A survey of projective geometric structures on 2-,3-manifolds

S. Choi A survey of projective geometric Outline Classical geometries structures on 2-,3-manifolds Euclidean geometry Spherical geometry

Manifolds with geometric S. Choi structures: manifolds need some canonical descriptions.. Department of Mathematical Science Manifolds with KAIST, Daejeon, South Korea (real) projective structures Tokyo Institute of Technology A survey of Outline projective geometric structures on 2-,3-manifolds

S. Choi

I Survey: Classical geometries Outline

I Euclidean geometry: Babylonians, Egyptians, Classical geometries Greeks, Chinese (Euclid’s axiomatic methods under Euclidean geometry Plato’s philosphy) Spherical geometry Manifolds with I Spherical geometry: Greek astronomy, Gauss, geometric Riemann structures: manifolds need I Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, some canonical Beltrami,Klein, Poincare descriptions.. Manifolds with I Conformal geometry (Mobius geometry or circle (real) projective geometry) structures I Projective geometry I Erlanger program, Cartan connections, Ehresmann connections A survey of Outline projective geometric structures on 2-,3-manifolds

S. Choi

I Survey: Classical geometries Outline

I Euclidean geometry: Babylonians, Egyptians, Classical geometries Greeks, Chinese (Euclid’s axiomatic methods under Euclidean geometry Plato’s philosphy) Spherical geometry Manifolds with I Spherical geometry: Greek astronomy, Gauss, geometric Riemann structures: manifolds need I Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, some canonical Beltrami,Klein, Poincare descriptions.. Manifolds with I Conformal geometry (Mobius geometry or circle (real) projective geometry) structures I Projective geometry I Erlanger program, Cartan connections, Ehresmann connections A survey of Outline projective geometric structures on 2-,3-manifolds

S. Choi

I Survey: Classical geometries Outline

I Euclidean geometry: Babylonians, Egyptians, Classical geometries Greeks, Chinese (Euclid’s axiomatic methods under Euclidean geometry Plato’s philosphy) Spherical geometry Manifolds with I Spherical geometry: Greek astronomy, Gauss, geometric Riemann structures: manifolds need I Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, some canonical Beltrami,Klein, Poincare descriptions.. Manifolds with I Conformal geometry (Mobius geometry or circle (real) projective geometry) structures I Projective geometry I Erlanger program, Cartan connections, Ehresmann connections A survey of Outline projective geometric structures on 2-,3-manifolds

S. Choi

I Survey: Classical geometries Outline

I Euclidean geometry: Babylonians, Egyptians, Classical geometries Greeks, Chinese (Euclid’s axiomatic methods under Euclidean geometry Plato’s philosphy) Spherical geometry Manifolds with I Spherical geometry: Greek astronomy, Gauss, geometric Riemann structures: manifolds need I Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, some canonical Beltrami,Klein, Poincare descriptions.. Manifolds with I Conformal geometry (Mobius geometry or circle (real) projective geometry) structures I Projective geometry I Erlanger program, Cartan connections, Ehresmann connections A survey of Outline projective geometric structures on 2-,3-manifolds

S. Choi

I Survey: Classical geometries Outline

I Euclidean geometry: Babylonians, Egyptians, Classical geometries Greeks, Chinese (Euclid’s axiomatic methods under Euclidean geometry Plato’s philosphy) Spherical geometry Manifolds with I Spherical geometry: Greek astronomy, Gauss, geometric Riemann structures: manifolds need I Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, some canonical Beltrami,Klein, Poincare descriptions.. Manifolds with I Conformal geometry (Mobius geometry or circle (real) projective geometry) structures I Projective geometry I Erlanger program, Cartan connections, Ehresmann connections A survey of Outline projective geometric structures on 2-,3-manifolds

S. Choi

I Survey: Classical geometries Outline

I Euclidean geometry: Babylonians, Egyptians, Classical geometries Greeks, Chinese (Euclid’s axiomatic methods under Euclidean geometry Plato’s philosphy) Spherical geometry Manifolds with I Spherical geometry: Greek astronomy, Gauss, geometric Riemann structures: manifolds need I Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, some canonical Beltrami,Klein, Poincare descriptions.. Manifolds with I Conformal geometry (Mobius geometry or circle (real) projective geometry) structures I Projective geometry I Erlanger program, Cartan connections, Ehresmann connections A survey of Outline projective geometric structures on 2-,3-manifolds

S. Choi

I Survey: Classical geometries Outline

I Euclidean geometry: Babylonians, Egyptians, Classical geometries Greeks, Chinese (Euclid’s axiomatic methods under Euclidean geometry Plato’s philosphy) Spherical geometry Manifolds with I Spherical geometry: Greek astronomy, Gauss, geometric Riemann structures: manifolds need I Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, some canonical Beltrami,Klein, Poincare descriptions.. Manifolds with I Conformal geometry (Mobius geometry or circle (real) projective geometry) structures I Projective geometry I Erlanger program, Cartan connections, Ehresmann connections A survey of Manifolds with geometric structures projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical geometries Euclidean geometry I Manifolds with geometric structures: manifolds need Spherical geometry some canonical descriptions. Manifolds with geometric I Manifolds with geometric structures. structures: manifolds need I Deformation spaces of geometric structures. some canonical descriptions.. I Examples Manifolds with (real) projective structures A survey of Manifolds with geometric structures projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical geometries Euclidean geometry I Manifolds with geometric structures: manifolds need Spherical geometry some canonical descriptions. Manifolds with geometric I Manifolds with geometric structures. structures: manifolds need I Deformation spaces of geometric structures. some canonical descriptions.. I Examples Manifolds with (real) projective structures A survey of Manifolds with geometric structures projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical geometries Euclidean geometry I Manifolds with geometric structures: manifolds need Spherical geometry some canonical descriptions. Manifolds with geometric I Manifolds with geometric structures. structures: manifolds need I Deformation spaces of geometric structures. some canonical descriptions.. I Examples Manifolds with (real) projective structures A survey of Manifolds with geometric structures projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical geometries Euclidean geometry I Manifolds with geometric structures: manifolds need Spherical geometry some canonical descriptions. Manifolds with geometric I Manifolds with geometric structures. structures: manifolds need I Deformation spaces of geometric structures. some canonical descriptions.. I Examples Manifolds with (real) projective structures A survey of Manifolds with projective structures projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical I Some history. geometries Euclidean geometry Spherical geometry I Projective manifolds: how many? Manifolds with I Projective surfaces geometric structures: I Projective surfaces and gauge theory. manifolds need some canonical I Labourie’s generalization descriptions..

I Projective surfaces and affine differential geometry. Manifolds with (real) projective I Projective 3-manifolds and deformations structures A survey of Manifolds with projective structures projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical I Some history. geometries Euclidean geometry Spherical geometry I Projective manifolds: how many? Manifolds with I Projective surfaces geometric structures: I Projective surfaces and gauge theory. manifolds need some canonical I Labourie’s generalization descriptions..

I Projective surfaces and affine differential geometry. Manifolds with (real) projective I Projective 3-manifolds and deformations structures A survey of Manifolds with projective structures projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical I Some history. geometries Euclidean geometry Spherical geometry I Projective manifolds: how many? Manifolds with I Projective surfaces geometric structures: I Projective surfaces and gauge theory. manifolds need some canonical I Labourie’s generalization descriptions..

I Projective surfaces and affine differential geometry. Manifolds with (real) projective I Projective 3-manifolds and deformations structures A survey of Manifolds with projective structures projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical I Some history. geometries Euclidean geometry Spherical geometry I Projective manifolds: how many? Manifolds with I Projective surfaces geometric structures: I Projective surfaces and gauge theory. manifolds need some canonical I Labourie’s generalization descriptions..

I Projective surfaces and affine differential geometry. Manifolds with (real) projective I Projective 3-manifolds and deformations structures A survey of Manifolds with projective structures projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical I Some history. geometries Euclidean geometry Spherical geometry I Projective manifolds: how many? Manifolds with I Projective surfaces geometric structures: I Projective surfaces and gauge theory. manifolds need some canonical I Labourie’s generalization descriptions..

I Projective surfaces and affine differential geometry. Manifolds with (real) projective I Projective 3-manifolds and deformations structures A survey of Manifolds with projective structures projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical I Some history. geometries Euclidean geometry Spherical geometry I Projective manifolds: how many? Manifolds with I Projective surfaces geometric structures: I Projective surfaces and gauge theory. manifolds need some canonical I Labourie’s generalization descriptions..

I Projective surfaces and affine differential geometry. Manifolds with (real) projective I Projective 3-manifolds and deformations structures A survey of Manifolds with projective structures projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical I Some history. geometries Euclidean geometry Spherical geometry I Projective manifolds: how many? Manifolds with I Projective surfaces geometric structures: I Projective surfaces and gauge theory. manifolds need some canonical I Labourie’s generalization descriptions..

I Projective surfaces and affine differential geometry. Manifolds with (real) projective I Projective 3-manifolds and deformations structures A survey of Euclidean geometry projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical geometries I Babylonian, Egyptian, Chinese, Greek... Euclidean geometry Spherical geometry I Euclid developed his axiomatic method to planar and Manifolds with geometric solid geometry under the influence of Plato, who structures: manifolds need thought that geometry should be the foundation of all some canonical thought after the Pythagorian attempt to understand descriptions.. Manifolds with the world using rational numbers failed.... (real) projective structures A survey of Euclidean geometry projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical geometries I Babylonian, Egyptian, Chinese, Greek... Euclidean geometry Spherical geometry I Euclid developed his axiomatic method to planar and Manifolds with geometric solid geometry under the influence of Plato, who structures: manifolds need thought that geometry should be the foundation of all some canonical thought after the Pythagorian attempt to understand descriptions.. Manifolds with the world using rational numbers failed.... (real) projective structures A survey of projective geometric structures on I Euclidean geometry consists of some notions such 2-,3-manifolds as lines, points, lengths, angle, and their interplay in S. Choi some space called a plane... Outline I The Euclid had five axioms Classical I D. Hilbert, and many others made a modern geometries Euclidean geometry foundation so that the Euclidean geometry was Spherical geometry

reduced to logic. Manifolds with geometric I Euclidean geometry has a notion of rigid structures: transformations which made the space manifolds need some canonical homogeneous. They form a group called a group of descriptions.. rigid motions. They preserve lines, length, angles, Manifolds with (real) projective and every geometric statements. structures I The group is useful in proving statements.... Turning it around, we see that actually the transformation group is more important. wallpaper groups I Notions of Euclidean subspaces... A survey of projective geometric structures on I Euclidean geometry consists of some notions such 2-,3-manifolds as lines, points, lengths, angle, and their interplay in S. Choi some space called a plane... Outline I The Euclid had five axioms Classical I D. Hilbert, and many others made a modern geometries Euclidean geometry foundation so that the Euclidean geometry was Spherical geometry

reduced to logic. Manifolds with geometric I Euclidean geometry has a notion of rigid structures: transformations which made the space manifolds need some canonical homogeneous. They form a group called a group of descriptions.. rigid motions. They preserve lines, length, angles, Manifolds with (real) projective and every geometric statements. structures I The group is useful in proving statements.... Turning it around, we see that actually the transformation group is more important. wallpaper groups I Notions of Euclidean subspaces... A survey of projective geometric structures on I Euclidean geometry consists of some notions such 2-,3-manifolds as lines, points, lengths, angle, and their interplay in S. Choi some space called a plane... Outline I The Euclid had five axioms Classical I D. Hilbert, and many others made a modern geometries Euclidean geometry foundation so that the Euclidean geometry was Spherical geometry

reduced to logic. Manifolds with geometric I Euclidean geometry has a notion of rigid structures: transformations which made the space manifolds need some canonical homogeneous. They form a group called a group of descriptions.. rigid motions. They preserve lines, length, angles, Manifolds with (real) projective and every geometric statements. structures I The group is useful in proving statements.... Turning it around, we see that actually the transformation group is more important. wallpaper groups I Notions of Euclidean subspaces... A survey of projective geometric structures on I Euclidean geometry consists of some notions such 2-,3-manifolds as lines, points, lengths, angle, and their interplay in S. Choi some space called a plane... Outline I The Euclid had five axioms Classical I D. Hilbert, and many others made a modern geometries Euclidean geometry foundation so that the Euclidean geometry was Spherical geometry

reduced to logic. Manifolds with geometric I Euclidean geometry has a notion of rigid structures: transformations which made the space manifolds need some canonical homogeneous. They form a group called a group of descriptions.. rigid motions. They preserve lines, length, angles, Manifolds with (real) projective and every geometric statements. structures I The group is useful in proving statements.... Turning it around, we see that actually the transformation group is more important. wallpaper groups I Notions of Euclidean subspaces... A survey of projective geometric structures on I Euclidean geometry consists of some notions such 2-,3-manifolds as lines, points, lengths, angle, and their interplay in S. Choi some space called a plane... Outline I The Euclid had five axioms Classical I D. Hilbert, and many others made a modern geometries Euclidean geometry foundation so that the Euclidean geometry was Spherical geometry

reduced to logic. Manifolds with geometric I Euclidean geometry has a notion of rigid structures: transformations which made the space manifolds need some canonical homogeneous. They form a group called a group of descriptions.. rigid motions. They preserve lines, length, angles, Manifolds with (real) projective and every geometric statements. structures I The group is useful in proving statements.... Turning it around, we see that actually the transformation group is more important. wallpaper groups I Notions of Euclidean subspaces... A survey of projective geometric structures on I Euclidean geometry consists of some notions such 2-,3-manifolds as lines, points, lengths, angle, and their interplay in S. Choi some space called a plane... Outline I The Euclid had five axioms Classical I D. Hilbert, and many others made a modern geometries Euclidean geometry foundation so that the Euclidean geometry was Spherical geometry

reduced to logic. Manifolds with geometric I Euclidean geometry has a notion of rigid structures: transformations which made the space manifolds need some canonical homogeneous. They form a group called a group of descriptions.. rigid motions. They preserve lines, length, angles, Manifolds with (real) projective and every geometric statements. structures I The group is useful in proving statements.... Turning it around, we see that actually the transformation group is more important. wallpaper groups I Notions of Euclidean subspaces... A survey of Spherical geometry. projective geometric structures on 2-,3-manifolds

I Greeks: astronomical, navigational... Arabs,.. S. Choi

I From astronomical viewpoint, it is nice to view the Outline 2 sky as a unit sphere S in the Euclidean 3-space. Classical geometries (See spherical.cdy) Euclidean geometry Spherical geometry I The great circles replaced lines and angles are Manifolds with measured in the tangential sense. Lengths are geometric structures: measured by taking arcs in the great circles. manifolds need some canonical descriptions.. I Geometric objects such as triangles behave a little Manifolds with bit different. (real) projective n structures I Higher dimensional spherical geometry S can be easily constructed.

I The group of orthogonal transformations O(n + 1) acts on Sn preserving every spherical geometric notions. The geometry of sphere A survey of Spherical geometry. projective geometric structures on 2-,3-manifolds

I Greeks: astronomical, navigational... Arabs,.. S. Choi

I From astronomical viewpoint, it is nice to view the Outline 2 sky as a unit sphere S in the Euclidean 3-space. Classical geometries (See spherical.cdy) Euclidean geometry Spherical geometry I The great circles replaced lines and angles are Manifolds with measured in the tangential sense. Lengths are geometric structures: measured by taking arcs in the great circles. manifolds need some canonical descriptions.. I Geometric objects such as triangles behave a little Manifolds with bit different. (real) projective n structures I Higher dimensional spherical geometry S can be easily constructed.

I The group of orthogonal transformations O(n + 1) acts on Sn preserving every spherical geometric notions. The geometry of sphere A survey of Spherical geometry. projective geometric structures on 2-,3-manifolds

I Greeks: astronomical, navigational... Arabs,.. S. Choi

I From astronomical viewpoint, it is nice to view the Outline 2 sky as a unit sphere S in the Euclidean 3-space. Classical geometries (See spherical.cdy) Euclidean geometry Spherical geometry I The great circles replaced lines and angles are Manifolds with measured in the tangential sense. Lengths are geometric structures: measured by taking arcs in the great circles. manifolds need some canonical descriptions.. I Geometric objects such as triangles behave a little Manifolds with bit different. (real) projective n structures I Higher dimensional spherical geometry S can be easily constructed.

I The group of orthogonal transformations O(n + 1) acts on Sn preserving every spherical geometric notions. The geometry of sphere A survey of Spherical geometry. projective geometric structures on 2-,3-manifolds

I Greeks: astronomical, navigational... Arabs,.. S. Choi

I From astronomical viewpoint, it is nice to view the Outline 2 sky as a unit sphere S in the Euclidean 3-space. Classical geometries (See spherical.cdy) Euclidean geometry Spherical geometry I The great circles replaced lines and angles are Manifolds with measured in the tangential sense. Lengths are geometric structures: measured by taking arcs in the great circles. manifolds need some canonical descriptions.. I Geometric objects such as triangles behave a little Manifolds with bit different. (real) projective n structures I Higher dimensional spherical geometry S can be easily constructed.

I The group of orthogonal transformations O(n + 1) acts on Sn preserving every spherical geometric notions. The geometry of sphere A survey of Spherical geometry. projective geometric structures on 2-,3-manifolds

I Greeks: astronomical, navigational... Arabs,.. S. Choi

I From astronomical viewpoint, it is nice to view the Outline 2 sky as a unit sphere S in the Euclidean 3-space. Classical geometries (See spherical.cdy) Euclidean geometry Spherical geometry I The great circles replaced lines and angles are Manifolds with measured in the tangential sense. Lengths are geometric structures: measured by taking arcs in the great circles. manifolds need some canonical descriptions.. I Geometric objects such as triangles behave a little Manifolds with bit different. (real) projective n structures I Higher dimensional spherical geometry S can be easily constructed.

I The group of orthogonal transformations O(n + 1) acts on Sn preserving every spherical geometric notions. The geometry of sphere A survey of Spherical geometry. projective geometric structures on 2-,3-manifolds

I Greeks: astronomical, navigational... Arabs,.. S. Choi

I From astronomical viewpoint, it is nice to view the Outline 2 sky as a unit sphere S in the Euclidean 3-space. Classical geometries (See spherical.cdy) Euclidean geometry Spherical geometry I The great circles replaced lines and angles are Manifolds with measured in the tangential sense. Lengths are geometric structures: measured by taking arcs in the great circles. manifolds need some canonical descriptions.. I Geometric objects such as triangles behave a little Manifolds with bit different. (real) projective n structures I Higher dimensional spherical geometry S can be easily constructed.

I The group of orthogonal transformations O(n + 1) acts on Sn preserving every spherical geometric notions. The geometry of sphere A survey of Hyperbolic geometry. projective geometric structures on I Lobachevsky and Bolyai tried to build a geometry 2-,3-manifolds

that did not satisfy the fifth axiom of Euclid. (See S. Choi hyperbolic.cdy) Outline

I Their attempts were justified by Beltrami-Klein model Classical geometries which is a disk and lines were replaced by chords Euclidean geometry and lengths were given by the logarithms of cross Spherical geometry Manifolds with ratios. See Beltrami-Klein model. Later other models geometric structures: such as Poincare half space model and Poincare manifolds need some canonical disk model were developed. Poincare model (Inst. descriptions..

figuring). Manifolds with (real) projective I Here the group of rigid motions is the Lie group structures SL(2, R). I Higher-dimensional hyperbolic spaces were later constructed. Actually, an upper part of a hyperboloid in the Lorentzian space would be a model and PO(1, n) forms the group of rigid motions. Minkowsky model A survey of Hyperbolic geometry. projective geometric structures on I Lobachevsky and Bolyai tried to build a geometry 2-,3-manifolds

that did not satisfy the fifth axiom of Euclid. (See S. Choi hyperbolic.cdy) Outline

I Their attempts were justified by Beltrami-Klein model Classical geometries which is a disk and lines were replaced by chords Euclidean geometry and lengths were given by the logarithms of cross Spherical geometry Manifolds with ratios. See Beltrami-Klein model. Later other models geometric structures: such as Poincare half space model and Poincare manifolds need some canonical disk model were developed. Poincare model (Inst. descriptions..

figuring). Manifolds with (real) projective I Here the group of rigid motions is the Lie group structures SL(2, R). I Higher-dimensional hyperbolic spaces were later constructed. Actually, an upper part of a hyperboloid in the Lorentzian space would be a model and PO(1, n) forms the group of rigid motions. Minkowsky model A survey of Hyperbolic geometry. projective geometric structures on I Lobachevsky and Bolyai tried to build a geometry 2-,3-manifolds

that did not satisfy the fifth axiom of Euclid. (See S. Choi hyperbolic.cdy) Outline

I Their attempts were justified by Beltrami-Klein model Classical geometries which is a disk and lines were replaced by chords Euclidean geometry and lengths were given by the logarithms of cross Spherical geometry Manifolds with ratios. See Beltrami-Klein model. Later other models geometric structures: such as Poincare half space model and Poincare manifolds need some canonical disk model were developed. Poincare model (Inst. descriptions..

figuring). Manifolds with (real) projective I Here the group of rigid motions is the Lie group structures SL(2, R). I Higher-dimensional hyperbolic spaces were later constructed. Actually, an upper part of a hyperboloid in the Lorentzian space would be a model and PO(1, n) forms the group of rigid motions. Minkowsky model A survey of Hyperbolic geometry. projective geometric structures on I Lobachevsky and Bolyai tried to build a geometry 2-,3-manifolds

that did not satisfy the fifth axiom of Euclid. (See S. Choi hyperbolic.cdy) Outline

I Their attempts were justified by Beltrami-Klein model Classical geometries which is a disk and lines were replaced by chords Euclidean geometry and lengths were given by the logarithms of cross Spherical geometry Manifolds with ratios. See Beltrami-Klein model. Later other models geometric structures: such as Poincare half space model and Poincare manifolds need some canonical disk model were developed. Poincare model (Inst. descriptions..

figuring). Manifolds with (real) projective I Here the group of rigid motions is the Lie group structures SL(2, R). I Higher-dimensional hyperbolic spaces were later constructed. Actually, an upper part of a hyperboloid in the Lorentzian space would be a model and PO(1, n) forms the group of rigid motions. Minkowsky model A survey of Conformal geometry projective geometric structures on I Suppose that we use to study circles and spheres 2-,3-manifolds only... We allow all transformations that preserves S. Choi

circles... Outline This geometry looses a notion of lengths but has a Classical I geometries notion of angles. There are no lines or geodesics but Euclidean geometry Spherical geometry

there are circles. Manifolds with geometric I The Euclidean plane is compactifed by adding a structures: manifolds need unique point as an infinity. The group of motions is some canonical generated by inversions in circles. The group is descriptions.. Manifolds with called the Mobius transformation group. That is, the (real) projective group of transformations of form structures

az + b az¯ + b z → , cz + d cz¯ + d

I The space itself is considered as a complex sphere, i.e., the complex plane with the infinity added. A survey of Conformal geometry projective geometric structures on I Suppose that we use to study circles and spheres 2-,3-manifolds only... We allow all transformations that preserves S. Choi

circles... Outline This geometry looses a notion of lengths but has a Classical I geometries notion of angles. There are no lines or geodesics but Euclidean geometry Spherical geometry

there are circles. Manifolds with geometric I The Euclidean plane is compactifed by adding a structures: manifolds need unique point as an infinity. The group of motions is some canonical generated by inversions in circles. The group is descriptions.. Manifolds with called the Mobius transformation group. That is, the (real) projective group of transformations of form structures

az + b az¯ + b z → , cz + d cz¯ + d

I The space itself is considered as a complex sphere, i.e., the complex plane with the infinity added. A survey of Conformal geometry projective geometric structures on I Suppose that we use to study circles and spheres 2-,3-manifolds only... We allow all transformations that preserves S. Choi

circles... Outline This geometry looses a notion of lengths but has a Classical I geometries notion of angles. There are no lines or geodesics but Euclidean geometry Spherical geometry

there are circles. Manifolds with geometric I The Euclidean plane is compactifed by adding a structures: manifolds need unique point as an infinity. The group of motions is some canonical generated by inversions in circles. The group is descriptions.. Manifolds with called the Mobius transformation group. That is, the (real) projective group of transformations of form structures

az + b az¯ + b z → , cz + d cz¯ + d

I The space itself is considered as a complex sphere, i.e., the complex plane with the infinity added. A survey of Conformal geometry projective geometric structures on I Suppose that we use to study circles and spheres 2-,3-manifolds only... We allow all transformations that preserves S. Choi

circles... Outline This geometry looses a notion of lengths but has a Classical I geometries notion of angles. There are no lines or geodesics but Euclidean geometry Spherical geometry

there are circles. Manifolds with geometric I The Euclidean plane is compactifed by adding a structures: manifolds need unique point as an infinity. The group of motions is some canonical generated by inversions in circles. The group is descriptions.. Manifolds with called the Mobius transformation group. That is, the (real) projective group of transformations of form structures

az + b az¯ + b z → , cz + d cz¯ + d

I The space itself is considered as a complex sphere, i.e., the complex plane with the infinity added. A survey of projective geometric structures on 2-,3-manifolds

S. Choi

Outline I Conformal geometry is useful in solving PDEs in Classical complex analysis. (The huge work of Ahlfors-Bers geometries Euclidean geometry school). Spherical geometry

I Recent work on discretization of complex analytic Manifolds with geometric (conformal) maps... (The circle packing theorem: structures: manifolds need Koebe-Andreev-Thurston theorem) some canonical descriptions..

I For higher-dimensions, the conformal geometry can Manifolds with (real) projective be defined similarly. structures A survey of projective geometric structures on 2-,3-manifolds

S. Choi

Outline I Conformal geometry is useful in solving PDEs in Classical complex analysis. (The huge work of Ahlfors-Bers geometries Euclidean geometry school). Spherical geometry

I Recent work on discretization of complex analytic Manifolds with geometric (conformal) maps... (The circle packing theorem: structures: manifolds need Koebe-Andreev-Thurston theorem) some canonical descriptions..

I For higher-dimensions, the conformal geometry can Manifolds with (real) projective be defined similarly. structures A survey of projective geometric structures on 2-,3-manifolds

S. Choi

Outline I Conformal geometry is useful in solving PDEs in Classical complex analysis. (The huge work of Ahlfors-Bers geometries Euclidean geometry school). Spherical geometry

I Recent work on discretization of complex analytic Manifolds with geometric (conformal) maps... (The circle packing theorem: structures: manifolds need Koebe-Andreev-Thurston theorem) some canonical descriptions..

I For higher-dimensions, the conformal geometry can Manifolds with (real) projective be defined similarly. structures A survey of Projective geometry projective geometric structures on I Projective geometry naturally arose in fine art 2-,3-manifolds drawing perspectives during Renascence. S. Choi

Perspective drawings Outline Desargues, Kepler first tried to add infinite points Classical I geometries corresponding to each direction that a line in a plane Euclidean geometry Spherical geometry

can take. Thus, a plane with infinite point for each Manifolds with geometric direction form a projective plane. structures: manifolds need I Transformations are ones generated by change of some canonical perspectives. In fact, when we are taking x-rays or descriptions.. Manifolds with other scans. (real) projective structures I The notions such as lengths, angles lose meaning. But notions of lines or geodesics are preserved. The Greeks discovered that the cross ratios, i.e., ratios of ratios, are preserved.

I The infinite points are just like the ordinary points under transformations. A survey of Projective geometry projective geometric structures on I Projective geometry naturally arose in fine art 2-,3-manifolds drawing perspectives during Renascence. S. Choi

Perspective drawings Outline Desargues, Kepler first tried to add infinite points Classical I geometries corresponding to each direction that a line in a plane Euclidean geometry Spherical geometry

can take. Thus, a plane with infinite point for each Manifolds with geometric direction form a projective plane. structures: manifolds need I Transformations are ones generated by change of some canonical perspectives. In fact, when we are taking x-rays or descriptions.. Manifolds with other scans. (real) projective structures I The notions such as lengths, angles lose meaning. But notions of lines or geodesics are preserved. The Greeks discovered that the cross ratios, i.e., ratios of ratios, are preserved.

I The infinite points are just like the ordinary points under transformations. A survey of Projective geometry projective geometric structures on I Projective geometry naturally arose in fine art 2-,3-manifolds drawing perspectives during Renascence. S. Choi

Perspective drawings Outline Desargues, Kepler first tried to add infinite points Classical I geometries corresponding to each direction that a line in a plane Euclidean geometry Spherical geometry

can take. Thus, a plane with infinite point for each Manifolds with geometric direction form a projective plane. structures: manifolds need I Transformations are ones generated by change of some canonical perspectives. In fact, when we are taking x-rays or descriptions.. Manifolds with other scans. (real) projective structures I The notions such as lengths, angles lose meaning. But notions of lines or geodesics are preserved. The Greeks discovered that the cross ratios, i.e., ratios of ratios, are preserved.

I The infinite points are just like the ordinary points under transformations. A survey of Projective geometry projective geometric structures on I Projective geometry naturally arose in fine art 2-,3-manifolds drawing perspectives during Renascence. S. Choi

Perspective drawings Outline Desargues, Kepler first tried to add infinite points Classical I geometries corresponding to each direction that a line in a plane Euclidean geometry Spherical geometry

can take. Thus, a plane with infinite point for each Manifolds with geometric direction form a projective plane. structures: manifolds need I Transformations are ones generated by change of some canonical perspectives. In fact, when we are taking x-rays or descriptions.. Manifolds with other scans. (real) projective structures I The notions such as lengths, angles lose meaning. But notions of lines or geodesics are preserved. The Greeks discovered that the cross ratios, i.e., ratios of ratios, are preserved.

I The infinite points are just like the ordinary points under transformations. A survey of Projective geometry projective geometric structures on I Projective geometry naturally arose in fine art 2-,3-manifolds drawing perspectives during Renascence. S. Choi

Perspective drawings Outline Desargues, Kepler first tried to add infinite points Classical I geometries corresponding to each direction that a line in a plane Euclidean geometry Spherical geometry

can take. Thus, a plane with infinite point for each Manifolds with geometric direction form a projective plane. structures: manifolds need I Transformations are ones generated by change of some canonical perspectives. In fact, when we are taking x-rays or descriptions.. Manifolds with other scans. (real) projective structures I The notions such as lengths, angles lose meaning. But notions of lines or geodesics are preserved. The Greeks discovered that the cross ratios, i.e., ratios of ratios, are preserved.

I The infinite points are just like the ordinary points under transformations. A survey of Projective geometry projective geometric structures on 2-,3-manifolds

S. Choi

Outline I Klein finally formalized it by considering a projective Classical plane as the space of lines through origin in the geometries Euclidean geometry Euclidean 3-space. Here, a Euclidean plane is Spherical geometry recovered as a subspace of the lines not in the Manifolds with geometric xy-plane and the infinite points are the lines in the structures: manifolds need xy-plane. some canonical descriptions..

I Many interesting geometric theorems hold: Manifolds with (real) projective Desargues, Pappus,... Moreover, the theorems come structures in pairs: duality of lines and points: Pappus Theorem, Pascal Theorem A survey of Projective geometry projective geometric structures on 2-,3-manifolds

S. Choi

Outline I Klein finally formalized it by considering a projective Classical plane as the space of lines through origin in the geometries Euclidean geometry Euclidean 3-space. Here, a Euclidean plane is Spherical geometry recovered as a subspace of the lines not in the Manifolds with geometric xy-plane and the infinite points are the lines in the structures: manifolds need xy-plane. some canonical descriptions..

I Many interesting geometric theorems hold: Manifolds with (real) projective Desargues, Pappus,... Moreover, the theorems come structures in pairs: duality of lines and points: Pappus Theorem, Pascal Theorem A survey of Projective geometry projective geometric structures on 2-,3-manifolds

S. Choi I Many geometries are actually sub-geometries of projective (conformal) geometry. (The two Outline correspond to maximal finite-dimensional Lie Classical geometries algebras acting locally on manifolds...) Euclidean geometry n n Spherical geometry I Hyperbolic geometry: H imbeds in P and R Manifolds with PO(n, 1) in PGL(n + 1, R) in canonical way. geometric structures: I spherical, euclidean geometry. manifolds need n n some canonical I The affine geometry G = SL(n, ) · and X = . R R R descriptions.. I Six of eight 3-dimensional geometries: euclidean, ˜ Manifolds with spherical, hyperbolic, nil, sol, SL(2, R) (B. Thiel) (real) projective 2 2 structures I S × R and H × R. (almost) I Some symmetric spaces and matrix groups...

I Recently, some graphics applications.... image superposition,.. A survey of Projective geometry projective geometric structures on 2-,3-manifolds

S. Choi I Many geometries are actually sub-geometries of projective (conformal) geometry. (The two Outline correspond to maximal finite-dimensional Lie Classical geometries algebras acting locally on manifolds...) Euclidean geometry n n Spherical geometry I Hyperbolic geometry: H imbeds in P and R Manifolds with PO(n, 1) in PGL(n + 1, R) in canonical way. geometric structures: I spherical, euclidean geometry. manifolds need n n some canonical I The affine geometry G = SL(n, ) · and X = . R R R descriptions.. I Six of eight 3-dimensional geometries: euclidean, ˜ Manifolds with spherical, hyperbolic, nil, sol, SL(2, R) (B. Thiel) (real) projective 2 2 structures I S × R and H × R. (almost) I Some symmetric spaces and matrix groups...

I Recently, some graphics applications.... image superposition,.. A survey of Projective geometry projective geometric structures on 2-,3-manifolds

S. Choi I Many geometries are actually sub-geometries of projective (conformal) geometry. (The two Outline correspond to maximal finite-dimensional Lie Classical geometries algebras acting locally on manifolds...) Euclidean geometry n n Spherical geometry I Hyperbolic geometry: H imbeds in P and R Manifolds with PO(n, 1) in PGL(n + 1, R) in canonical way. geometric structures: I spherical, euclidean geometry. manifolds need n n some canonical I The affine geometry G = SL(n, ) · and X = . R R R descriptions.. I Six of eight 3-dimensional geometries: euclidean, ˜ Manifolds with spherical, hyperbolic, nil, sol, SL(2, R) (B. Thiel) (real) projective 2 2 structures I S × R and H × R. (almost) I Some symmetric spaces and matrix groups...

I Recently, some graphics applications.... image superposition,.. A survey of Projective geometry projective geometric structures on 2-,3-manifolds

S. Choi I Many geometries are actually sub-geometries of projective (conformal) geometry. (The two Outline correspond to maximal finite-dimensional Lie Classical geometries algebras acting locally on manifolds...) Euclidean geometry n n Spherical geometry I Hyperbolic geometry: H imbeds in P and R Manifolds with PO(n, 1) in PGL(n + 1, R) in canonical way. geometric structures: I spherical, euclidean geometry. manifolds need n n some canonical I The affine geometry G = SL(n, ) · and X = . R R R descriptions.. I Six of eight 3-dimensional geometries: euclidean, ˜ Manifolds with spherical, hyperbolic, nil, sol, SL(2, R) (B. Thiel) (real) projective 2 2 structures I S × R and H × R. (almost) I Some symmetric spaces and matrix groups...

I Recently, some graphics applications.... image superposition,.. A survey of Projective geometry projective geometric structures on 2-,3-manifolds

S. Choi I Many geometries are actually sub-geometries of projective (conformal) geometry. (The two Outline correspond to maximal finite-dimensional Lie Classical geometries algebras acting locally on manifolds...) Euclidean geometry n n Spherical geometry I Hyperbolic geometry: H imbeds in P and R Manifolds with PO(n, 1) in PGL(n + 1, R) in canonical way. geometric structures: I spherical, euclidean geometry. manifolds need n n some canonical I The affine geometry G = SL(n, ) · and X = . R R R descriptions.. I Six of eight 3-dimensional geometries: euclidean, ˜ Manifolds with spherical, hyperbolic, nil, sol, SL(2, R) (B. Thiel) (real) projective 2 2 structures I S × R and H × R. (almost) I Some symmetric spaces and matrix groups...

I Recently, some graphics applications.... image superposition,.. A survey of Projective geometry projective geometric structures on 2-,3-manifolds

S. Choi I Many geometries are actually sub-geometries of projective (conformal) geometry. (The two Outline correspond to maximal finite-dimensional Lie Classical geometries algebras acting locally on manifolds...) Euclidean geometry n n Spherical geometry I Hyperbolic geometry: H imbeds in P and R Manifolds with PO(n, 1) in PGL(n + 1, R) in canonical way. geometric structures: I spherical, euclidean geometry. manifolds need n n some canonical I The affine geometry G = SL(n, ) · and X = . R R R descriptions.. I Six of eight 3-dimensional geometries: euclidean, ˜ Manifolds with spherical, hyperbolic, nil, sol, SL(2, R) (B. Thiel) (real) projective 2 2 structures I S × R and H × R. (almost) I Some symmetric spaces and matrix groups...

I Recently, some graphics applications.... image superposition,.. A survey of Projective geometry projective geometric structures on 2-,3-manifolds

S. Choi I Many geometries are actually sub-geometries of projective (conformal) geometry. (The two Outline correspond to maximal finite-dimensional Lie Classical geometries algebras acting locally on manifolds...) Euclidean geometry n n Spherical geometry I Hyperbolic geometry: H imbeds in P and R Manifolds with PO(n, 1) in PGL(n + 1, R) in canonical way. geometric structures: I spherical, euclidean geometry. manifolds need n n some canonical I The affine geometry G = SL(n, ) · and X = . R R R descriptions.. I Six of eight 3-dimensional geometries: euclidean, ˜ Manifolds with spherical, hyperbolic, nil, sol, SL(2, R) (B. Thiel) (real) projective 2 2 structures I S × R and H × R. (almost) I Some symmetric spaces and matrix groups...

I Recently, some graphics applications.... image superposition,.. A survey of Projective geometry projective geometric structures on 2-,3-manifolds

S. Choi I Many geometries are actually sub-geometries of projective (conformal) geometry. (The two Outline correspond to maximal finite-dimensional Lie Classical geometries algebras acting locally on manifolds...) Euclidean geometry n n Spherical geometry I Hyperbolic geometry: H imbeds in P and R Manifolds with PO(n, 1) in PGL(n + 1, R) in canonical way. geometric structures: I spherical, euclidean geometry. manifolds need n n some canonical I The affine geometry G = SL(n, ) · and X = . R R R descriptions.. I Six of eight 3-dimensional geometries: euclidean, ˜ Manifolds with spherical, hyperbolic, nil, sol, SL(2, R) (B. Thiel) (real) projective 2 2 structures I S × R and H × R. (almost) I Some symmetric spaces and matrix groups...

I Recently, some graphics applications.... image superposition,.. A survey of Erlangen program of Klein projective geometric structures on 2-,3-manifolds

S. Choi

Outline

I A Lie group is a set of symmetries of some object Classical forming a manifold. geometries Euclidean geometry Spherical geometry I Klein worked out a general scheme to study almost Manifolds with all geometries... geometric structures: manifolds need I Klein proposed that "geometry" is actually a space some canonical with a Lie group acting on it transitively. descriptions.. Manifolds with I Essentially, the transformation group actually defines (real) projective the geometry by determining which properties are structures preserved. A survey of Erlangen program of Klein projective geometric structures on 2-,3-manifolds

S. Choi

Outline

I A Lie group is a set of symmetries of some object Classical forming a manifold. geometries Euclidean geometry Spherical geometry I Klein worked out a general scheme to study almost Manifolds with all geometries... geometric structures: manifolds need I Klein proposed that "geometry" is actually a space some canonical with a Lie group acting on it transitively. descriptions.. Manifolds with I Essentially, the transformation group actually defines (real) projective the geometry by determining which properties are structures preserved. A survey of Erlangen program of Klein projective geometric structures on 2-,3-manifolds

S. Choi

Outline

I A Lie group is a set of symmetries of some object Classical forming a manifold. geometries Euclidean geometry Spherical geometry I Klein worked out a general scheme to study almost Manifolds with all geometries... geometric structures: manifolds need I Klein proposed that "geometry" is actually a space some canonical with a Lie group acting on it transitively. descriptions.. Manifolds with I Essentially, the transformation group actually defines (real) projective the geometry by determining which properties are structures preserved. A survey of Erlangen program of Klein projective geometric structures on 2-,3-manifolds

S. Choi

Outline

I A Lie group is a set of symmetries of some object Classical forming a manifold. geometries Euclidean geometry Spherical geometry I Klein worked out a general scheme to study almost Manifolds with all geometries... geometric structures: manifolds need I Klein proposed that "geometry" is actually a space some canonical with a Lie group acting on it transitively. descriptions.. Manifolds with I Essentially, the transformation group actually defines (real) projective the geometry by determining which properties are structures preserved. A survey of projective I We extract properties from the pair: geometric structures on I For each pair (X, G) where G is a Lie group and X is 2-,3-manifolds a space. S. Choi n I Thus, Euclidean geometry is given by X = R and Outline G = Isom(Rn) = O(n) · Rn. Classical I The spherical geometry: G = O(n + 1, R) and geometries X = Sn. Euclidean geometry Spherical geometry

I The hyperbolic geometry: G = PO(n, 1) and X upper Manifolds with part of the hyperboloid t2 − x 2 − · · · − x 2 = 1. For geometric 1 n structures: n = 2, we also have G = PSL(2, R) and X the upper manifolds need some canonical half-plane. For n = 3, we also have G = PSL(2, C) descriptions.. 3 and X the upper half-space in R . Manifolds with I Lorentzian space-times.... de Sitter, anti-de-Sitter,... (real) projective structures I The conformal geometry: G = SO(n + 1, 1) and X the celestial sphere in Rn+1,1. n n I The affine geometry G = SL(n, R) · R and X = R . I The projective geometry G = PGL(n + 1, R) and X = RPn. (The conformal and projective geometries are “maximal” geometry in the sense of Klein.) A survey of projective I We extract properties from the pair: geometric structures on I For each pair (X, G) where G is a Lie group and X is 2-,3-manifolds a space. S. Choi n I Thus, Euclidean geometry is given by X = R and Outline G = Isom(Rn) = O(n) · Rn. Classical I The spherical geometry: G = O(n + 1, R) and geometries X = Sn. Euclidean geometry Spherical geometry

I The hyperbolic geometry: G = PO(n, 1) and X upper Manifolds with part of the hyperboloid t2 − x 2 − · · · − x 2 = 1. For geometric 1 n structures: n = 2, we also have G = PSL(2, R) and X the upper manifolds need some canonical half-plane. For n = 3, we also have G = PSL(2, C) descriptions.. 3 and X the upper half-space in R . Manifolds with I Lorentzian space-times.... de Sitter, anti-de-Sitter,... (real) projective structures I The conformal geometry: G = SO(n + 1, 1) and X the celestial sphere in Rn+1,1. n n I The affine geometry G = SL(n, R) · R and X = R . I The projective geometry G = PGL(n + 1, R) and X = RPn. (The conformal and projective geometries are “maximal” geometry in the sense of Klein.) A survey of projective I We extract properties from the pair: geometric structures on I For each pair (X, G) where G is a Lie group and X is 2-,3-manifolds a space. S. Choi n I Thus, Euclidean geometry is given by X = R and Outline G = Isom(Rn) = O(n) · Rn. Classical I The spherical geometry: G = O(n + 1, R) and geometries X = Sn. Euclidean geometry Spherical geometry

I The hyperbolic geometry: G = PO(n, 1) and X upper Manifolds with part of the hyperboloid t2 − x 2 − · · · − x 2 = 1. For geometric 1 n structures: n = 2, we also have G = PSL(2, R) and X the upper manifolds need some canonical half-plane. For n = 3, we also have G = PSL(2, C) descriptions.. 3 and X the upper half-space in R . Manifolds with I Lorentzian space-times.... de Sitter, anti-de-Sitter,... (real) projective structures I The conformal geometry: G = SO(n + 1, 1) and X the celestial sphere in Rn+1,1. n n I The affine geometry G = SL(n, R) · R and X = R . I The projective geometry G = PGL(n + 1, R) and X = RPn. (The conformal and projective geometries are “maximal” geometry in the sense of Klein.) A survey of projective I We extract properties from the pair: geometric structures on I For each pair (X, G) where G is a Lie group and X is 2-,3-manifolds a space. S. Choi n I Thus, Euclidean geometry is given by X = R and Outline G = Isom(Rn) = O(n) · Rn. Classical I The spherical geometry: G = O(n + 1, R) and geometries X = Sn. Euclidean geometry Spherical geometry

I The hyperbolic geometry: G = PO(n, 1) and X upper Manifolds with part of the hyperboloid t2 − x 2 − · · · − x 2 = 1. For geometric 1 n structures: n = 2, we also have G = PSL(2, R) and X the upper manifolds need some canonical half-plane. For n = 3, we also have G = PSL(2, C) descriptions.. 3 and X the upper half-space in R . Manifolds with I Lorentzian space-times.... de Sitter, anti-de-Sitter,... (real) projective structures I The conformal geometry: G = SO(n + 1, 1) and X the celestial sphere in Rn+1,1. n n I The affine geometry G = SL(n, R) · R and X = R . I The projective geometry G = PGL(n + 1, R) and X = RPn. (The conformal and projective geometries are “maximal” geometry in the sense of Klein.) A survey of projective I We extract properties from the pair: geometric structures on I For each pair (X, G) where G is a Lie group and X is 2-,3-manifolds a space. S. Choi n I Thus, Euclidean geometry is given by X = R and Outline G = Isom(Rn) = O(n) · Rn. Classical I The spherical geometry: G = O(n + 1, R) and geometries X = Sn. Euclidean geometry Spherical geometry

I The hyperbolic geometry: G = PO(n, 1) and X upper Manifolds with part of the hyperboloid t2 − x 2 − · · · − x 2 = 1. For geometric 1 n structures: n = 2, we also have G = PSL(2, R) and X the upper manifolds need some canonical half-plane. For n = 3, we also have G = PSL(2, C) descriptions.. 3 and X the upper half-space in R . Manifolds with I Lorentzian space-times.... de Sitter, anti-de-Sitter,... (real) projective structures I The conformal geometry: G = SO(n + 1, 1) and X the celestial sphere in Rn+1,1. n n I The affine geometry G = SL(n, R) · R and X = R . I The projective geometry G = PGL(n + 1, R) and X = RPn. (The conformal and projective geometries are “maximal” geometry in the sense of Klein.) A survey of projective I We extract properties from the pair: geometric structures on I For each pair (X, G) where G is a Lie group and X is 2-,3-manifolds a space. S. Choi n I Thus, Euclidean geometry is given by X = R and Outline G = Isom(Rn) = O(n) · Rn. Classical I The spherical geometry: G = O(n + 1, R) and geometries X = Sn. Euclidean geometry Spherical geometry

I The hyperbolic geometry: G = PO(n, 1) and X upper Manifolds with part of the hyperboloid t2 − x 2 − · · · − x 2 = 1. For geometric 1 n structures: n = 2, we also have G = PSL(2, R) and X the upper manifolds need some canonical half-plane. For n = 3, we also have G = PSL(2, C) descriptions.. 3 and X the upper half-space in R . Manifolds with I Lorentzian space-times.... de Sitter, anti-de-Sitter,... (real) projective structures I The conformal geometry: G = SO(n + 1, 1) and X the celestial sphere in Rn+1,1. n n I The affine geometry G = SL(n, R) · R and X = R . I The projective geometry G = PGL(n + 1, R) and X = RPn. (The conformal and projective geometries are “maximal” geometry in the sense of Klein.) A survey of projective I We extract properties from the pair: geometric structures on I For each pair (X, G) where G is a Lie group and X is 2-,3-manifolds a space. S. Choi n I Thus, Euclidean geometry is given by X = R and Outline G = Isom(Rn) = O(n) · Rn. Classical I The spherical geometry: G = O(n + 1, R) and geometries X = Sn. Euclidean geometry Spherical geometry

I The hyperbolic geometry: G = PO(n, 1) and X upper Manifolds with part of the hyperboloid t2 − x 2 − · · · − x 2 = 1. For geometric 1 n structures: n = 2, we also have G = PSL(2, R) and X the upper manifolds need some canonical half-plane. For n = 3, we also have G = PSL(2, C) descriptions.. 3 and X the upper half-space in R . Manifolds with I Lorentzian space-times.... de Sitter, anti-de-Sitter,... (real) projective structures I The conformal geometry: G = SO(n + 1, 1) and X the celestial sphere in Rn+1,1. n n I The affine geometry G = SL(n, R) · R and X = R . I The projective geometry G = PGL(n + 1, R) and X = RPn. (The conformal and projective geometries are “maximal” geometry in the sense of Klein.) A survey of projective I We extract properties from the pair: geometric structures on I For each pair (X, G) where G is a Lie group and X is 2-,3-manifolds a space. S. Choi n I Thus, Euclidean geometry is given by X = R and Outline G = Isom(Rn) = O(n) · Rn. Classical I The spherical geometry: G = O(n + 1, R) and geometries X = Sn. Euclidean geometry Spherical geometry

I The hyperbolic geometry: G = PO(n, 1) and X upper Manifolds with part of the hyperboloid t2 − x 2 − · · · − x 2 = 1. For geometric 1 n structures: n = 2, we also have G = PSL(2, R) and X the upper manifolds need some canonical half-plane. For n = 3, we also have G = PSL(2, C) descriptions.. 3 and X the upper half-space in R . Manifolds with I Lorentzian space-times.... de Sitter, anti-de-Sitter,... (real) projective structures I The conformal geometry: G = SO(n + 1, 1) and X the celestial sphere in Rn+1,1. n n I The affine geometry G = SL(n, R) · R and X = R . I The projective geometry G = PGL(n + 1, R) and X = RPn. (The conformal and projective geometries are “maximal” geometry in the sense of Klein.) A survey of projective I We extract properties from the pair: geometric structures on I For each pair (X, G) where G is a Lie group and X is 2-,3-manifolds a space. S. Choi n I Thus, Euclidean geometry is given by X = R and Outline G = Isom(Rn) = O(n) · Rn. Classical I The spherical geometry: G = O(n + 1, R) and geometries X = Sn. Euclidean geometry Spherical geometry

I The hyperbolic geometry: G = PO(n, 1) and X upper Manifolds with part of the hyperboloid t2 − x 2 − · · · − x 2 = 1. For geometric 1 n structures: n = 2, we also have G = PSL(2, R) and X the upper manifolds need some canonical half-plane. For n = 3, we also have G = PSL(2, C) descriptions.. 3 and X the upper half-space in R . Manifolds with I Lorentzian space-times.... de Sitter, anti-de-Sitter,... (real) projective structures I The conformal geometry: G = SO(n + 1, 1) and X the celestial sphere in Rn+1,1. n n I The affine geometry G = SL(n, R) · R and X = R . I The projective geometry G = PGL(n + 1, R) and X = RPn. (The conformal and projective geometries are “maximal” geometry in the sense of Klein.) A survey of Cartan, Ehresmann connections: Most projective geometric structures on general geometry possible 2-,3-manifolds

I Cartan proposed that for each pair (X, G), we can S. Choi

impart their properties to at each point of the Outline

manifolds so that they vary also. The flatness implies Classical geometries that the (X, G)-geometry is actually recovered. Euclidean geometry Spherical geometry I Ehresmann introduced the most general notion of Manifolds with connections by generalizing Cartan connections. geometric structures: I For example, for euclidean geometry, a Cartan manifolds need some canonical connection gives us Riemannian geometry. descriptions.. I For projective geometry, a Cartan connection Manifolds with (real) projective corresponds to a projectively flat torsion-free affine structures connections and conversely. I Wilson Stothers’ Geometry Pages I Reference: Projective and Cayley-Klein Geometries (Springer Monographs in Mathematics) (Hardcover) by Arkadij L. Onishchik (Author), Rolf Sulanke (Author) A survey of Cartan, Ehresmann connections: Most projective geometric structures on general geometry possible 2-,3-manifolds

I Cartan proposed that for each pair (X, G), we can S. Choi

impart their properties to at each point of the Outline

manifolds so that they vary also. The flatness implies Classical geometries that the (X, G)-geometry is actually recovered. Euclidean geometry Spherical geometry I Ehresmann introduced the most general notion of Manifolds with connections by generalizing Cartan connections. geometric structures: I For example, for euclidean geometry, a Cartan manifolds need some canonical connection gives us Riemannian geometry. descriptions.. I For projective geometry, a Cartan connection Manifolds with (real) projective corresponds to a projectively flat torsion-free affine structures connections and conversely. I Wilson Stothers’ Geometry Pages I Reference: Projective and Cayley-Klein Geometries (Springer Monographs in Mathematics) (Hardcover) by Arkadij L. Onishchik (Author), Rolf Sulanke (Author) A survey of Cartan, Ehresmann connections: Most projective geometric structures on general geometry possible 2-,3-manifolds

I Cartan proposed that for each pair (X, G), we can S. Choi

impart their properties to at each point of the Outline

manifolds so that they vary also. The flatness implies Classical geometries that the (X, G)-geometry is actually recovered. Euclidean geometry Spherical geometry I Ehresmann introduced the most general notion of Manifolds with connections by generalizing Cartan connections. geometric structures: I For example, for euclidean geometry, a Cartan manifolds need some canonical connection gives us Riemannian geometry. descriptions.. I For projective geometry, a Cartan connection Manifolds with (real) projective corresponds to a projectively flat torsion-free affine structures connections and conversely. I Wilson Stothers’ Geometry Pages I Reference: Projective and Cayley-Klein Geometries (Springer Monographs in Mathematics) (Hardcover) by Arkadij L. Onishchik (Author), Rolf Sulanke (Author) A survey of Cartan, Ehresmann connections: Most projective geometric structures on general geometry possible 2-,3-manifolds

I Cartan proposed that for each pair (X, G), we can S. Choi

impart their properties to at each point of the Outline

manifolds so that they vary also. The flatness implies Classical geometries that the (X, G)-geometry is actually recovered. Euclidean geometry Spherical geometry I Ehresmann introduced the most general notion of Manifolds with connections by generalizing Cartan connections. geometric structures: I For example, for euclidean geometry, a Cartan manifolds need some canonical connection gives us Riemannian geometry. descriptions.. I For projective geometry, a Cartan connection Manifolds with (real) projective corresponds to a projectively flat torsion-free affine structures connections and conversely. I Wilson Stothers’ Geometry Pages I Reference: Projective and Cayley-Klein Geometries (Springer Monographs in Mathematics) (Hardcover) by Arkadij L. Onishchik (Author), Rolf Sulanke (Author) A survey of Cartan, Ehresmann connections: Most projective geometric structures on general geometry possible 2-,3-manifolds

I Cartan proposed that for each pair (X, G), we can S. Choi

impart their properties to at each point of the Outline

manifolds so that they vary also. The flatness implies Classical geometries that the (X, G)-geometry is actually recovered. Euclidean geometry Spherical geometry I Ehresmann introduced the most general notion of Manifolds with connections by generalizing Cartan connections. geometric structures: I For example, for euclidean geometry, a Cartan manifolds need some canonical connection gives us Riemannian geometry. descriptions.. I For projective geometry, a Cartan connection Manifolds with (real) projective corresponds to a projectively flat torsion-free affine structures connections and conversely. I Wilson Stothers’ Geometry Pages I Reference: Projective and Cayley-Klein Geometries (Springer Monographs in Mathematics) (Hardcover) by Arkadij L. Onishchik (Author), Rolf Sulanke (Author) A survey of Cartan, Ehresmann connections: Most projective geometric structures on general geometry possible 2-,3-manifolds

I Cartan proposed that for each pair (X, G), we can S. Choi

impart their properties to at each point of the Outline

manifolds so that they vary also. The flatness implies Classical geometries that the (X, G)-geometry is actually recovered. Euclidean geometry Spherical geometry I Ehresmann introduced the most general notion of Manifolds with connections by generalizing Cartan connections. geometric structures: I For example, for euclidean geometry, a Cartan manifolds need some canonical connection gives us Riemannian geometry. descriptions.. I For projective geometry, a Cartan connection Manifolds with (real) projective corresponds to a projectively flat torsion-free affine structures connections and conversely. I Wilson Stothers’ Geometry Pages I Reference: Projective and Cayley-Klein Geometries (Springer Monographs in Mathematics) (Hardcover) by Arkadij L. Onishchik (Author), Rolf Sulanke (Author) A survey of Manifolds with geometric structures: projective geometric structures on 2-,3-manifolds

S. Choi Topology of manifolds Outline

Classical I Manifolds come up in many areas of technology and geometries Euclidean geometry sciences. Spherical geometry I Studying the topological structures of manifolds is Manifolds with geometric complicated by the fact that there is no uniform way structures: manifolds need to describe many topologically important features some canonical descriptions.. and provides useful coordinates. Manifolds with (real) projective I We would like to find some good descriptions and structures perhaps even classify collections of manifolds.

I Of course these are for pure-mathematical uses for now.... A survey of Manifolds with geometric structures: projective geometric structures on 2-,3-manifolds

S. Choi Topology of manifolds Outline

Classical I Manifolds come up in many areas of technology and geometries Euclidean geometry sciences. Spherical geometry I Studying the topological structures of manifolds is Manifolds with geometric complicated by the fact that there is no uniform way structures: manifolds need to describe many topologically important features some canonical descriptions.. and provides useful coordinates. Manifolds with (real) projective I We would like to find some good descriptions and structures perhaps even classify collections of manifolds.

I Of course these are for pure-mathematical uses for now.... A survey of Manifolds with geometric structures: projective geometric structures on 2-,3-manifolds

S. Choi Topology of manifolds Outline

Classical I Manifolds come up in many areas of technology and geometries Euclidean geometry sciences. Spherical geometry I Studying the topological structures of manifolds is Manifolds with geometric complicated by the fact that there is no uniform way structures: manifolds need to describe many topologically important features some canonical descriptions.. and provides useful coordinates. Manifolds with (real) projective I We would like to find some good descriptions and structures perhaps even classify collections of manifolds.

I Of course these are for pure-mathematical uses for now.... A survey of Manifolds with geometric structures: projective geometric structures on 2-,3-manifolds

S. Choi Topology of manifolds Outline

Classical I Manifolds come up in many areas of technology and geometries Euclidean geometry sciences. Spherical geometry I Studying the topological structures of manifolds is Manifolds with geometric complicated by the fact that there is no uniform way structures: manifolds need to describe many topologically important features some canonical descriptions.. and provides useful coordinates. Manifolds with (real) projective I We would like to find some good descriptions and structures perhaps even classify collections of manifolds.

I Of course these are for pure-mathematical uses for now.... A survey of projective geometric structures on 2-,3-manifolds

S. Choi I An (X, G)-geometric structure on a manifold M is given by an atlas of charts to X where the transition Outline maps are in G. Classical geometries This equips M with all of the local (X, G)-geometrical Euclidean geometry I Spherical geometry

notions. Manifolds with geometric I If the geometry admits notions such as geodesic, structures: manifolds need length, angle, cross ratio, then M now has such some canonical notions... descriptions.. Manifolds with (real) projective I So the central question is: which manifolds admit structures which structures and how many and if geometric structures do not exists, why not? A survey of projective geometric structures on 2-,3-manifolds

S. Choi I An (X, G)-geometric structure on a manifold M is given by an atlas of charts to X where the transition Outline maps are in G. Classical geometries This equips M with all of the local (X, G)-geometrical Euclidean geometry I Spherical geometry

notions. Manifolds with geometric I If the geometry admits notions such as geodesic, structures: manifolds need length, angle, cross ratio, then M now has such some canonical notions... descriptions.. Manifolds with (real) projective I So the central question is: which manifolds admit structures which structures and how many and if geometric structures do not exists, why not? A survey of projective geometric structures on 2-,3-manifolds

S. Choi I An (X, G)-geometric structure on a manifold M is given by an atlas of charts to X where the transition Outline maps are in G. Classical geometries This equips M with all of the local (X, G)-geometrical Euclidean geometry I Spherical geometry

notions. Manifolds with geometric I If the geometry admits notions such as geodesic, structures: manifolds need length, angle, cross ratio, then M now has such some canonical notions... descriptions.. Manifolds with (real) projective I So the central question is: which manifolds admit structures which structures and how many and if geometric structures do not exists, why not? A survey of projective geometric structures on 2-,3-manifolds

S. Choi I An (X, G)-geometric structure on a manifold M is given by an atlas of charts to X where the transition Outline maps are in G. Classical geometries This equips M with all of the local (X, G)-geometrical Euclidean geometry I Spherical geometry

notions. Manifolds with geometric I If the geometry admits notions such as geodesic, structures: manifolds need length, angle, cross ratio, then M now has such some canonical notions... descriptions.. Manifolds with (real) projective I So the central question is: which manifolds admit structures which structures and how many and if geometric structures do not exists, why not? A survey of projective geometric

H structures on 2-,3-manifolds b1' G a2 K S. Choi

b2 a1' Outline

Classical F A B<)b1a1'= 0.248 a2' geometries

b1 Euclidean geometry Spherical geometry

Manifolds with E b2' C geometric a1 structures: D manifolds need some canonical descriptions..

Manifolds with (real) projective structures A survey of Deformation spaces of geometric structures projective geometric structures on 2-,3-manifolds I In major cases, M = X/Γ for a discrete subgroup Γ in the Lie group G. S. Choi Outline I Thus, X provides a global coordinate system and the Classical classification of discrete subgroups of G provides geometries Euclidean geometry classifications of manifolds like M. Spherical geometry Manifolds with I Here Γ tends to be the fundamental group of M. geometric structures: Thus, a geometric structure can be considered a manifolds need π ( ) some canonical discrete representation of 1 M in G. descriptions.. I Often, there are cases when Γ is unique up to Manifolds with (real) projective conjugations and there is a unique (X, G)-structure. structures (Rigidity)

I Given an (X, G)-manifold (orbifold) M, the deformation space D(X,G)(M) is locally homeomorphic to the G-representation space Hom(π1(M), G)/G of π1(M). A survey of Deformation spaces of geometric structures projective geometric structures on 2-,3-manifolds I In major cases, M = X/Γ for a discrete subgroup Γ in the Lie group G. S. Choi Outline I Thus, X provides a global coordinate system and the Classical classification of discrete subgroups of G provides geometries Euclidean geometry classifications of manifolds like M. Spherical geometry Manifolds with I Here Γ tends to be the fundamental group of M. geometric structures: Thus, a geometric structure can be considered a manifolds need π ( ) some canonical discrete representation of 1 M in G. descriptions.. I Often, there are cases when Γ is unique up to Manifolds with (real) projective conjugations and there is a unique (X, G)-structure. structures (Rigidity)

I Given an (X, G)-manifold (orbifold) M, the deformation space D(X,G)(M) is locally homeomorphic to the G-representation space Hom(π1(M), G)/G of π1(M). A survey of Deformation spaces of geometric structures projective geometric structures on 2-,3-manifolds I In major cases, M = X/Γ for a discrete subgroup Γ in the Lie group G. S. Choi Outline I Thus, X provides a global coordinate system and the Classical classification of discrete subgroups of G provides geometries Euclidean geometry classifications of manifolds like M. Spherical geometry Manifolds with I Here Γ tends to be the fundamental group of M. geometric structures: Thus, a geometric structure can be considered a manifolds need π ( ) some canonical discrete representation of 1 M in G. descriptions.. I Often, there are cases when Γ is unique up to Manifolds with (real) projective conjugations and there is a unique (X, G)-structure. structures (Rigidity)

I Given an (X, G)-manifold (orbifold) M, the deformation space D(X,G)(M) is locally homeomorphic to the G-representation space Hom(π1(M), G)/G of π1(M). A survey of Deformation spaces of geometric structures projective geometric structures on 2-,3-manifolds I In major cases, M = X/Γ for a discrete subgroup Γ in the Lie group G. S. Choi Outline I Thus, X provides a global coordinate system and the Classical classification of discrete subgroups of G provides geometries Euclidean geometry classifications of manifolds like M. Spherical geometry Manifolds with I Here Γ tends to be the fundamental group of M. geometric structures: Thus, a geometric structure can be considered a manifolds need π ( ) some canonical discrete representation of 1 M in G. descriptions.. I Often, there are cases when Γ is unique up to Manifolds with (real) projective conjugations and there is a unique (X, G)-structure. structures (Rigidity)

I Given an (X, G)-manifold (orbifold) M, the deformation space D(X,G)(M) is locally homeomorphic to the G-representation space Hom(π1(M), G)/G of π1(M). A survey of Deformation spaces of geometric structures projective geometric structures on 2-,3-manifolds I In major cases, M = X/Γ for a discrete subgroup Γ in the Lie group G. S. Choi Outline I Thus, X provides a global coordinate system and the Classical classification of discrete subgroups of G provides geometries Euclidean geometry classifications of manifolds like M. Spherical geometry Manifolds with I Here Γ tends to be the fundamental group of M. geometric structures: Thus, a geometric structure can be considered a manifolds need π ( ) some canonical discrete representation of 1 M in G. descriptions.. I Often, there are cases when Γ is unique up to Manifolds with (real) projective conjugations and there is a unique (X, G)-structure. structures (Rigidity)

I Given an (X, G)-manifold (orbifold) M, the deformation space D(X,G)(M) is locally homeomorphic to the G-representation space Hom(π1(M), G)/G of π1(M). A survey of Examples projective geometric I Closed surfaces have either a spherical, euclidean, structures on or hyperbolic structures depending on genus. We 2-,3-manifolds can classify these to form deformation spaces such S. Choi as Teichmuller spaces. Outline 2 I A hyperbolic surface equals H /Γ for the image Γ of Classical the representation π → PSL(2, ). geometries R Euclidean geometry I A Teichmuller space can be identified with a Spherical geometry component of the space Hom(π, PSL(2, ))/ ∼ of Manifolds with R geometric conjugacy classes of representations. The structures: manifolds need component consists of discrete faithful some canonical representations. descriptions.. Manifolds with I For closed 3-manifolds, it is recently proved that they (real) projective decompose into pieces admitting one of eight structures geometrical structures including hyperbolic, euclidean, spherical ones... Thus, these manifolds are now being classified... (Proof of the geometrization conjecture by Perelman). I For higher-dimensional manifolds, there are other types of geometric structures... A survey of Examples projective geometric I Closed surfaces have either a spherical, euclidean, structures on or hyperbolic structures depending on genus. We 2-,3-manifolds can classify these to form deformation spaces such S. Choi as Teichmuller spaces. Outline 2 I A hyperbolic surface equals H /Γ for the image Γ of Classical the representation π → PSL(2, ). geometries R Euclidean geometry I A Teichmuller space can be identified with a Spherical geometry component of the space Hom(π, PSL(2, ))/ ∼ of Manifolds with R geometric conjugacy classes of representations. The structures: manifolds need component consists of discrete faithful some canonical representations. descriptions.. Manifolds with I For closed 3-manifolds, it is recently proved that they (real) projective decompose into pieces admitting one of eight structures geometrical structures including hyperbolic, euclidean, spherical ones... Thus, these manifolds are now being classified... (Proof of the geometrization conjecture by Perelman). I For higher-dimensional manifolds, there are other types of geometric structures... A survey of Examples projective geometric I Closed surfaces have either a spherical, euclidean, structures on or hyperbolic structures depending on genus. We 2-,3-manifolds can classify these to form deformation spaces such S. Choi as Teichmuller spaces. Outline 2 I A hyperbolic surface equals H /Γ for the image Γ of Classical the representation π → PSL(2, ). geometries R Euclidean geometry I A Teichmuller space can be identified with a Spherical geometry component of the space Hom(π, PSL(2, ))/ ∼ of Manifolds with R geometric conjugacy classes of representations. The structures: manifolds need component consists of discrete faithful some canonical representations. descriptions.. Manifolds with I For closed 3-manifolds, it is recently proved that they (real) projective decompose into pieces admitting one of eight structures geometrical structures including hyperbolic, euclidean, spherical ones... Thus, these manifolds are now being classified... (Proof of the geometrization conjecture by Perelman). I For higher-dimensional manifolds, there are other types of geometric structures... A survey of Examples projective geometric I Closed surfaces have either a spherical, euclidean, structures on or hyperbolic structures depending on genus. We 2-,3-manifolds can classify these to form deformation spaces such S. Choi as Teichmuller spaces. Outline 2 I A hyperbolic surface equals H /Γ for the image Γ of Classical the representation π → PSL(2, ). geometries R Euclidean geometry I A Teichmuller space can be identified with a Spherical geometry component of the space Hom(π, PSL(2, ))/ ∼ of Manifolds with R geometric conjugacy classes of representations. The structures: manifolds need component consists of discrete faithful some canonical representations. descriptions.. Manifolds with I For closed 3-manifolds, it is recently proved that they (real) projective decompose into pieces admitting one of eight structures geometrical structures including hyperbolic, euclidean, spherical ones... Thus, these manifolds are now being classified... (Proof of the geometrization conjecture by Perelman). I For higher-dimensional manifolds, there are other types of geometric structures... A survey of Examples projective geometric I Closed surfaces have either a spherical, euclidean, structures on or hyperbolic structures depending on genus. We 2-,3-manifolds can classify these to form deformation spaces such S. Choi as Teichmuller spaces. Outline 2 I A hyperbolic surface equals H /Γ for the image Γ of Classical the representation π → PSL(2, ). geometries R Euclidean geometry I A Teichmuller space can be identified with a Spherical geometry component of the space Hom(π, PSL(2, ))/ ∼ of Manifolds with R geometric conjugacy classes of representations. The structures: manifolds need component consists of discrete faithful some canonical representations. descriptions.. Manifolds with I For closed 3-manifolds, it is recently proved that they (real) projective decompose into pieces admitting one of eight structures geometrical structures including hyperbolic, euclidean, spherical ones... Thus, these manifolds are now being classified... (Proof of the geometrization conjecture by Perelman). I For higher-dimensional manifolds, there are other types of geometric structures... A survey of Manifolds with (real) projective structures projective geometric structures on 2-,3-manifolds

S. Choi I Cartan first considered projective structures on surfaces as projectively flat torsion-free connections Outline Classical on surfaces. geometries Euclidean geometry Spherical geometry I Chern worked on general type of projective Manifolds with structures in the differential geometry point of view. geometric structures: I The study of affine manifolds precede that of manifolds need some canonical projective manifolds. descriptions..

I There were extensive work on affine manifolds, and Manifolds with (real) projective affine Lie groups by Chern, Auslander, Goldman, structures Fried, Smillie, Nagano, Yagi, Shima, Carriere, Margulis,... I There are outstanding questions such as the Chern conjecture, Auslander conjecture,... A survey of Manifolds with (real) projective structures projective geometric structures on 2-,3-manifolds

S. Choi I Cartan first considered projective structures on surfaces as projectively flat torsion-free connections Outline Classical on surfaces. geometries Euclidean geometry Spherical geometry I Chern worked on general type of projective Manifolds with structures in the differential geometry point of view. geometric structures: I The study of affine manifolds precede that of manifolds need some canonical projective manifolds. descriptions..

I There were extensive work on affine manifolds, and Manifolds with (real) projective affine Lie groups by Chern, Auslander, Goldman, structures Fried, Smillie, Nagano, Yagi, Shima, Carriere, Margulis,... I There are outstanding questions such as the Chern conjecture, Auslander conjecture,... A survey of Manifolds with (real) projective structures projective geometric structures on 2-,3-manifolds

S. Choi I Cartan first considered projective structures on surfaces as projectively flat torsion-free connections Outline Classical on surfaces. geometries Euclidean geometry Spherical geometry I Chern worked on general type of projective Manifolds with structures in the differential geometry point of view. geometric structures: I The study of affine manifolds precede that of manifolds need some canonical projective manifolds. descriptions..

I There were extensive work on affine manifolds, and Manifolds with (real) projective affine Lie groups by Chern, Auslander, Goldman, structures Fried, Smillie, Nagano, Yagi, Shima, Carriere, Margulis,... I There are outstanding questions such as the Chern conjecture, Auslander conjecture,... A survey of Manifolds with (real) projective structures projective geometric structures on 2-,3-manifolds

S. Choi I Cartan first considered projective structures on surfaces as projectively flat torsion-free connections Outline Classical on surfaces. geometries Euclidean geometry Spherical geometry I Chern worked on general type of projective Manifolds with structures in the differential geometry point of view. geometric structures: I The study of affine manifolds precede that of manifolds need some canonical projective manifolds. descriptions..

I There were extensive work on affine manifolds, and Manifolds with (real) projective affine Lie groups by Chern, Auslander, Goldman, structures Fried, Smillie, Nagano, Yagi, Shima, Carriere, Margulis,... I There are outstanding questions such as the Chern conjecture, Auslander conjecture,... A survey of Manifolds with (real) projective structures projective geometric structures on 2-,3-manifolds

S. Choi I Cartan first considered projective structures on surfaces as projectively flat torsion-free connections Outline Classical on surfaces. geometries Euclidean geometry Spherical geometry I Chern worked on general type of projective Manifolds with structures in the differential geometry point of view. geometric structures: I The study of affine manifolds precede that of manifolds need some canonical projective manifolds. descriptions..

I There were extensive work on affine manifolds, and Manifolds with (real) projective affine Lie groups by Chern, Auslander, Goldman, structures Fried, Smillie, Nagano, Yagi, Shima, Carriere, Margulis,... I There are outstanding questions such as the Chern conjecture, Auslander conjecture,... A survey of Projective manifolds: how many? projective geometric structures on I By the Klein-Beltrami model of hyperbolic space, we 2-,3-manifolds can consider Hn as a unit ball B in an affine S. Choi n subspace of RP and PO(n + 1, R) as a subgroup of Outline PGL(n + 1, R) acting on B. Thus, a (complete) Classical geometries hyperbolic manifold has a projective structure. (In Euclidean geometry fact any closed surface has one.) Spherical geometry Manifolds with geometric I In fact, 3-manifolds with six types of geometric structures: manifolds need structures have real projective structures. (In fact, some canonical up to coverings of order two, 3-manifolds with eight descriptions.. Manifolds with geometric structures have real projective structures.) (real) projective structures I A question arises whether these projective structures can be deformed to purely projective structures.

I Deformed projective manifolds from closed hyperbolic ones are convex in the sense that any path can be homotoped to a geodesic path. (Koszul’s openness) A survey of Projective manifolds: how many? projective geometric structures on I By the Klein-Beltrami model of hyperbolic space, we 2-,3-manifolds can consider Hn as a unit ball B in an affine S. Choi n subspace of RP and PO(n + 1, R) as a subgroup of Outline PGL(n + 1, R) acting on B. Thus, a (complete) Classical geometries hyperbolic manifold has a projective structure. (In Euclidean geometry fact any closed surface has one.) Spherical geometry Manifolds with geometric I In fact, 3-manifolds with six types of geometric structures: manifolds need structures have real projective structures. (In fact, some canonical up to coverings of order two, 3-manifolds with eight descriptions.. Manifolds with geometric structures have real projective structures.) (real) projective structures I A question arises whether these projective structures can be deformed to purely projective structures.

I Deformed projective manifolds from closed hyperbolic ones are convex in the sense that any path can be homotoped to a geodesic path. (Koszul’s openness) A survey of Projective manifolds: how many? projective geometric structures on I By the Klein-Beltrami model of hyperbolic space, we 2-,3-manifolds can consider Hn as a unit ball B in an affine S. Choi n subspace of RP and PO(n + 1, R) as a subgroup of Outline PGL(n + 1, R) acting on B. Thus, a (complete) Classical geometries hyperbolic manifold has a projective structure. (In Euclidean geometry fact any closed surface has one.) Spherical geometry Manifolds with geometric I In fact, 3-manifolds with six types of geometric structures: manifolds need structures have real projective structures. (In fact, some canonical up to coverings of order two, 3-manifolds with eight descriptions.. Manifolds with geometric structures have real projective structures.) (real) projective structures I A question arises whether these projective structures can be deformed to purely projective structures.

I Deformed projective manifolds from closed hyperbolic ones are convex in the sense that any path can be homotoped to a geodesic path. (Koszul’s openness) A survey of Projective manifolds: how many? projective geometric structures on I By the Klein-Beltrami model of hyperbolic space, we 2-,3-manifolds can consider Hn as a unit ball B in an affine S. Choi n subspace of RP and PO(n + 1, R) as a subgroup of Outline PGL(n + 1, R) acting on B. Thus, a (complete) Classical geometries hyperbolic manifold has a projective structure. (In Euclidean geometry fact any closed surface has one.) Spherical geometry Manifolds with geometric I In fact, 3-manifolds with six types of geometric structures: manifolds need structures have real projective structures. (In fact, some canonical up to coverings of order two, 3-manifolds with eight descriptions.. Manifolds with geometric structures have real projective structures.) (real) projective structures I A question arises whether these projective structures can be deformed to purely projective structures.

I Deformed projective manifolds from closed hyperbolic ones are convex in the sense that any path can be homotoped to a geodesic path. (Koszul’s openness) A survey of projective geometric structures on 2-,3-manifolds I In 1960s, Benzecri started working on strictly convex S. Choi domain Ω where a projective transformation group Γ acted with a compact quotient. That is, M = Ω/Γ is a Outline Classical compact manifold (orbifold). (Related to convex geometries Euclidean geometry cones and group transformations (Kuiper, Spherical geometry

Koszul,Vinberg,...)). He showed that the boundary is Manifolds with 1 geometric either C or is an ellipsoid. structures: manifolds need I Kac-Vinberg found the first example for n = 2 for a some canonical descriptions.. triangle reflection group associated with Kac-Moody Manifolds with algebra. (real) projective structures I Recently, Benoist found many interesting properties such as the fact that Ω is strictly convex if and only if the group Γ is Gromov-hyperbolic. A survey of projective geometric structures on 2-,3-manifolds I In 1960s, Benzecri started working on strictly convex S. Choi domain Ω where a projective transformation group Γ acted with a compact quotient. That is, M = Ω/Γ is a Outline Classical compact manifold (orbifold). (Related to convex geometries Euclidean geometry cones and group transformations (Kuiper, Spherical geometry

Koszul,Vinberg,...)). He showed that the boundary is Manifolds with 1 geometric either C or is an ellipsoid. structures: manifolds need I Kac-Vinberg found the first example for n = 2 for a some canonical descriptions.. triangle reflection group associated with Kac-Moody Manifolds with algebra. (real) projective structures I Recently, Benoist found many interesting properties such as the fact that Ω is strictly convex if and only if the group Γ is Gromov-hyperbolic. A survey of projective geometric structures on 2-,3-manifolds I In 1960s, Benzecri started working on strictly convex S. Choi domain Ω where a projective transformation group Γ acted with a compact quotient. That is, M = Ω/Γ is a Outline Classical compact manifold (orbifold). (Related to convex geometries Euclidean geometry cones and group transformations (Kuiper, Spherical geometry

Koszul,Vinberg,...)). He showed that the boundary is Manifolds with 1 geometric either C or is an ellipsoid. structures: manifolds need I Kac-Vinberg found the first example for n = 2 for a some canonical descriptions.. triangle reflection group associated with Kac-Moody Manifolds with algebra. (real) projective structures I Recently, Benoist found many interesting properties such as the fact that Ω is strictly convex if and only if the group Γ is Gromov-hyperbolic. A survey of Projective surfaces projective geometric structures on 2-,3-manifolds

S. Choi

Outline I For closed surfaces, in 80s, Goldman found a Classical geometries general dimension counting method for the Euclidean geometry nonsingular part of the surface-group representation Spherical geometry Manifolds with space into the Lie group G which is dim G × (2g − 2) geometric structures: if g ≥ 2. manifolds need some canonical Since the deformation space D 2 (S) for a descriptions.. I (RP ,PGL(3,R)) Manifolds with closed surface S is locally homeomorphic to (real) projective structures Hom(π1(S), PGL(3, R))/PGL(3, R), we know the dimension of the deformation space to be 8(2g − 2). A survey of Projective surfaces projective geometric structures on 2-,3-manifolds

S. Choi

Outline I For closed surfaces, in 80s, Goldman found a Classical geometries general dimension counting method for the Euclidean geometry nonsingular part of the surface-group representation Spherical geometry Manifolds with space into the Lie group G which is dim G × (2g − 2) geometric structures: if g ≥ 2. manifolds need some canonical Since the deformation space D 2 (S) for a descriptions.. I (RP ,PGL(3,R)) Manifolds with closed surface S is locally homeomorphic to (real) projective structures Hom(π1(S), PGL(3, R))/PGL(3, R), we know the dimension of the deformation space to be 8(2g − 2). A survey of projective geometric I Goldman found that the the deformation space of structures on 2-,3-manifolds convex projective structures is a cell of dimension S. Choi 16g − 16. Deformations (The real pro. str. on hyp. mflds. Outline Classical I Choi showed that if g ≥ 2, then a projective surface geometries Euclidean geometry always decomposes into convex projective surfaces Spherical geometry and annuli along disjoint closed geodesics. (Some Manifolds with geometric are not convex) structures: manifolds need some canonical I The annuli were classified by Nagano, Yagi, and descriptions..

Goldman earlier. Manifolds with (real) projective I Using this, we have a complete classification of structures projective structures on closed surfaces (even constructive one).

I For 2-orbifolds, Goldman and Choi completed the classification. developing images A survey of projective geometric I Goldman found that the the deformation space of structures on 2-,3-manifolds convex projective structures is a cell of dimension S. Choi 16g − 16. Deformations (The real pro. str. on hyp. mflds. Outline Classical I Choi showed that if g ≥ 2, then a projective surface geometries Euclidean geometry always decomposes into convex projective surfaces Spherical geometry and annuli along disjoint closed geodesics. (Some Manifolds with geometric are not convex) structures: manifolds need some canonical I The annuli were classified by Nagano, Yagi, and descriptions..

Goldman earlier. Manifolds with (real) projective I Using this, we have a complete classification of structures projective structures on closed surfaces (even constructive one).

I For 2-orbifolds, Goldman and Choi completed the classification. developing images A survey of projective geometric I Goldman found that the the deformation space of structures on 2-,3-manifolds convex projective structures is a cell of dimension S. Choi 16g − 16. Deformations (The real pro. str. on hyp. mflds. Outline Classical I Choi showed that if g ≥ 2, then a projective surface geometries Euclidean geometry always decomposes into convex projective surfaces Spherical geometry and annuli along disjoint closed geodesics. (Some Manifolds with geometric are not convex) structures: manifolds need some canonical I The annuli were classified by Nagano, Yagi, and descriptions..

Goldman earlier. Manifolds with (real) projective I Using this, we have a complete classification of structures projective structures on closed surfaces (even constructive one).

I For 2-orbifolds, Goldman and Choi completed the classification. developing images A survey of projective geometric I Goldman found that the the deformation space of structures on 2-,3-manifolds convex projective structures is a cell of dimension S. Choi 16g − 16. Deformations (The real pro. str. on hyp. mflds. Outline Classical I Choi showed that if g ≥ 2, then a projective surface geometries Euclidean geometry always decomposes into convex projective surfaces Spherical geometry and annuli along disjoint closed geodesics. (Some Manifolds with geometric are not convex) structures: manifolds need some canonical I The annuli were classified by Nagano, Yagi, and descriptions..

Goldman earlier. Manifolds with (real) projective I Using this, we have a complete classification of structures projective structures on closed surfaces (even constructive one).

I For 2-orbifolds, Goldman and Choi completed the classification. developing images A survey of projective geometric I Goldman found that the the deformation space of structures on 2-,3-manifolds convex projective structures is a cell of dimension S. Choi 16g − 16. Deformations (The real pro. str. on hyp. mflds. Outline Classical I Choi showed that if g ≥ 2, then a projective surface geometries Euclidean geometry always decomposes into convex projective surfaces Spherical geometry and annuli along disjoint closed geodesics. (Some Manifolds with geometric are not convex) structures: manifolds need some canonical I The annuli were classified by Nagano, Yagi, and descriptions..

Goldman earlier. Manifolds with (real) projective I Using this, we have a complete classification of structures projective structures on closed surfaces (even constructive one).

I For 2-orbifolds, Goldman and Choi completed the classification. developing images A survey of Projective surfaces and gauge theory projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical I Atiyah and Hitchin studied self-dual connections on geometries surfaces (70s) Euclidean geometry Spherical geometry I Given a Lie group G, a representation of π1(S) → G Manifolds with geometric can be thought of as a flat G-connection on a structures: manifolds need principal G-bundle over S and vice versa. some canonical descriptions..

I Corlette showed that flat G-connections for manifolds Manifolds with (real) projective (80s) can be realized as harmonic maps to certain structures associated symmetric space bundles. A survey of Projective surfaces and gauge theory projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical I Atiyah and Hitchin studied self-dual connections on geometries surfaces (70s) Euclidean geometry Spherical geometry I Given a Lie group G, a representation of π1(S) → G Manifolds with geometric can be thought of as a flat G-connection on a structures: manifolds need principal G-bundle over S and vice versa. some canonical descriptions..

I Corlette showed that flat G-connections for manifolds Manifolds with (real) projective (80s) can be realized as harmonic maps to certain structures associated symmetric space bundles. A survey of Projective surfaces and gauge theory projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical I Atiyah and Hitchin studied self-dual connections on geometries surfaces (70s) Euclidean geometry Spherical geometry I Given a Lie group G, a representation of π1(S) → G Manifolds with geometric can be thought of as a flat G-connection on a structures: manifolds need principal G-bundle over S and vice versa. some canonical descriptions..

I Corlette showed that flat G-connections for manifolds Manifolds with (real) projective (80s) can be realized as harmonic maps to certain structures associated symmetric space bundles. A survey of projective geometric structures on 2-,3-manifolds I Hitchin used Higgs field on principal G-bundles over S. Choi surfaces to obtain parametrizations of flat Outline

G-connections over surfaces. (G is a real split form Classical of a reductive group.) (90s) geometries Euclidean geometry Spherical geometry I A Teichmüller space Manifolds with geometric structures: T (Σ) = {hyperbolic structures on Σ}/isotopies manifolds need some canonical descriptions.. is a component of Manifolds with (real) projective + structures Hom (π1(Σ), PSL(2, R))/PSL(2, R)

of Fuchian representations. A survey of projective geometric structures on 2-,3-manifolds I Hitchin used Higgs field on principal G-bundles over S. Choi surfaces to obtain parametrizations of flat Outline

G-connections over surfaces. (G is a real split form Classical of a reductive group.) (90s) geometries Euclidean geometry Spherical geometry I A Teichmüller space Manifolds with geometric structures: T (Σ) = {hyperbolic structures on Σ}/isotopies manifolds need some canonical descriptions.. is a component of Manifolds with (real) projective + structures Hom (π1(Σ), PSL(2, R))/PSL(2, R)

of Fuchian representations. A survey of projective geometric structures on I Hitchin-Teichmüller components: 2-,3-manifolds

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

Classical gives a component of geometries Euclidean geometry + Spherical geometry Hom (π1(Σ), G)/G. Manifolds with geometric structures: manifolds need some canonical I The Hitchin-Teichmüller component is descriptions.. r homeomorphic to a cell of dimension (2g − 2) dim G . Manifolds with (real) projective I For n > 2, structures

+ Hom (π1(Σ), PGL(n, R))/PGL(n, R) has three connected components if n is odd and six components if n is even. A survey of projective geometric structures on I Hitchin-Teichmüller components: 2-,3-manifolds

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

Classical gives a component of geometries Euclidean geometry + Spherical geometry Hom (π1(Σ), G)/G. Manifolds with geometric structures: manifolds need some canonical I The Hitchin-Teichmüller component is descriptions.. r homeomorphic to a cell of dimension (2g − 2) dim G . Manifolds with (real) projective I For n > 2, structures

+ Hom (π1(Σ), PGL(n, R))/PGL(n, R) has three connected components if n is odd and six components if n is even. A survey of projective geometric structures on I Hitchin-Teichmüller components: 2-,3-manifolds

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

Classical gives a component of geometries Euclidean geometry + Spherical geometry Hom (π1(Σ), G)/G. Manifolds with geometric structures: manifolds need some canonical I The Hitchin-Teichmüller component is descriptions.. r homeomorphic to a cell of dimension (2g − 2) dim G . Manifolds with (real) projective I For n > 2, structures

+ Hom (π1(Σ), PGL(n, R))/PGL(n, R) has three connected components if n is odd and six components if n is even. A survey of projective geometric structures on I A convex projective surface is of form Ω/Γ. Hence, 2-,3-manifolds there is a representation π1(Σ) → Γ determined only S. Choi up to conjugation by PGL(3, ). This gives us a map R Outline

Classical hol : D(Σ) → Hom(π1(Σ), PGL(3, R))/ ∼ . geometries Euclidean geometry Spherical geometry

This map is known to be a local-homeomorphism Manifolds with geometric (Ehresmann, Thurston) and is injective (Goldman) structures: manifolds need I The map is in fact a homeomorphism onto some canonical descriptions.. Hitchin-Teichmüller component (Goldman, Choi) The Manifolds with main idea for proof is to show that the image of the (real) projective structures map is closed.

I As a consequence, we know that the Hitchin-Teichmuller component for PGL(3, R)) consists of discrete faithful representations only. A survey of projective geometric structures on I A convex projective surface is of form Ω/Γ. Hence, 2-,3-manifolds there is a representation π1(Σ) → Γ determined only S. Choi up to conjugation by PGL(3, ). This gives us a map R Outline

Classical hol : D(Σ) → Hom(π1(Σ), PGL(3, R))/ ∼ . geometries Euclidean geometry Spherical geometry

This map is known to be a local-homeomorphism Manifolds with geometric (Ehresmann, Thurston) and is injective (Goldman) structures: manifolds need I The map is in fact a homeomorphism onto some canonical descriptions.. Hitchin-Teichmüller component (Goldman, Choi) The Manifolds with main idea for proof is to show that the image of the (real) projective structures map is closed.

I As a consequence, we know that the Hitchin-Teichmuller component for PGL(3, R)) consists of discrete faithful representations only. A survey of projective geometric structures on I A convex projective surface is of form Ω/Γ. Hence, 2-,3-manifolds there is a representation π1(Σ) → Γ determined only S. Choi up to conjugation by PGL(3, ). This gives us a map R Outline

Classical hol : D(Σ) → Hom(π1(Σ), PGL(3, R))/ ∼ . geometries Euclidean geometry Spherical geometry

This map is known to be a local-homeomorphism Manifolds with geometric (Ehresmann, Thurston) and is injective (Goldman) structures: manifolds need I The map is in fact a homeomorphism onto some canonical descriptions.. Hitchin-Teichmüller component (Goldman, Choi) The Manifolds with main idea for proof is to show that the image of the (real) projective structures map is closed.

I As a consequence, we know that the Hitchin-Teichmuller component for PGL(3, R)) consists of discrete faithful representations only. A survey of Generalizations projective geometric structures on 2-,3-manifolds

S. Choi

Outline I Labourie recently generalized this to n ≥ 2, 3: That Classical the component of representations that can be geometries Euclidean geometry deformed to the Fuschian representation acts on a Spherical geometry n−1 hyperconvex curve in RP . Manifolds with geometric structures: I Corollary: Every Hitchin representation is a discrete manifolds need faithful and “purely loxodromic”. The mapping class some canonical descriptions..

group acts properly on H(n). Manifolds with (real) projective I Burger, Iozzi, Labourie, Wienhard generalized this structures result to maximal representations into other Lie groups. A survey of Generalizations projective geometric structures on 2-,3-manifolds

S. Choi

Outline I Labourie recently generalized this to n ≥ 2, 3: That Classical the component of representations that can be geometries Euclidean geometry deformed to the Fuschian representation acts on a Spherical geometry n−1 hyperconvex curve in RP . Manifolds with geometric structures: I Corollary: Every Hitchin representation is a discrete manifolds need faithful and “purely loxodromic”. The mapping class some canonical descriptions..

group acts properly on H(n). Manifolds with (real) projective I Burger, Iozzi, Labourie, Wienhard generalized this structures result to maximal representations into other Lie groups. A survey of Generalizations projective geometric structures on 2-,3-manifolds

S. Choi

Outline I Labourie recently generalized this to n ≥ 2, 3: That Classical the component of representations that can be geometries Euclidean geometry deformed to the Fuschian representation acts on a Spherical geometry n−1 hyperconvex curve in RP . Manifolds with geometric structures: I Corollary: Every Hitchin representation is a discrete manifolds need faithful and “purely loxodromic”. The mapping class some canonical descriptions..

group acts properly on H(n). Manifolds with (real) projective I Burger, Iozzi, Labourie, Wienhard generalized this structures result to maximal representations into other Lie groups. A survey of Projective surfaces and affine differential projective geometric structures on geometry 2-,3-manifolds

S. Choi I John Loftin (simultaneously Labourie) showed using the work of Calabi, Cheng-Yau, and C.P. Wang on Outline affine spheres: Let Mn be a closed projectively flat Classical geometries manifold. Euclidean geometry Spherical geometry I Then M is convex if and only if M admits an affine n+1 Manifolds with sphere structure in R . geometric structures: I For n = 2, the affine sphere structure is equivalent to manifolds need some canonical the conformal structure on M with a holomorphic descriptions.. cubic-differential. Manifolds with (real) projective I In particular, this shows that the deformation space structures D(Σ) of convex projective structures on Σ admits a complex structure, which is preserved under the moduli group actions. (Is it Kahler?) I Loftin also worked out Mumford type compactifications of the moduli space M(Σ) of convex projective structures. A survey of Projective surfaces and affine differential projective geometric structures on geometry 2-,3-manifolds

S. Choi I John Loftin (simultaneously Labourie) showed using the work of Calabi, Cheng-Yau, and C.P. Wang on Outline affine spheres: Let Mn be a closed projectively flat Classical geometries manifold. Euclidean geometry Spherical geometry I Then M is convex if and only if M admits an affine n+1 Manifolds with sphere structure in R . geometric structures: I For n = 2, the affine sphere structure is equivalent to manifolds need some canonical the conformal structure on M with a holomorphic descriptions.. cubic-differential. Manifolds with (real) projective I In particular, this shows that the deformation space structures D(Σ) of convex projective structures on Σ admits a complex structure, which is preserved under the moduli group actions. (Is it Kahler?) I Loftin also worked out Mumford type compactifications of the moduli space M(Σ) of convex projective structures. A survey of Projective surfaces and affine differential projective geometric structures on geometry 2-,3-manifolds

S. Choi I John Loftin (simultaneously Labourie) showed using the work of Calabi, Cheng-Yau, and C.P. Wang on Outline affine spheres: Let Mn be a closed projectively flat Classical geometries manifold. Euclidean geometry Spherical geometry I Then M is convex if and only if M admits an affine n+1 Manifolds with sphere structure in R . geometric structures: I For n = 2, the affine sphere structure is equivalent to manifolds need some canonical the conformal structure on M with a holomorphic descriptions.. cubic-differential. Manifolds with (real) projective I In particular, this shows that the deformation space structures D(Σ) of convex projective structures on Σ admits a complex structure, which is preserved under the moduli group actions. (Is it Kahler?) I Loftin also worked out Mumford type compactifications of the moduli space M(Σ) of convex projective structures. A survey of Projective surfaces and affine differential projective geometric structures on geometry 2-,3-manifolds

S. Choi I John Loftin (simultaneously Labourie) showed using the work of Calabi, Cheng-Yau, and C.P. Wang on Outline affine spheres: Let Mn be a closed projectively flat Classical geometries manifold. Euclidean geometry Spherical geometry I Then M is convex if and only if M admits an affine n+1 Manifolds with sphere structure in R . geometric structures: I For n = 2, the affine sphere structure is equivalent to manifolds need some canonical the conformal structure on M with a holomorphic descriptions.. cubic-differential. Manifolds with (real) projective I In particular, this shows that the deformation space structures D(Σ) of convex projective structures on Σ admits a complex structure, which is preserved under the moduli group actions. (Is it Kahler?) I Loftin also worked out Mumford type compactifications of the moduli space M(Σ) of convex projective structures. A survey of Projective surfaces and affine differential projective geometric structures on geometry 2-,3-manifolds

S. Choi I John Loftin (simultaneously Labourie) showed using the work of Calabi, Cheng-Yau, and C.P. Wang on Outline affine spheres: Let Mn be a closed projectively flat Classical geometries manifold. Euclidean geometry Spherical geometry I Then M is convex if and only if M admits an affine n+1 Manifolds with sphere structure in R . geometric structures: I For n = 2, the affine sphere structure is equivalent to manifolds need some canonical the conformal structure on M with a holomorphic descriptions.. cubic-differential. Manifolds with (real) projective I In particular, this shows that the deformation space structures D(Σ) of convex projective structures on Σ admits a complex structure, which is preserved under the moduli group actions. (Is it Kahler?) I Loftin also worked out Mumford type compactifications of the moduli space M(Σ) of convex projective structures. A survey of Projective 3-manifolds and deformations projective geometric structures on 2-,3-manifolds I We saw that many 3-manifolds has projective S. Choi structures. 3 3 Outline I Cooper showed that RP #RP does not admit a real Classical projective structures. (answers Bill Goldman’s geometries Euclidean geometry question). Other examples? Spherical geometry Manifolds with I We wish to understand the deformation spaces for geometric structures: higher-dimensional manifolds and so on. (The manifolds need some canonical deformation theory is still very difficult for conformal descriptions..

structures as well.) Manifolds with (real) projective I There is a well-known construction of Apanasov structures using bending: deforming along a closed totally geodesic hypersurface.

I Johnson and Millson found deformations of higher-dimensional hyperbolic manifolds which are locally singular. (also bending constructions) A survey of Projective 3-manifolds and deformations projective geometric structures on 2-,3-manifolds I We saw that many 3-manifolds has projective S. Choi structures. 3 3 Outline I Cooper showed that RP #RP does not admit a real Classical projective structures. (answers Bill Goldman’s geometries Euclidean geometry question). Other examples? Spherical geometry Manifolds with I We wish to understand the deformation spaces for geometric structures: higher-dimensional manifolds and so on. (The manifolds need some canonical deformation theory is still very difficult for conformal descriptions..

structures as well.) Manifolds with (real) projective I There is a well-known construction of Apanasov structures using bending: deforming along a closed totally geodesic hypersurface.

I Johnson and Millson found deformations of higher-dimensional hyperbolic manifolds which are locally singular. (also bending constructions) A survey of Projective 3-manifolds and deformations projective geometric structures on 2-,3-manifolds I We saw that many 3-manifolds has projective S. Choi structures. 3 3 Outline I Cooper showed that RP #RP does not admit a real Classical projective structures. (answers Bill Goldman’s geometries Euclidean geometry question). Other examples? Spherical geometry Manifolds with I We wish to understand the deformation spaces for geometric structures: higher-dimensional manifolds and so on. (The manifolds need some canonical deformation theory is still very difficult for conformal descriptions..

structures as well.) Manifolds with (real) projective I There is a well-known construction of Apanasov structures using bending: deforming along a closed totally geodesic hypersurface.

I Johnson and Millson found deformations of higher-dimensional hyperbolic manifolds which are locally singular. (also bending constructions) A survey of Projective 3-manifolds and deformations projective geometric structures on 2-,3-manifolds I We saw that many 3-manifolds has projective S. Choi structures. 3 3 Outline I Cooper showed that RP #RP does not admit a real Classical projective structures. (answers Bill Goldman’s geometries Euclidean geometry question). Other examples? Spherical geometry Manifolds with I We wish to understand the deformation spaces for geometric structures: higher-dimensional manifolds and so on. (The manifolds need some canonical deformation theory is still very difficult for conformal descriptions..

structures as well.) Manifolds with (real) projective I There is a well-known construction of Apanasov structures using bending: deforming along a closed totally geodesic hypersurface.

I Johnson and Millson found deformations of higher-dimensional hyperbolic manifolds which are locally singular. (also bending constructions) A survey of Projective 3-manifolds and deformations projective geometric structures on 2-,3-manifolds I We saw that many 3-manifolds has projective S. Choi structures. 3 3 Outline I Cooper showed that RP #RP does not admit a real Classical projective structures. (answers Bill Goldman’s geometries Euclidean geometry question). Other examples? Spherical geometry Manifolds with I We wish to understand the deformation spaces for geometric structures: higher-dimensional manifolds and so on. (The manifolds need some canonical deformation theory is still very difficult for conformal descriptions..

structures as well.) Manifolds with (real) projective I There is a well-known construction of Apanasov structures using bending: deforming along a closed totally geodesic hypersurface.

I Johnson and Millson found deformations of higher-dimensional hyperbolic manifolds which are locally singular. (also bending constructions) A survey of projective I Cooper,Long and Thistlethwaite found a numerical geometric structures on (some exact algebraic) evidences that out of the first 2-,3-manifolds

1000 closed hyperbolic 3-manifolds in the S. Choi Hodgson-Weeks census, a handful admit non-trivial Outline deformations of their SO+(3, 1)-representations into Classical SL(4, ); each resulting representation variety then geometries R Euclidean geometry gives rise to a family of real projective structures on Spherical geometry Manifolds with the manifold. geometric structures: I Benoist and Choi found some class of 3-dimensional manifolds need some canonical reflection orbifold that has positive dimensional descriptions.. deformation spaces. But we haven’t been able to Manifolds with (real) projective study the wider class of reflection orbifolds. (prisms, structures pyramids, icosahedron,...)

I We have been working algebraically and numerically for simple polytopes.

I So far, there are no Gauge theoretic or affine sphere approach to 3-dimensional projective manifolds. A survey of projective I Cooper,Long and Thistlethwaite found a numerical geometric structures on (some exact algebraic) evidences that out of the first 2-,3-manifolds

1000 closed hyperbolic 3-manifolds in the S. Choi Hodgson-Weeks census, a handful admit non-trivial Outline deformations of their SO+(3, 1)-representations into Classical SL(4, ); each resulting representation variety then geometries R Euclidean geometry gives rise to a family of real projective structures on Spherical geometry Manifolds with the manifold. geometric structures: I Benoist and Choi found some class of 3-dimensional manifolds need some canonical reflection orbifold that has positive dimensional descriptions.. deformation spaces. But we haven’t been able to Manifolds with (real) projective study the wider class of reflection orbifolds. (prisms, structures pyramids, icosahedron,...)

I We have been working algebraically and numerically for simple polytopes.

I So far, there are no Gauge theoretic or affine sphere approach to 3-dimensional projective manifolds. A survey of projective I Cooper,Long and Thistlethwaite found a numerical geometric structures on (some exact algebraic) evidences that out of the first 2-,3-manifolds

1000 closed hyperbolic 3-manifolds in the S. Choi Hodgson-Weeks census, a handful admit non-trivial Outline deformations of their SO+(3, 1)-representations into Classical SL(4, ); each resulting representation variety then geometries R Euclidean geometry gives rise to a family of real projective structures on Spherical geometry Manifolds with the manifold. geometric structures: I Benoist and Choi found some class of 3-dimensional manifolds need some canonical reflection orbifold that has positive dimensional descriptions.. deformation spaces. But we haven’t been able to Manifolds with (real) projective study the wider class of reflection orbifolds. (prisms, structures pyramids, icosahedron,...)

I We have been working algebraically and numerically for simple polytopes.

I So far, there are no Gauge theoretic or affine sphere approach to 3-dimensional projective manifolds. A survey of projective I Cooper,Long and Thistlethwaite found a numerical geometric structures on (some exact algebraic) evidences that out of the first 2-,3-manifolds

1000 closed hyperbolic 3-manifolds in the S. Choi Hodgson-Weeks census, a handful admit non-trivial Outline deformations of their SO+(3, 1)-representations into Classical SL(4, ); each resulting representation variety then geometries R Euclidean geometry gives rise to a family of real projective structures on Spherical geometry Manifolds with the manifold. geometric structures: I Benoist and Choi found some class of 3-dimensional manifolds need some canonical reflection orbifold that has positive dimensional descriptions.. deformation spaces. But we haven’t been able to Manifolds with (real) projective study the wider class of reflection orbifolds. (prisms, structures pyramids, icosahedron,...)

I We have been working algebraically and numerically for simple polytopes.

I So far, there are no Gauge theoretic or affine sphere approach to 3-dimensional projective manifolds. A survey of projective geometric structures on 2-,3-manifolds

S. Choi

Outline

Classical geometries Euclidean geometry Spherical geometry

Manifolds with geometric structures: manifolds need some canonical descriptions..

Manifolds with (real) projective structures