of Lstrongly depends on S(L). Intrinsic geometry of L ⊂ M The 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 manifolds Introduction Klein and Cartan Geometries Lightlike manifolds It will be useful an intrinsic definition of lightlike manifold L: Lightlike manifolds A lightlike manifold is a pair (Ln+1,g) where g∈ T0,2(L) is a symmetric tensor (to simplify we assume gis positive semidefinite) Rad(g) is a 1-dimensional distribution on L. For a lightlike manifold (L,g), the distribution Rad(g) is Killing when every vector field ξ∈Rad(g) is Killing. That is, the Lie derivative of gsatisfies Lξg= 0. Duggal-Jin, 2007 A torsion free linear connection on a lightlike manifold Ln+1 compatible with g exists if and only if Rad(g) is a Killing distribution. In this case, there is an infinitude of connections with none distinguished. Rad(g) is a Killing distribution. m There are coordinates systems (r,x1, ..., xn) with ∂ ∂rspans Rad(g) and ∂gij ∂r= 0. Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Summing up... from my point of view: 1The Screen distribution construction could be a good approach to study extrinsic properties but not for intrinsic geometry. 2The lightlike hypersurfaces are conformal invariants. It would be desirable certain conformal invariance. Natural questions Is it possible to construct an intrinsic geometric structure on L? This intrinsic geometric structure should be independent of any arbitrary election... ... and should provide local invariants which permit to distinguish locally two lightlike manifolds. Where can we look for such geometric structures ? Although I do not have a definitive answer to the above questions, let me introduce you the tangle of ideas I have developed hoping to find these geometric structures. The main ideas come from the notion of Cartan geometry. Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Klein Geometries Cartan geometries KLEIN AND CARTAN GEOMETRIES In the early 1920s, Elie Cartan found a common generalization for the Klein’s Erlangen program and Riemann geometry. He called Espaces g´en´eraliz´es and now we call Cartan Geometries. Elie Cartan (Wikimedia Commons) From a Klein Geometry G/H ⇓ A Cartan Geometry with model (G,H) on Mis a differential geometric structure on Mwhose objects may be thought as curved analogs of the homogeneous model. Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Klein Geometries Cartan geometries Around 1950, Charles Ehresmann gave for the first time a rigorous global definition of a Cartan connection as a particular case of a more general notion now called Ehresmann connection. Charles Ehresmann This is the point of view of the influential book Foundations of Differential Geometry, Volumes I and II by S. Kobayashi and K. Nomizu. Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Klein Geometries Cartan geometries The M¨obius sphere (S(n),[g]) is a |1|-graded Klein geometry It is better to consider Ln+2 =En⊕⊥L2 This yields to write the usual scalar Lorentzian product as the corresponding to the matrix 001 0In0 100.Recall G=O(n+ 1,1)/{±Id} P=     λ−λwtC−λ 2hw,wi 0C w 0 0 λ−1 :λ∈R∗,w∈Rn,C∈O(n)   Thus, g=     az0 xA−zt 0−xt−a  :a∈R,x∈Rn,z∈(Rn)∗,A∈o(n)   g=g−1⊕g0⊕g1and p=g0⊕g1 Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Klein Geometries Cartan geometries Looking for the model of lightlike manifolds L=G/H 1The isometries of the degenerate tensor gon Lshould be exactly the left multiplications by elements of G. 2(Liouville Theorem) Suppose that L=G/His connected. Then any isometry between two connected open subset of Luniquely globalizes to an isometry of L Why not L=Rn+1? Why don’t we use (Rn+1 =R⊕En,0⊕ h ,i) as the model for lightlike manifolds? The group of isometric tranformations is infinite dimensional in this case!!! What could it be the model for lightlike manifolds? Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Klein Geometries Cartan geometries L: The upper lightlike cone Qn+1 ⊂Ln+2 Recall that Ln+2 =En⊕⊥L2 Fix `, η ∈L2lightlike vectors such that g(`, η) = 1 and take T=1 √2(`−η). Qn+1 =nv∈Ln+2 :g(v,v) = 0,g(v,T)<0o≈Cn+1/Z2 is a lightlike hypersurface (gis the Lorentz metric). The natural action, O(n+ 1,1) ×Cn+1 →Cn+1 induces a transitive action PO(n+ 1,1) ×Qn+1 →Qn+1 Thus, we can identify Qn+1 =PO(n+ 1,1)/Hwhere H⊂PO(n+ 1,1) is the isotropy group of the first basis vector `. H=     1−wtC−1 2hw,wi 0C w 0 0 1  :w∈Rn,C∈O(n)   ≡Euc(n). (here h,iis the Euclidean scalar product) Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Klein Geometries Cartan geometries We have the above required properties 1, 1For n≥3 Iso(Qn+1) = PO(n+ 1,1) ≈O+(n+ 1,1) 2This is true even locally: any isometry between two connected open subset of Qn+1 is the restriction of an element in PO(n+ 1,1). 1Bekkara, Frances and Zeghib, 2009 Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Klein Geometries Cartan geometries Taking a look at the model Qn+1 =G/Hat an algebraical level The 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/His not reductive. Summing-up: g=g−1⊕g0⊕g1,p=g0⊕g1conformal model g=g−1⊕g0⊕g1,h= [g0,g0]⊕g1≤plightlike model? h=     0z0 0A−zt 0 0 0  :z∈(Rn)∗,A∈o(n)   = [g0,g0]⊕g1≤p≤g Now 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 manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Klein Geometries Cartan geometries Cartan Geometry of type (G,H) on M A principal fiber bundle p:P → Mwith structure group H. Ag-valuated one form ω∈Ω1(P,g), called the Cartan connection such that: 1ω(u) : TuP → gisalinear isomorphism for all u∈ P. 2For ξX, the fundamental vector field corresponding to X∈h, ω(ξX) = XξX(u) := d dt |0(u·exp(tX)) 3For every h∈H, let rhbe the corresponding right multiplication on P. Then (rh)∗ω=Ad(h−1)◦ω That is, the following diagram commutes. TuPω(u) −−−−−→ g Turh  y  yAd(h−1) TuhP −−−−−→ ω(uh)g dim(P) = dim(G),dim(M) = dim(G/H) Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Klein Geometries Cartan geometries The curvature of a Cartan connection Let (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 geometry The principal fiber bundle: π:G→M∼ =G/H,g7→ gH; The Cartan connection: the Maurer-Cartan form ωG∈Ω1(G,g): ωG(g) : TgG→g, ξ ∈TgG7→ 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 manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Klein Geometries Cartan geometries The tangent bundle of a Cartan geometry of type (G,H) Consider the representation of Hgiven by Ad :H→Gl(g/h),h7→ Ad(h)(X+h) = Ad(h)(X) + h. For each u∈ P with p(u) = x∈M, there is a canonical linear isomorphism φu:TxM→g/h such that the following diagram conmutes TuPω(u) −−−−−→ g Tup  y  y ρ TxM−−−−−→ φu∼ =g/h with φuh =Ad(h−1)φufor all h∈H. TM ∼ =P ×H(g/h),(x,v)∈TM 7→ [u, φu(v)],where p(u) = x. Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Lightlike manifolds as Cartan geometries Correspondence spaces LIGHTLIKE MANIFOLDS From now on, we return to the upper lightlike cone Qn+1 ⊂Ln+2 and consider our chain of Lie groups. H⊂P⊂G⇒Qn+1 =G/H−→ S(n)=G/P G=Iso(Qn+1),G=Conf(S(n)). Proposition Let (π:P → L, ω) be a Cartan geometry with model (G,H). Then, Lis a lightlike manifold with tensor gand there is a global vector field ξ∈X(L) which spans Rad(g). For every u∈ P with π(u) = x, consider φu:TxL → g/h∼ =Rn⊕R. Ad :H→Gl(g/h),h7→ Ad(h)(X,a) = (CX,a− hC−1w,Xi), where h↔(C,w)∈O(n)×Rn. Note that the natural hyperplane of Rn⊕Ris not invariant by Ad(H). In order to get a first approach to the converse... Francisco J. Palomo Lightlike manifolds Introduction Klein and Cartan Geometries Lightlike manifolds Lightlike manifolds as Cartan geometries Correspondence spaces Correspondence spaces Let (p:P → M, ω) be a Cartan geometry with arbitrary model (G,P). We define the correspondence space C(M)of Mfor H⊂Pto be the quotient space P/H: P ↓ & P/H=C(M)Π −→ M=P/P The projection Π : C(M)→Mis a fiber bundle with fiber the homogeneous space P/Hand (π: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 manifolds