A survey of projective geometric structures on...

A survey ofprojectivegeometric

structures on2-,3-manifolds

S. Choi

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ClassicalgeometriesEuclidean geometry

Spherical geometry

Manifolds withgeometricstructures:manifolds needsome canonicaldescriptions..

Manifolds with(real) projectivestructures

A survey of projective geometricstructures on 2-,3-manifolds

S. Choi

Department of Mathematical ScienceKAIST, Daejeon, South Korea

Tokyo Institute of Technology

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Spherical geometry

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I Survey: Classical geometriesI Euclidean geometry: Babylonians, Egyptians,

Greeks, Chinese (Euclid’s axiomatic methods underPlato’s philosphy)

I Spherical geometry: Greek astronomy, Gauss,Riemann

I Hyperbolic geometry: Bolyai, Lobachevsky, Gauss,Beltrami,Klein, Poincare

I Conformal geometry (Mobius geometry or circlegeometry)

I Projective geometryI Erlanger program, Cartan connections, Ehresmann

connections

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Spherical geometry

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connections

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Spherical geometry

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connections

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Spherical geometry

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connections

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Spherical geometry

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connections

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Spherical geometry

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connections

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Spherical geometry

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connections

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Spherical geometry

Manifolds with geometric structures

I Manifolds with geometric structures: manifolds needsome canonical descriptions.

I Manifolds with geometric structures.I Deformation spaces of geometric structures.I Examples

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Spherical geometry

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Spherical geometry

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Spherical geometry

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Spherical geometry

Manifolds with projective structures

I Some history.I Projective manifolds: how many?I Projective surfaces

I Projective surfaces and gauge theory.I Labourie’s generalizationI Projective surfaces and affine differential geometry.

I Projective 3-manifolds and deformations

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Spherical geometry

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Spherical geometry

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Spherical geometry

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Spherical geometry

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Spherical geometry

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Spherical geometry

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Spherical geometry

Euclidean geometry

I Babylonian, Egyptian, Chinese, Greek...I Euclid developed his axiomatic method to planar and

solid geometry under the influence of Plato, whothought that geometry should be the foundation of allthought after the Pythagorian attempt to understandthe world using rational numbers failed....

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Spherical geometry

Euclidean geometry

I Babylonian, Egyptian, Chinese, Greek...I Euclid developed his axiomatic method to planar and

solid geometry under the influence of Plato, whothought that geometry should be the foundation of allthought after the Pythagorian attempt to understandthe world using rational numbers failed....

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Spherical geometry

I Euclidean geometry consists of some notions suchas lines, points, lengths, angle, and their interplay insome space called a plane...

I The Euclid had five axiomsI D. Hilbert, and many others made a modern

foundation so that the Euclidean geometry wasreduced to logic.

I Euclidean geometry has a notion of rigidtransformations which made the spacehomogeneous. They form a group called a group ofrigid motions. They preserve lines, length, angles,and every geometric statements.

I The group is useful in proving statements.... Turningit around, we see that actually the transformationgroup is more important. wallpaper groups

I Notions of Euclidean subspaces...

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Spherical geometry

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Spherical geometry

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Spherical geometry

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Spherical geometry

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Spherical geometry

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Spherical geometry

Spherical geometry.

I Greeks: astronomical, navigational... Arabs,..I From astronomical viewpoint, it is nice to view the

sky as a unit sphere S2 in the Euclidean 3-space.(See spherical.cdy)

I The great circles replaced lines and angles aremeasured in the tangential sense. Lengths aremeasured by taking arcs in the great circles.

I Geometric objects such as triangles behave a littlebit different.

I Higher dimensional spherical geometry Sn can beeasily constructed.

I The group of orthogonal transformations O(n + 1)acts on Sn preserving every spherical geometricnotions. The geometry of sphere

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Spherical geometry

Spherical geometry.

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Spherical geometry

Spherical geometry.

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Spherical geometry

Spherical geometry.

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Spherical geometry.

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Spherical geometry

Spherical geometry.

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Spherical geometry

Hyperbolic geometry.I Lobachevsky and Bolyai tried to build a geometry

that did not satisfy the fifth axiom of Euclid. (Seehyperbolic.cdy)

I Their attempts were justified by Beltrami-Klein modelwhich is a disk and lines were replaced by chordsand lengths were given by the logarithms of crossratios. See Beltrami-Klein model. Later other modelssuch as Poincare half space model and Poincaredisk model were developed. Poincare model (Inst.figuring).

I Here the group of rigid motions is the Lie groupSL(2,R).

I Higher-dimensional hyperbolic spaces were laterconstructed. Actually, an upper part of a hyperboloidin the Lorentzian space would be a model andPO(1,n) forms the group of rigid motions.Minkowsky model

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Spherical geometry

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Spherical geometry

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Spherical geometry

Conformal geometryI Suppose that we use to study circles and spheres

only... We allow all transformations that preservescircles...

I This geometry looses a notion of lengths but has anotion of angles. There are no lines or geodesics butthere are circles.

I The Euclidean plane is compactifed by adding aunique point as an infinity. The group of motions isgenerated by inversions in circles. The group iscalled the Mobius transformation group. That is, thegroup of transformations of form

z → az + bcz + d

,az + bcz + d

I The space itself is considered as a complex sphere,i.e., the complex plane with the infinity added.

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Spherical geometry

z → az + bcz + d

,az + bcz + d

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Spherical geometry

z → az + bcz + d

,az + bcz + d

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Spherical geometry

z → az + bcz + d

,az + bcz + d

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Spherical geometry

I Conformal geometry is useful in solving PDEs incomplex analysis. (The huge work of Ahlfors-Bersschool).

I Recent work on discretization of complex analytic(conformal) maps... (The circle packing theorem:Koebe-Andreev-Thurston theorem)

I For higher-dimensions, the conformal geometry canbe defined similarly.

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Spherical geometry

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Spherical geometry

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Spherical geometry

Projective geometryI Projective geometry naturally arose in fine art

drawing perspectives during Renascence.Perspective drawings

I Desargues, Kepler first tried to add infinite pointscorresponding to each direction that a line in a planecan take. Thus, a plane with infinite point for eachdirection form a projective plane.

I Transformations are ones generated by change ofperspectives. In fact, when we are taking x-rays orother scans.

I The notions such as lengths, angles lose meaning.But notions of lines or geodesics are preserved. TheGreeks discovered that the cross ratios, i.e., ratios ofratios, are preserved.

I The infinite points are just like the ordinary pointsunder transformations.

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Spherical geometry

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Spherical geometry

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Spherical geometry

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Spherical geometry

Projective geometry

I Klein finally formalized it by considering a projectiveplane as the space of lines through origin in theEuclidean 3-space. Here, a Euclidean plane isrecovered as a subspace of the lines not in thexy -plane and the infinite points are the lines in thexy -plane.

I Many interesting geometric theorems hold:Desargues, Pappus,... Moreover, the theorems comein pairs: duality of lines and points: PappusTheorem, Pascal Theorem

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Spherical geometry

Projective geometry

I Klein finally formalized it by considering a projectiveplane as the space of lines through origin in theEuclidean 3-space. Here, a Euclidean plane isrecovered as a subspace of the lines not in thexy -plane and the infinite points are the lines in thexy -plane.

I Many interesting geometric theorems hold:Desargues, Pappus,... Moreover, the theorems comein pairs: duality of lines and points: PappusTheorem, Pascal Theorem

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Spherical geometry

Projective geometry

I Many geometries are actually sub-geometries ofprojective (conformal) geometry. (The twocorrespond to maximal finite-dimensional Liealgebras acting locally on manifolds...)

I Hyperbolic geometry: Hn imbeds in RPn andPO(n,1) in PGL(n + 1,R) in canonical way.

I spherical, euclidean geometry.I The affine geometry G = SL(n,R) · Rn and X = Rn.I Six of eight 3-dimensional geometries: euclidean,

spherical, hyperbolic, nil, sol, SL(2,R) (B. Thiel)I S2 × R and H2 × R. (almost)I Some symmetric spaces and matrix groups...

I Recently, some graphics applications.... imagesuperposition,..

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Spherical geometry

Projective geometry

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Spherical geometry

Projective geometry

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Projective geometry

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Projective geometry

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Projective geometry

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Projective geometry

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Projective geometry

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Erlangen program of Klein

I A Lie group is a set of symmetries of some objectforming a manifold.

I Klein worked out a general scheme to study almostall geometries...

I Klein proposed that "geometry" is actually a spacewith a Lie group acting on it transitively.

I Essentially, the transformation group actually definesthe geometry by determining which properties arepreserved.

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Spherical geometry

I We extract properties from the pair:I For each pair (X ,G) where G is a Lie group and X is

a space.I Thus, Euclidean geometry is given by X = Rn and

G = Isom(Rn) = O(n) · Rn.I The spherical geometry: G = O(n + 1,R) and

X = Sn.I The hyperbolic geometry: G = PO(n,1) and X upper

part of the hyperboloid t2 − x21 − · · · − x2

n = 1. Forn = 2, we also have G = PSL(2,R) and X the upperhalf-plane. For n = 3, we also have G = PSL(2,C)and X the upper half-space in R3.

I Lorentzian space-times.... de Sitter, anti-de-Sitter,...I The conformal geometry: G = SO(n + 1,1) and X

the celestial sphere in Rn+1,1.I The affine geometry G = SL(n,R) · Rn and X = Rn.I The projective geometry G = PGL(n + 1,R) and

X = RPn. (The conformal and projective geometriesare “maximal” geometry in the sense of Klein.)

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Cartan, Ehresmann connections: Mostgeneral geometry possible

I Cartan proposed that for each pair (X ,G), we canimpart their properties to at each point of themanifolds so that they vary also. The flatness impliesthat the (X ,G)-geometry is actually recovered.

I Ehresmann introduced the most general notion ofconnections by generalizing Cartan connections.

I For example, for euclidean geometry, a Cartanconnection gives us Riemannian geometry.

I For projective geometry, a Cartan connectioncorresponds to a projectively flat torsion-free affineconnections and conversely.

I Wilson Stothers’ Geometry PagesI Reference: Projective and Cayley-Klein Geometries

(Springer Monographs in Mathematics) (Hardcover)by Arkadij L. Onishchik (Author), Rolf Sulanke(Author)

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Manifolds with geometric structures:

Topology of manifolds

I Manifolds come up in many areas of technology andsciences.

I Studying the topological structures of manifolds iscomplicated by the fact that there is no uniform wayto describe many topologically important featuresand provides useful coordinates.

I We would like to find some good descriptions andperhaps even classify collections of manifolds.

I Of course these are for pure-mathematical uses fornow....

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Spherical geometry

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I An (X ,G)-geometric structure on a manifold M isgiven by an atlas of charts to X where the transitionmaps are in G.

I This equips M with all of the local (X ,G)-geometricalnotions.

I If the geometry admits notions such as geodesic,length, angle, cross ratio, then M now has suchnotions...

I So the central question is: which manifolds admitwhich structures and how many and if geometricstructures do not exists, why not?

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Spherical geometry

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<)b1a1'= 0.248

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Spherical geometry

Deformation spaces of geometric structures

I In major cases, M = X/Γ for a discrete subgroup Γ inthe Lie group G.

I Thus, X provides a global coordinate system and theclassification of discrete subgroups of G providesclassifications of manifolds like M.

I Here Γ tends to be the fundamental group of M.Thus, a geometric structure can be considered adiscrete representation of π1(M) in G.

I Often, there are cases when Γ is unique up toconjugations and there is a unique (X ,G)-structure.(Rigidity)

I Given an (X ,G)-manifold (orbifold) M, thedeformation space D(X ,G)(M) is locallyhomeomorphic to the G-representation spaceHom(π1(M),G)/G of π1(M).

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ExamplesI Closed surfaces have either a spherical, euclidean,

or hyperbolic structures depending on genus. Wecan classify these to form deformation spaces suchas Teichmuller spaces.

I A hyperbolic surface equals H2/Γ for the image Γ ofthe representation π → PSL(2,R).

I A Teichmuller space can be identified with acomponent of the space Hom(π,PSL(2,R))/ ∼ ofconjugacy classes of representations. Thecomponent consists of discrete faithfulrepresentations.

I For closed 3-manifolds, it is recently proved that theydecompose into pieces admitting one of eightgeometrical structures including hyperbolic,euclidean, spherical ones... Thus, these manifoldsare now being classified... (Proof of thegeometrization conjecture by Perelman).

I For higher-dimensional manifolds, there are othertypes of geometric structures...

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Spherical geometry

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Manifolds with (real) projective structures

I Cartan first considered projective structures onsurfaces as projectively flat torsion-free connectionson surfaces.

I Chern worked on general type of projectivestructures in the differential geometry point of view.

I The study of affine manifolds precede that ofprojective manifolds.

I There were extensive work on affine manifolds, andaffine Lie groups by Chern, Auslander, Goldman,Fried, Smillie, Nagano, Yagi, Shima, Carriere,Margulis,...

I There are outstanding questions such as the Chernconjecture, Auslander conjecture,...

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Projective manifolds: how many?I By the Klein-Beltrami model of hyperbolic space, we

can consider Hn as a unit ball B in an affinesubspace of RPn and PO(n + 1,R) as a subgroup ofPGL(n + 1,R) acting on B. Thus, a (complete)hyperbolic manifold has a projective structure. (Infact any closed surface has one.)

I In fact, 3-manifolds with six types of geometricstructures have real projective structures. (In fact,up to coverings of order two, 3-manifolds with eightgeometric structures have real projective structures.)

I A question arises whether these projective structurescan be deformed to purely projective structures.

I Deformed projective manifolds from closedhyperbolic ones are convex in the sense that anypath can be homotoped to a geodesic path. (Koszul’sopenness)

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I In 1960s, Benzecri started working on strictly convexdomain Ω where a projective transformation group Γacted with a compact quotient. That is, M = Ω/Γ is acompact manifold (orbifold). (Related to convexcones and group transformations (Kuiper,Koszul,Vinberg,...)). He showed that the boundary iseither C1 or is an ellipsoid.

I Kac-Vinberg found the first example for n = 2 for atriangle reflection group associated with Kac-Moodyalgebra.

I Recently, Benoist found many interesting propertiessuch as the fact that Ω is strictly convex if and only ifthe group Γ is Gromov-hyperbolic.

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Projective surfaces

I For closed surfaces, in 80s, Goldman found ageneral dimension counting method for thenonsingular part of the surface-group representationspace into the Lie group G which is dim G× (2g − 2)if g ≥ 2.

I Since the deformation space D(RP2,PGL(3,R))(S) for aclosed surface S is locally homeomorphic toHom(π1(S),PGL(3,R))/PGL(3,R), we know thedimension of the deformation space to be 8(2g − 2).

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Spherical geometry

Projective surfaces

I For closed surfaces, in 80s, Goldman found ageneral dimension counting method for thenonsingular part of the surface-group representationspace into the Lie group G which is dim G× (2g − 2)if g ≥ 2.

I Since the deformation space D(RP2,PGL(3,R))(S) for aclosed surface S is locally homeomorphic toHom(π1(S),PGL(3,R))/PGL(3,R), we know thedimension of the deformation space to be 8(2g − 2).

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Spherical geometry

I Goldman found that the the deformation space ofconvex projective structures is a cell of dimension16g − 16. Deformations (The real pro. str. on hyp.mflds.

I Choi showed that if g ≥ 2, then a projective surfacealways decomposes into convex projective surfacesand annuli along disjoint closed geodesics. (Someare not convex)

I The annuli were classified by Nagano, Yagi, andGoldman earlier.

I Using this, we have a complete classification ofprojective structures on closed surfaces (evenconstructive one).

I For 2-orbifolds, Goldman and Choi completed theclassification. developing images

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Projective surfaces and gauge theory

I Atiyah and Hitchin studied self-dual connections onsurfaces (70s)

I Given a Lie group G, a representation of π1(S)→ Gcan be thought of as a flat G-connection on aprincipal G-bundle over S and vice versa.

I Corlette showed that flat G-connections for manifolds(80s) can be realized as harmonic maps to certainassociated symmetric space bundles.

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I Hitchin used Higgs field on principal G-bundles oversurfaces to obtain parametrizations of flatG-connections over surfaces. (G is a real split formof a reductive group.) (90s)

I A Teichmüller space

T (Σ) = hyperbolic structures on Σ/isotopies

is a component of

Hom+(π1(Σ),PSL(2,R))/PSL(2,R)

of Fuchian representations.

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Spherical geometry

I Hitchin used Higgs field on principal G-bundles oversurfaces to obtain parametrizations of flatG-connections over surfaces. (G is a real split formof a reductive group.) (90s)

I A Teichmüller space

T (Σ) = hyperbolic structures on Σ/isotopies

is a component of

Hom+(π1(Σ),PSL(2,R))/PSL(2,R)

of Fuchian representations.

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I Hitchin-Teichmüller components:

ΓFuchsian→ PSL(2,R)

irreducible→ G. (1)

gives a component of

Hom+(π1(Σ),G)/G.

I The Hitchin-Teichmüller component ishomeomorphic to a cell of dimension (2g− 2) dim Gr .

I For n > 2,

Hom+(π1(Σ),PGL(n,R))/PGL(n,R)

has three connected components if n is odd and sixcomponents if n is even.

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Hom+(π1(Σ),G)/G.

I For n > 2,

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Hom+(π1(Σ),G)/G.

I For n > 2,

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I A convex projective surface is of form Ω/Γ. Hence,there is a representation π1(Σ)→ Γ determined onlyup to conjugation by PGL(3,R). This gives us a map

hol : D(Σ)→ Hom(π1(Σ),PGL(3,R))/ ∼ .

This map is known to be a local-homeomorphism(Ehresmann, Thurston) and is injective (Goldman)

I The map is in fact a homeomorphism ontoHitchin-Teichmüller component (Goldman, Choi) Themain idea for proof is to show that the image of themap is closed.

I As a consequence, we know that theHitchin-Teichmuller component for PGL(3,R))consists of discrete faithful representations only.

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hol : D(Σ)→ Hom(π1(Σ),PGL(3,R))/ ∼ .

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hol : D(Σ)→ Hom(π1(Σ),PGL(3,R))/ ∼ .

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Generalizations

I Labourie recently generalized this to n ≥ 2,3: Thatthe component of representations that can bedeformed to the Fuschian representation acts on ahyperconvex curve in RPn−1.

I Corollary: Every Hitchin representation is a discretefaithful and “purely loxodromic”. The mapping classgroup acts properly on H(n).

I Burger, Iozzi, Labourie, Wienhard generalized thisresult to maximal representations into other Liegroups.

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Generalizations

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Generalizations

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Projective surfaces and affine differentialgeometry

I John Loftin (simultaneously Labourie) showed usingthe work of Calabi, Cheng-Yau, and C.P. Wang onaffine spheres: Let Mn be a closed projectively flatmanifold.

I Then M is convex if and only if M admits an affinesphere structure in Rn+1.

I For n = 2, the affine sphere structure is equivalent tothe conformal structure on M with a holomorphiccubic-differential.

I In particular, this shows that the deformation spaceD(Σ) of convex projective structures on Σ admits acomplex structure, which is preserved under themoduli group actions. (Is it Kahler?)

I Loftin also worked out Mumford typecompactifications of the moduli space M(Σ) ofconvex projective structures.

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Projective 3-manifolds and deformations

I We saw that many 3-manifolds has projectivestructures.

I Cooper showed that RP3#RP3 does not admit a realprojective structures. (answers Bill Goldman’squestion). Other examples?

I We wish to understand the deformation spaces forhigher-dimensional manifolds and so on. (Thedeformation theory is still very difficult for conformalstructures as well.)

I There is a well-known construction of Apanasovusing bending: deforming along a closed totallygeodesic hypersurface.

I Johnson and Millson found deformations ofhigher-dimensional hyperbolic manifolds which arelocally singular. (also bending constructions)

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I Cooper,Long and Thistlethwaite found a numerical(some exact algebraic) evidences that out of the first1000 closed hyperbolic 3-manifolds in theHodgson-Weeks census, a handful admit non-trivialdeformations of their SO+(3,1)-representations intoSL(4,R); each resulting representation variety thengives rise to a family of real projective structures onthe manifold.

I Benoist and Choi found some class of 3-dimensionalreflection orbifold that has positive dimensionaldeformation spaces. But we haven’t been able tostudy the wider class of reflection orbifolds. (prisms,pyramids, icosahedron,...)

I We have been working algebraically and numericallyfor simple polytopes.

I So far, there are no Gauge theoretic or affine sphereapproach to 3-dimensional projective manifolds.

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