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