A survey of projectively flat structureson 2- and 3-manifolds.

Suhyoung ChoiDepartment of Mathematics

KAIST305-701 Daejeon, South Koreaschoi@math.kaist.ac.krmathx.kaist.ac.kr/̃schoi

August 2, 2005

A survey of projectively flat structureson 2- and 3-manifolds.

Suhyoung ChoiDepartment of Mathematics

KAIST305-701 Daejeon, South Koreaschoi@math.kaist.ac.krmathx.kaist.ac.kr/̃schoi

August 2, 2005

Abstract• A projectively flat manifold is a manifold with

an atlas of charts to the projective space RPn withtransition maps in the projective automorphism groups.These objects are closely related to the representationsof groups into the projective group PGL(n + 1, R).

A survey of projectively flat structureson 2- and 3-manifolds.

Suhyoung ChoiDepartment of Mathematics

KAIST305-701 Daejeon, South Koreaschoi@math.kaist.ac.krmathx.kaist.ac.kr/̃schoi

August 2, 2005

Abstract• A projectively flat manifold is a manifold with

an atlas of charts to the projective space RPn withtransition maps in the projective automorphism groups.These objects are closely related to the representationsof groups into the projective group PGL(n + 1, R).

1

• We will give a partial survey of this area including theclassical results and many recent results by Goldman,Loftin, Labourie, Benoist, and myself using manydiverse methods from low-dimensional topology, grouprepresentations, and affine differential geometry.

1

• We will give a partial survey of this area including theclassical results and many recent results by Goldman,Loftin, Labourie, Benoist, and myself using manydiverse methods from low-dimensional topology, grouprepresentations, and affine differential geometry.

Projective structureson manifolds

Geometric structures

Geometry

Differential Geometry

Gauge theory

Group theory

Representations

deformations

TheoreticalPhysics

Topologist’s approach:

conformally flat structuresParallel Universe of

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What are the objectives?

General remarks:

• What kind of geometric (projective) structures are on a givenmanifold?

• Determine the topology and geometry of the deformationspace

D(M) = {geometric (projective) structures on M}/isotopies.

2

What are the objectives?

General remarks:

• What kind of geometric (projective) structures are on a givenmanifold?

• Determine the topology and geometry of the deformationspace

D(M) = {geometric (projective) structures on M}/isotopies.

(Notation: projectively flat structure = projective structure =real projective structure = RPn-structure.)

Almost, we have

PGL(n, R) ∼ SL(n, R) ∼ PSL(n, R).

3

Origins in Geometry

• Cartan defined projectively flat structures on manifolds as:

? “geodesically Euclidean but with no metrics”? torsion-free? projectively flat (i.e., same geodesics structures as flat

metrics)? affine connection on manifolds.

• Chern made deep contributions to projective differentialgeometry before his celebrated results.

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Origins in Geometry

• Cartan defined projectively flat structures on manifolds as:

? “geodesically Euclidean but with no metrics”? torsion-free? projectively flat (i.e., same geodesics structures as flat

metrics)? affine connection on manifolds.

• Chern made deep contributions to projective differentialgeometry before his celebrated results.

• Ehresmann identifies this structure as having a maximal atlasof charts

? to RPn? with transition maps in PGL(n + 1, R).

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• Most examples of projective manifolds are by taking quotientsof a domain in RPn by a discrete subgroup of PGL(n+1, R).

? The domains are usually convex and we call the quotientconvex projective manifold.

? Let Hn be the interior of an ellipsoid. Then Hn isthe hyperbolic space and Aut(Hn) is the isometry group.Hn/Γ has a canonical projective structure.

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• Most examples of projective manifolds are by taking quotientsof a domain in RPn by a discrete subgroup of PGL(n+1, R).

? The domains are usually convex and we call the quotientconvex projective manifold.

? Let Hn be the interior of an ellipsoid. Then Hn isthe hyperbolic space and Aut(Hn) is the isometry group.Hn/Γ has a canonical projective structure.

? The deformation space D(M) of convex projectivestructures contains the (generalized) Teichmüller space

T (Σ) = {hyperbolic structures on M}/isotopies

as an imbedded subset: actually the set of fixed points ofduality involution

ι : D(Σ) → D(Σ)

given byD/Γ → D∗/ΓT,−1.

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• J.L. Koszul showed that convexity is preserved under the smallperturbations of the projective structures.

• Kac-Vinberg were first to find examples of convex projectivesurfaces that are not hyperbolic. The examples are based onCoxeter groups.

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7

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Kobayashi and convex projective manifolds

• Kobayashi studied metrics on projective manifolds: Heconsiders maps

l ⊂ RP1 → Mand take maximal ones. (l a proper interval or a completereal line. This defines a pseudo-metric.

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Kobayashi and convex projective manifolds

• Kobayashi studied metrics on projective manifolds: Heconsiders maps

l ⊂ RP1 → Mand take maximal ones. (l a proper interval or a completereal line. This defines a pseudo-metric.

• Kobayashi metric is a metric if and only if

? there are no lines of length π if and only if? M = Ω/Γ where Ω is a convex domain in RPn.

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Kobayashi and convex projective manifolds

• Kobayashi studied metrics on projective manifolds: Heconsiders maps

l ⊂ RP1 → Mand take maximal ones. (l a proper interval or a completereal line. This defines a pseudo-metric.

• Kobayashi metric is a metric if and only if

? there are no lines of length π if and only if? M = Ω/Γ where Ω is a convex domain in RPn.

• In this case, Kobayashi metric is Finsler and is a Hilbert metric

d(p, q) = log(o, s, q, p).

If Ω = Hn, the metric is the standard hyperbolic metric.

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p

q

o

s

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Affine differential geometry and projectivemanifolds

• Projective structures on manifold M induces an affinely flatstructure on M × R and conversely:

? Given a convex domain Ω in RPn, we can obtain aconvex cone V in Rn+1 by taking only the positive rayscorresponding to Ω.

? Conversely, given a convex cone V in Rn+1, we obtain aconvex domain Ω in RPn.

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Affine differential geometry and projectivemanifolds

• Projective structures on manifold M induces an affinely flatstructure on M × R and conversely:

? Given a convex domain Ω in RPn, we can obtain aconvex cone V in Rn+1 by taking only the positive rayscorresponding to Ω.

? Conversely, given a convex cone V in Rn+1, we obtain aconvex domain Ω in RPn.

• An affine-sphere structure on a closed manifold M is animbedding of M̃

? as an affine sphere asymptotic to the cone V —corresponding to the convex domain Ω

? where the deck transformation group acts as a discretelinear group.

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• John Loftin (simultaneously Labourie) showed using Calabi,Cheng-Yau’s work on affine spheres: Let Mn be a closedprojective manifold.

? Then M is convex if and only if M admits an affine-spherestructure in Rn+1.

? The duality construction corresponds to the conjugateconnection construction. (2001)

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• John Loftin (simultaneously Labourie) showed using Calabi,Cheng-Yau’s work on affine spheres: Let Mn be a closedprojective manifold.

? Then M is convex if and only if M admits an affine-spherestructure in Rn+1.

? The duality construction corresponds to the conjugateconnection construction. (2001)

• C.P. Wang showed much earlier that for a closed surface S

? (a conformal structure on S, a holomorphic section of K3

over S) ⇔? an affine-sphere structure on S.

This shows that:

? The deformation space D(Σ) of convex projectivestructures on Σ admits a complex structure.

? D(Σ) fibers over the Teichmüller space T (Σ) with cells asfibers.

? Loftin also worked out the Mumford type compactificationsof the moduli space M(Σ) of convex projective structures.

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Gauge theory and projective structures

• Atiyah and Hitchin studied self-dual connections on surfaces(70s)

• Corlette showed that flat connections for manifolds (80s) canbe realized as harmonic maps to certain symmetric spacebundles.

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Gauge theory and projective structures

• Atiyah and Hitchin studied self-dual connections on surfaces(70s)

• Corlette showed that flat connections for manifolds (80s) canbe realized as harmonic maps to certain symmetric spacebundles.

• Hitchin used Higgs fields on principal G-bundles over surfacesto obtain parametrizations of flat G-connections over surfaces.(G is a real split form of a reductive group.) (90s)

? A Higgs bundle is a pair (V,Φ) where V is a holomorphicvector bundle over Σ and Φ is a holomorphic section ofEndV ⊗K.

? A Teichmüller space

T (Σ) = {hyperbolic structures on Σ}/isotopies

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is a component of

Hom+(π1(Σ),PSL(2, R))/PSL(2, R)

of Fuchian representations.? Hitchin-Teichmüller components:

Γ Fuchsian→ PSL(2, R) irreducible→ G. (1)

gives a component of

Hom+(π1(Σ), G)/G.

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is a component of

Hom+(π1(Σ),PSL(2, R))/PSL(2, R)

of Fuchian representations.? Hitchin-Teichmüller components:

Γ Fuchsian→ PSL(2, R) irreducible→ G. (1)

gives a component of

Hom+(π1(Σ), G)/G.

? To find a flat connection given a Higgs bundle, we solvefor A

FA + [Φ,Φ∗] = 0.∗ The Hitchin-Teichmüler component is homeomorphic to

a cell of dimension (2g − 2) dim Gr.∗ For n > 2,

Hom+(π1(Σ),PGL(n, R))/PGL(n, R)

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

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Topologist’s approach

• Benzecri (and Milnor) showed a closed affine 2-manifold hasthe Euler characteristic = 0 (Chern conjecture).

• Benzecri studied convex domains that arise for convexprojective manifolds:

He showed that the boundary of Ω is C1 or is an ellipsoid fora closed convex projective surface.

• Nagano and Yagi classified affinely flat structures on tori.

• Goldman classified projective structures on tori. (His seniorthesis)

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Topologist’s approach

• Benzecri (and Milnor) showed a closed affine 2-manifold hasthe Euler characteristic = 0 (Chern conjecture).

• Benzecri studied convex domains that arise for convexprojective manifolds:

He showed that the boundary of Ω is C1 or is an ellipsoid fora closed convex projective surface.

• Nagano and Yagi classified affinely flat structures on tori.

• Goldman classified projective structures on tori. (His seniorthesis)

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• Grafting: One can insert this type of annuli into a convexprojective surface to obtain a non-convex projective surface.

• Convex decomposition theorem: actually, one can show thatthese are all that can happen. (—)

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• Goldman’s classification of convex projective structures onsurfaces: D(Σ) is a cell of dimension 8(2g− 2).

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• Goldman’s classification of convex projective structures onsurfaces: D(Σ) is a cell of dimension 8(2g− 2). Determiningthe deformation space D(Σ):

? first cut up the surface into pairs of pants.? Each pair of pants is a union of two open triangles.? We realize the triangles as geodesic ones.? We investigate the projective invariants of union of four

triangles in RP2.

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? The needed key is that

D(P ) → D(∂P )

for a pair of pants P is a principle fibration for a pair ofpants P .

[0,1,0]

[0,0,1]

[1,0,0]

[a ,−1, c ]

[a ,b ,−1]

a

c

A

0

b

∆

∆

∆

∆

B

C

[−1, b , c ]11

3 3

22

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? The needed key is that

D(P ) → D(∂P )

for a pair of pants P is a principle fibration for a pair ofpants P .

[0,1,0]

[0,0,1]

[1,0,0]

[a ,−1, c ]

[a ,b ,−1]

a

c

A

0

b

∆

∆

∆

∆

B

C

[−1, b , c ]11

3 3

22

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Hitchin’s conjecture and the generalizations

• A convex projective surface is of form Ω/Γ. Hence, there is arepresentation π1(Σ) → Γ determined only up to conjugationby PGL(3, R). This gives us a map

hol : D(Σ) → Hom(π1(Σ),PGL(3, R))/ ∼ .

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Hitchin’s conjecture and the generalizations

• A convex projective surface is of form Ω/Γ. Hence, there is arepresentation π1(Σ) → Γ determined only up to conjugationby 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)

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Hitchin’s conjecture and the generalizations

• A convex projective surface is of form Ω/Γ. Hence, there is arepresentation π1(Σ) → Γ determined only up to conjugationby 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)

• The map is in fact a homeomorphism onto Hitchin-Teichmüller component (Goldman, —) The main idea forproof is to show that the image of the map is closed.

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• Labourie recently generalized this to n ≥ 2, 3.

? A Fuchsian representations: π1(Σ) → PSL(2, R) withimage Γ such that H2/Γ is homeomorphic to Σ.

? Hitchin representations in PSL(n, R): a representationwhich can be deformed to a Fuchsian representation. i.e.,those in the Hitchin-Teichmüller component.

Γ Fuchsian→ PSL(2, R) irreducible→ PSL(n, R).

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• Labourie recently generalized this to n ≥ 2, 3.

? A Fuchsian representations: π1(Σ) → PSL(2, R) withimage Γ such that H2/Γ is homeomorphic to Σ.

? Hitchin representations in PSL(n, R): a representationwhich can be deformed to a Fuchsian representation. i.e.,those in the Hitchin-Teichmüller component.

Γ Fuchsian→ PSL(2, R) irreducible→ PSL(n, R). (2)

? Theorem: If ρ is a Hitchin representation, then there existsa (unique) ρ-equivariant hyperconvex curve ζ, the limitcurve, from ∂∞Γ → RPn−1.

? A continuous curve η : S1 → RPn−1 is hyperconvex if forany distinct points (x1, . . . , xn) in S1,

η(x1) + · · ·+ η(xn)

is a direct sum.

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? The proof uses Anosov type dynamical geometricalstructures and stability argument.

? Corollary: Every Hitchin representation is a discrete faithfuland “purely loxodromic”. The mapping class group actsproperly on H(n).

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Group theory and representations

• As stated earlier, Kac, Vinberg, and Koszul started to studythe deformations of representations Γ → PGL(n, R).

• There is a well-known deformation (Apanasov) called bendingfor projective and conformal structures.

• Johnson and Millson found that a certain hyperbolic manifoldhas a deformation space of projective structures that issingular.

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Group theory and representations

• As stated earlier, Kac, Vinberg, and Koszul started to studythe deformations of representations Γ → PGL(n, R).

• There is a well-known deformation (Apanasov) called bendingfor projective and conformal structures.

• Johnson and Millson found that a certain hyperbolic manifoldhas a deformation space of projective structures that issingular.

• Recent work of Benoist (papers “Convex divisibles I-IV”):

? Theorem. Γ an irreducible torsion-free subgroup ofSL(m, R). Then Γ divides a proper convex cone C ifand only if Γ is positive proximal.

? If C is not a Lorentzian cone, then Γ is Zariski dense inSL(m, R). (i.e., not in any analystic subgroup of G.)

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? Theorem. Let Γ be a discrete torsion-free subgroup ofSL(m, R) dividing an open convex domain in RPm−1. LetC be the corresponding cone on Rm. Then one of thefollowing holds.∗ C is a product∗ C is homogeneous∗ or Γ is Zariski dense in SL(m, R).

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? Theorem. Let Γ be a discrete torsion-free subgroup ofSL(m, R) dividing an open convex domain in RPm−1. LetC be the corresponding cone on Rm. Then one of thefollowing holds.∗ C is a product∗ C is homogeneous∗ or Γ is Zariski dense in SL(m, R).

? If the virtual center of Γ0 is trivial, then

EΓ0 = {ρ ∈ HΓ0| The image of ρdivides a convex open domain in RPn−1.}

is closed in

HΓ0 := Hom(Γ0,PGL(m, R))

. Openness was obtained by Koszul.

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? Theorem. Let Γ be a discrete torsion-free subgroup ofSL(m, R) dividing an open convex domain in RPm−1. LetC be the corresponding cone on Rm. Then one of thefollowing holds.∗ C is a product∗ C is homogeneous∗ or Γ is Zariski dense in SL(m, R).

? If the virtual center of Γ0 is trivial, then

EΓ0 = {ρ ∈ HΓ0| The image of ρdivides a convex open domain in RPn−1.}

is closed in

HΓ0 := Hom(Γ0,PGL(m, R))

. Openness was obtained by Koszul.

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? Let Γ be as above. Then the following conditions areequivalent:∗ Ω is strictly convex.∗ ∂Ω is C1.∗ Γ is Gromov-hyperbolic.∗ Geodesic flow on Ω/Γ is Anosov.

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A relation to mathematical physics

• Fock’s quantization of Teichmüller space as well as thedeformation space D of convex projective structures.

• The relationship between the universal Teichmüller space andvirasoro algebra is conjectured to hold between universal Dand W3-algebra.

• Fock and Goncharev’s recent work on cluster algerbra andhigher Teichmüller theory.

• The relationship is expected to hold for any PGL(n, R).

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A relation to mathematical physics

• Fock’s quantization of Teichmüller space as well as thedeformation space D of convex projective structures.

• The relationship between the universal Teichmüller space andvirasoro algebra is conjectured to hold between universal Dand W3-algebra.

• Fock and Goncharev’s recent work on cluster algerbra andhigher Teichmüller theory.

• The relationship is expected to hold for any PGL(n, R).

• Ovsienko, Duval, Lecompte introduced equivariant projectivequantization of manifolds for projective manifolds (also forconformal manifolds).

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Some questions related to Knot theory and3-manifold theory

• Morwen Thistlethwaite (jointly Cooper and Long) showedthat: out of the first 1000 closed hyperbolic 3-manifolds inthe Hodgson-Weeks census, a handful admit a non-trivialfamily of deformations projective structures on the manifold.

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Some questions related to Knot theory and3-manifold theory

• Morwen Thistlethwaite (jointly Cooper and Long) showedthat: out of the first 1000 closed hyperbolic 3-manifolds inthe Hodgson-Weeks census, a handful admit a non-trivialfamily of deformations projective structures on the manifold.

• Cooper showed that RP3#RP3 does not admit a projectivestructure. (answers Bill Goldman’s question).

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Some questions related to Knot theory and3-manifold theory

• Morwen Thistlethwaite (jointly Cooper and Long) showedthat: out of the first 1000 closed hyperbolic 3-manifolds inthe Hodgson-Weeks census, a handful admit a non-trivialfamily of deformations projective structures on the manifold.

• Cooper showed that RP3#RP3 does not admit a projectivestructure. (answers Bill Goldman’s question).

• Questions: Calculate the dimension of the deformation spacesof projective structures on each of the knot complements andrelate the result with the results of CLT.

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References

[Benoist] Yves Benoist, Convexes divisible, C.R. Acad. Sci.332 (2003) 387–390. (There are more references here (seealso Convex divisibles I, II, III, IV))

[CG] S. Choi and W. M. Goldman, The deformation spaces ofprojectively flat structures on 2-orbifold. to appear in Amer.J. Mathematics.

[Goldman 90] W. Goldman, Convex real projective structureson compact surfaces, J. Differential Geometry 31 791-845(1990).

[Goldman88] W. Goldman, Projective geometry on manifolds,Lecture notes, 1988.

[Hitchin1] , N.J. Hitchin, Lie groups and Teichmuller spaces,Topology 31, 449-473 (1992)

[Johnson and Millson 86] D. Johnson and J.J. Millson,Deformation spaces associated to compact hyperbolic

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manifolds. In Discrete Groups in Geometry and Analysis,Proceedings of a conference held at Yale University in honorof G. D. Mostow, 1986.

[Labourie1] F. Labourie, Anosov flows, Surface groups andcurves in projective space, preprint (2003)

[Labourie2] F. Labourie, Cross ratios, Surface groups andSL(n, R), preprint (2003)

[Loftin] J. Loftin, Riemannian metrics on locally projectivelyflat manifolds, American J. of Mathematics 124, 595-609(2002)