A Survey Of Projective Geometric Structures On 2-,3-manifolds - KAIST
A survey of projective geometric structures on 2-,3-manifolds S. Choi A survey of projective geometric structures on 2-,3-manifolds Outline Classical geometries Euclidean geometry Spherical geometry S. Choi Department of Mathematical Science KAIST, Daejeon, South Korea Tokyo Institute of Technology Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
A survey of projective geometric structures on 2-,3-manifolds Outline S. Choi I Survey: Classical geometries I I I I I I Euclidean geometry: Babylonians, Egyptians, Greeks, Chinese (Euclid’s axiomatic methods under Plato’s philosphy) Spherical geometry: Greek astronomy, Gauss, Riemann Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, Beltrami,Klein, Poincare Conformal geometry (Mobius geometry or circle geometry) Projective geometry Erlanger program, Cartan connections, Ehresmann connections Outline Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
A survey of projective geometric structures on 2-,3-manifolds Outline S. Choi I Survey: Classical geometries I I I I I I Euclidean geometry: Babylonians, Egyptians, Greeks, Chinese (Euclid’s axiomatic methods under Plato’s philosphy) Spherical geometry: Greek astronomy, Gauss, Riemann Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, Beltrami,Klein, Poincare Conformal geometry (Mobius geometry or circle geometry) Projective geometry Erlanger program, Cartan connections, Ehresmann connections Outline Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
A survey of projective geometric structures on 2-,3-manifolds Outline S. Choi I Survey: Classical geometries I I I I I I Euclidean geometry: Babylonians, Egyptians, Greeks, Chinese (Euclid’s axiomatic methods under Plato’s philosphy) Spherical geometry: Greek astronomy, Gauss, Riemann Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, Beltrami,Klein, Poincare Conformal geometry (Mobius geometry or circle geometry) Projective geometry Erlanger program, Cartan connections, Ehresmann connections Outline Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
A survey of projective geometric structures on 2-,3-manifolds Outline S. Choi I Survey: Classical geometries I I I I I I Euclidean geometry: Babylonians, Egyptians, Greeks, Chinese (Euclid’s axiomatic methods under Plato’s philosphy) Spherical geometry: Greek astronomy, Gauss, Riemann Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, Beltrami,Klein, Poincare Conformal geometry (Mobius geometry or circle geometry) Projective geometry Erlanger program, Cartan connections, Ehresmann connections Outline Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
A survey of projective geometric structures on 2-,3-manifolds Outline S. Choi I Survey: Classical geometries I I I I I I Euclidean geometry: Babylonians, Egyptians, Greeks, Chinese (Euclid’s axiomatic methods under Plato’s philosphy) Spherical geometry: Greek astronomy, Gauss, Riemann Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, Beltrami,Klein, Poincare Conformal geometry (Mobius geometry or circle geometry) Projective geometry Erlanger program, Cartan connections, Ehresmann connections Outline Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
A survey of projective geometric structures on 2-,3-manifolds Outline S. Choi I Survey: Classical geometries I I I I I I Euclidean geometry: Babylonians, Egyptians, Greeks, Chinese (Euclid’s axiomatic methods under Plato’s philosphy) Spherical geometry: Greek astronomy, Gauss, Riemann Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, Beltrami,Klein, Poincare Conformal geometry (Mobius geometry or circle geometry) Projective geometry Erlanger program, Cartan connections, Ehresmann connections Outline Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
A survey of projective geometric structures on 2-,3-manifolds Outline S. Choi I Survey: Classical geometries I I I I I I Euclidean geometry: Babylonians, Egyptians, Greeks, Chinese (Euclid’s axiomatic methods under Plato’s philosphy) Spherical geometry: Greek astronomy, Gauss, Riemann Hyperbolic geometry: Bolyai, Lobachevsky, Gauss, Beltrami,Klein, Poincare Conformal geometry (Mobius geometry or circle geometry) Projective geometry Erlanger program, Cartan connections, Ehresmann connections Outline Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Manifolds with geometric structures A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline Classical geometries Euclidean geometry I Manifolds with geometric structures: manifolds need some canonical descriptions. I I I Manifolds with geometric structures. Deformation spaces of geometric structures. Examples Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Manifolds with geometric structures A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline Classical geometries Euclidean geometry I Manifolds with geometric structures: manifolds need some canonical descriptions. I I I Manifolds with geometric structures. Deformation spaces of geometric structures. Examples Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Manifolds with geometric structures A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline Classical geometries Euclidean geometry I Manifolds with geometric structures: manifolds need some canonical descriptions. I I I Manifolds with geometric structures. Deformation spaces of geometric structures. Examples Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Manifolds with geometric structures A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline Classical geometries Euclidean geometry I Manifolds with geometric structures: manifolds need some canonical descriptions. I I I Manifolds with geometric structures. Deformation spaces of geometric structures. Examples Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Manifolds with projective structures A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline I Some history. I Projective manifolds: how many? Projective surfaces I I I I I Projective surfaces and gauge theory. Labourie’s generalization Projective surfaces and affine differential geometry. Projective 3-manifolds and deformations Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Manifolds with projective structures A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline I Some history. I Projective manifolds: how many? Projective surfaces I I I I I Projective surfaces and gauge theory. Labourie’s generalization Projective surfaces and affine differential geometry. Projective 3-manifolds and deformations Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Manifolds with projective structures A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline I Some history. I Projective manifolds: how many? Projective surfaces I I I I I Projective surfaces and gauge theory. Labourie’s generalization Projective surfaces and affine differential geometry. Projective 3-manifolds and deformations Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Manifolds with projective structures A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline I Some history. I Projective manifolds: how many? Projective surfaces I I I I I Projective surfaces and gauge theory. Labourie’s generalization Projective surfaces and affine differential geometry. Projective 3-manifolds and deformations Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Manifolds with projective structures A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline I Some history. I Projective manifolds: how many? Projective surfaces I I I I I Projective surfaces and gauge theory. Labourie’s generalization Projective surfaces and affine differential geometry. Projective 3-manifolds and deformations Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Manifolds with projective structures A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline I Some history. I Projective manifolds: how many? Projective surfaces I I I I I Projective surfaces and gauge theory. Labourie’s generalization Projective surfaces and affine differential geometry. Projective 3-manifolds and deformations Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Manifolds with projective structures A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline I Some history. I Projective manifolds: how many? Projective surfaces I I I I I Projective surfaces and gauge theory. Labourie’s generalization Projective surfaces and affine differential geometry. Projective 3-manifolds and deformations Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Euclidean geometry A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline I Babylonian, Egyptian, Chinese, Greek. I Euclid developed his axiomatic method to planar and solid geometry under the influence of Plato, who thought that geometry should be the foundation of all thought after the Pythagorian attempt to understand the world using rational numbers failed. Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Euclidean geometry A survey of projective geometric structures on 2-,3-manifolds S. Choi Outline I Babylonian, Egyptian, Chinese, Greek. I Euclid developed his axiomatic method to planar and solid geometry under the influence of Plato, who thought that geometry should be the foundation of all thought after the Pythagorian attempt to understand the world using rational numbers failed. Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
I Euclidean geometry consists of some notions such as lines, points, lengths, angle, and their interplay in some space called a plane. I I I I I The Euclid had five axioms D. Hilbert, and many others made a modern foundation so that the Euclidean geometry was reduced to logic. Euclidean geometry has a notion of rigid transformations which made the space homogeneous. They form a group called a group of rigid motions. They preserve lines, length, angles, and every geometric statements. The group is useful in proving statements. Turning it around, we see that actually the transformation group is more important. wallpaper groups Notions of Euclidean subspaces. 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
I Euclidean geometry consists of some notions such as lines, points, lengths, angle, and their interplay in some space called a plane. I I I I I The Euclid had five axioms D. Hilbert, and many others made a modern foundation so that the Euclidean geometry was reduced to logic. Euclidean geometry has a notion of rigid transformations which made the space homogeneous. They form a group called a group of rigid motions. They preserve lines, length, angles, and every geometric statements. The group is useful in proving statements. Turning it around, we see that actually the transformation group is more important. wallpaper groups Notions of Euclidean subspaces. 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
I Euclidean geometry consists of some notions such as lines, points, lengths, angle, and their interplay in some space called a plane. I I I I I The Euclid had five axioms D. Hilbert, and many others made a modern foundation so that the Euclidean geometry was reduced to logic. Euclidean geometry has a notion of rigid transformations which made the space homogeneous. They form a group called a group of rigid motions. They preserve lines, length, angles, and every geometric statements. The group is useful in proving statements. Turning it around, we see that actually the transformation group is more important. wallpaper groups Notions of Euclidean subspaces. 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
I Euclidean geometry consists of some notions such as lines, points, lengths, angle, and their interplay in some space called a plane. I I I I I The Euclid had five axioms D. Hilbert, and many others made a modern foundation so that the Euclidean geometry was reduced to logic. Euclidean geometry has a notion of rigid transformations which made the space homogeneous. They form a group called a group of rigid motions. They preserve lines, length, angles, and every geometric statements. The group is useful in proving statements. Turning it around, we see that actually the transformation group is more important. wallpaper groups Notions of Euclidean subspaces. 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
I Euclidean geometry consists of some notions such as lines, points, lengths, angle, and their interplay in some space called a plane. I I I I I The Euclid had five axioms D. Hilbert, and many others made a modern foundation so that the Euclidean geometry was reduced to logic. Euclidean geometry has a notion of rigid transformations which made the space homogeneous. They form a group called a group of rigid motions. They preserve lines, length, angles, and every geometric statements. The group is useful in proving statements. Turning it around, we see that actually the transformation group is more important. wallpaper groups Notions of Euclidean subspaces. 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
I Euclidean geometry consists of some notions such as lines, points, lengths, angle, and their interplay in some space called a plane. I I I I I The Euclid had five axioms D. Hilbert, and many others made a modern foundation so that the Euclidean geometry was reduced to logic. Euclidean geometry has a notion of rigid transformations which made the space homogeneous. They form a group called a group of rigid motions. They preserve lines, length, angles, and every geometric statements. The group is useful in proving statements. Turning it around, we see that actually the transformation group is more important. wallpaper groups Notions of Euclidean subspaces. 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
Spherical geometry. I Greeks: astronomical, navigational. Arabs,. I From astronomical viewpoint, it is nice to view the sky as a unit sphere S2 in the Euclidean 3-space. (See spherical.cdy) I The great circles replaced lines and angles are measured in the tangential sense. Lengths are measured by taking arcs in the great circles. I Geometric objects such as triangles behave a little bit different. I Higher dimensional spherical geometry Sn 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 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
Spherical geometry. I Greeks: astronomical, navigational. Arabs,. I From astronomical viewpoint, it is nice to view the sky as a unit sphere S2 in the Euclidean 3-space. (See spherical.cdy) I The great circles replaced lines and angles are measured in the tangential sense. Lengths are measured by taking arcs in the great circles. I Geometric objects such as triangles behave a little bit different. I Higher dimensional spherical geometry Sn 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 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
Spherical geometry. I Greeks: astronomical, navigational. Arabs,. I From astronomical viewpoint, it is nice to view the sky as a unit sphere S2 in the Euclidean 3-space. (See spherical.cdy) I The great circles replaced lines and angles are measured in the tangential sense. Lengths are measured by taking arcs in the great circles. I Geometric objects such as triangles behave a little bit different. I Higher dimensional spherical geometry Sn 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 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
Spherical geometry. I Greeks: astronomical, navigational. Arabs,. I From astronomical viewpoint, it is nice to view the sky as a unit sphere S2 in the Euclidean 3-space. (See spherical.cdy) I The great circles replaced lines and angles are measured in the tangential sense. Lengths are measured by taking arcs in the great circles. I Geometric objects such as triangles behave a little bit different. I Higher dimensional spherical geometry Sn 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 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
Spherical geometry. I Greeks: astronomical, navigational. Arabs,. I From astronomical viewpoint, it is nice to view the sky as a unit sphere S2 in the Euclidean 3-space. (See spherical.cdy) I The great circles replaced lines and angles are measured in the tangential sense. Lengths are measured by taking arcs in the great circles. I Geometric objects such as triangles behave a little bit different. I Higher dimensional spherical geometry Sn 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 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
Spherical geometry. I Greeks: astronomical, navigational. Arabs,. I From astronomical viewpoint, it is nice to view the sky as a unit sphere S2 in the Euclidean 3-space. (See spherical.cdy) I The great circles replaced lines and angles are measured in the tangential sense. Lengths are measured by taking arcs in the great circles. I Geometric objects such as triangles behave a little bit different. I Higher dimensional spherical geometry Sn 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 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
Hyperbolic geometry. I I I I Lobachevsky and Bolyai tried to build a geometry that did not satisfy the fifth axiom of Euclid. (See hyperbolic.cdy) Their attempts were justified by Beltrami-Klein model which is a disk and lines were replaced by chords and lengths were given by the logarithms of cross ratios. See Beltrami-Klein model. Later other models such as Poincare half space model and Poincare disk model were developed. Poincare model (Inst. figuring). Here the group of rigid motions is the Lie group SL(2, R). 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 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
Hyperbolic geometry. I I I I Lobachevsky and Bolyai tried to build a geometry that did not satisfy the fifth axiom of Euclid. (See hyperbolic.cdy) Their attempts were justified by Beltrami-Klein model which is a disk and lines were replaced by chords and lengths were given by the logarithms of cross ratios. See Beltrami-Klein model. Later other models such as Poincare half space model and Poincare disk model were developed. Poincare model (Inst. figuring). Here the group of rigid motions is the Lie group SL(2, R). 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 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
Hyperbolic geometry. I I I I Lobachevsky and Bolyai tried to build a geometry that did not satisfy the fifth axiom of Euclid. (See hyperbolic.cdy) Their attempts were justified by Beltrami-Klein model which is a disk and lines were replaced by chords and lengths were given by the logarithms of cross ratios. See Beltrami-Klein model. Later other models such as Poincare half space model and Poincare disk model were developed. Poincare model (Inst. figuring). Here the group of rigid motions is the Lie group SL(2, R). 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 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
Hyperbolic geometry. I I I I Lobachevsky and Bolyai tried to build a geometry that did not satisfy the fifth axiom of Euclid. (See hyperbolic.cdy) Their attempts were justified by Beltrami-Klein model which is a disk and lines were replaced by chords and lengths were given by the logarithms of cross ratios. See Beltrami-Klein model. Later other models such as Poincare half space model and Poincare disk model were developed. Poincare model (Inst. figuring). Here the group of rigid motions is the Lie group SL(2, R). 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 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
Conformal geometry I I I Suppose that we use to study circles and spheres only. We allow all transformations that preserves circles. This geometry looses a notion of lengths but has a notion of angles. There are no lines or geodesics but there are circles. The Euclidean plane is compactifed by adding a unique point as an infinity. The group of motions is generated by inversions in circles. The group is called the Mobius transformation group. That is, the group of transformations of form z I az b az̄ b , cz d c z̄ d 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 Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Conformal geometry I I I Suppose that we use to study circles and spheres only. We allow all transformations that preserves circles. This geometry looses a notion of lengths but has a notion of angles. There are no lines or geodesics but there are circles. The Euclidean plane is compactifed by adding a unique point as an infinity. The group of motions is generated by inversions in circles. The group is called the Mobius transformation group. That is, the group of transformations of form z I az b az̄ b , cz d c z̄ d 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 Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Conformal geometry I I I Suppose that we use to study circles and spheres only. We allow all transformations that preserves circles. This geometry looses a notion of lengths but has a notion of angles. There are no lines or geodesics but there are circles. The Euclidean plane is compactifed by adding a unique point as an infinity. The group of motions is generated by inversions in circles. The group is called the Mobius transformation group. That is, the group of transformations of form z I az b az̄ b , cz d c z̄ d 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 Classical geometries Euclidean geometry Spherical geometry Manifolds with geometric structures: manifolds need some canonical descriptions. Manifolds with (real) projective structures
Conformal geometry I I I Suppose that we use to study circles and spheres only. We allow all transformations that preserves circles. This geometry looses a notion of lengths but has a notion of angles. There are no lines or geodesics but there ar
I Projective geometry I Erlanger program, Cartan connections, Ehresmann connections. 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
projective properties of gures and the invariance by projection. This is the rst treaty on projective geometry: a projective property is a prop-erty invariant by projection. Chasles et M obius study the most general Grenoble Universities 3File Size: 725KBPage Count: 27Explore furtherChapter 5 Basics of Projective Geometrywww.cis.upenn.eduLecture 1: Introduction to Projective Geometrymath.fsu.edu1 Projective spaces - Queen Mary University of Londonwww.maths.qmul.ac.ukProjective Spaceswww.fen.bilkent.edu.trReal Projective Space: An Abstract Manifoldmath.uchicago.eduRecommended to you b
Projective Geometry In projective geometry there are no parallel lines. Any two lines in a common plane must intersect! The usual Euclidean plane is contained in what we call the real projective plane. To construct the real projective plane we need to introduce several new points and one new line which contains them all to the Euclidean plane.
What are the orders such that projective planes can be constructed? IIf n is a prime power then projective planes can always be constructed. IIf not, then we have no idea. Conjecture If n is not prime power then there is no projective plane with order n. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, 2016 15 / 22
maximality that projective di erential invariants, such as the Schwarzian derivative, are uniquely determined by their invariance properties. Once projective geometry used to be a core subject in university curricu-lum and, as late as the rst half of the XX-th century, projective di erential geometry was a cutting edge geometric research.
5.2 Projective Spaces 107 5.2 Projective Spaces As in the case of affine geometry, our presentation of projective geometry is rather sketchy and biased toward the algorithmic geometry of curvesandsurfaces.Fora systematic treatment of File Size: 849KB
The Beltrami-Klein model of the real hyperbolic space H2 is an open disk in an a ne chart in P(R3) where the distance between points is determined by projective cross-ratios. This is a motivating example in convex projective geometry. Convex projective geometry is a generalization of real hyperbolic geometry where we replace
1.5. Volume in affine geometry 8 1.6. Centers of gravity 9 1.7. Affine manifolds 10 2. Projective geometry 11 2.1. Ideal points 11 2.2. Homogeneous coordinates 12 2.3. The basic dictionary 15 2.4. Affine patches 18 2.5. Projective reflections 19 2.6. Fundamental theorem of projective geometry 20 3. Duality, non-Euclidean geometry and .
Because children are more vulnerable than adults, and quite often are not able to make decisions for themselves, the Code protects them more than adults. The Code makes it clear that children’s interests can only be overridden if there is an exceptional reason to do so. In the Code, there are some specific clauses which protect children and
