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 A survey of projective geometric structures on 2-,3-manifolds S. Choi Department of Mathematical Science KAIST, Daejeon, South Korea Tokyo Institute of Technology
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I Babylonian, Egyptian, Chinese, Greek...I Euclid developed his axiomatic method to planar and
solid geometry under the influence of Plato, whothought that geometry should be the foundation of allthought after the Pythagorian attempt to understandthe world using rational numbers failed....
I Babylonian, Egyptian, Chinese, Greek...I Euclid developed his axiomatic method to planar and
solid geometry under the influence of Plato, whothought that geometry should be the foundation of allthought after the Pythagorian attempt to understandthe world using rational numbers failed....
I Euclidean geometry consists of some notions suchas lines, points, lengths, angle, and their interplay insome space called a plane...
I The Euclid had five axiomsI D. Hilbert, and many others made a modern
foundation so that the Euclidean geometry wasreduced to logic.
I Euclidean geometry has a notion of rigidtransformations which made the spacehomogeneous. They form a group called a group ofrigid motions. They preserve lines, length, angles,and every geometric statements.
I The group is useful in proving statements.... Turningit around, we see that actually the transformationgroup is more important. wallpaper groups
I Euclidean geometry consists of some notions suchas lines, points, lengths, angle, and their interplay insome space called a plane...
I The Euclid had five axiomsI D. Hilbert, and many others made a modern
foundation so that the Euclidean geometry wasreduced to logic.
I Euclidean geometry has a notion of rigidtransformations which made the spacehomogeneous. They form a group called a group ofrigid motions. They preserve lines, length, angles,and every geometric statements.
I The group is useful in proving statements.... Turningit around, we see that actually the transformationgroup is more important. wallpaper groups
I Euclidean geometry consists of some notions suchas lines, points, lengths, angle, and their interplay insome space called a plane...
I The Euclid had five axiomsI D. Hilbert, and many others made a modern
foundation so that the Euclidean geometry wasreduced to logic.
I Euclidean geometry has a notion of rigidtransformations which made the spacehomogeneous. They form a group called a group ofrigid motions. They preserve lines, length, angles,and every geometric statements.
I The group is useful in proving statements.... Turningit around, we see that actually the transformationgroup is more important. wallpaper groups
I Euclidean geometry consists of some notions suchas lines, points, lengths, angle, and their interplay insome space called a plane...
I The Euclid had five axiomsI D. Hilbert, and many others made a modern
foundation so that the Euclidean geometry wasreduced to logic.
I Euclidean geometry has a notion of rigidtransformations which made the spacehomogeneous. They form a group called a group ofrigid motions. They preserve lines, length, angles,and every geometric statements.
I The group is useful in proving statements.... Turningit around, we see that actually the transformationgroup is more important. wallpaper groups
I Euclidean geometry consists of some notions suchas lines, points, lengths, angle, and their interplay insome space called a plane...
I The Euclid had five axiomsI D. Hilbert, and many others made a modern
foundation so that the Euclidean geometry wasreduced to logic.
I Euclidean geometry has a notion of rigidtransformations which made the spacehomogeneous. They form a group called a group ofrigid motions. They preserve lines, length, angles,and every geometric statements.
I The group is useful in proving statements.... Turningit around, we see that actually the transformationgroup is more important. wallpaper groups
I Euclidean geometry consists of some notions suchas lines, points, lengths, angle, and their interplay insome space called a plane...
I The Euclid had five axiomsI D. Hilbert, and many others made a modern
foundation so that the Euclidean geometry wasreduced to logic.
I Euclidean geometry has a notion of rigidtransformations which made the spacehomogeneous. They form a group called a group ofrigid motions. They preserve lines, length, angles,and every geometric statements.
I The group is useful in proving statements.... Turningit around, we see that actually the transformationgroup is more important. wallpaper groups
Hyperbolic geometry.I Lobachevsky and Bolyai tried to build a geometry
that did not satisfy the fifth axiom of Euclid. (Seehyperbolic.cdy)
I Their attempts were justified by Beltrami-Klein modelwhich is a disk and lines were replaced by chordsand lengths were given by the logarithms of crossratios. See Beltrami-Klein model. Later other modelssuch as Poincare half space model and Poincaredisk model were developed. Poincare model (Inst.figuring).
I Here the group of rigid motions is the Lie groupSL(2,R).
I Higher-dimensional hyperbolic spaces were laterconstructed. Actually, an upper part of a hyperboloidin the Lorentzian space would be a model andPO(1,n) forms the group of rigid motions.Minkowsky model
Hyperbolic geometry.I Lobachevsky and Bolyai tried to build a geometry
that did not satisfy the fifth axiom of Euclid. (Seehyperbolic.cdy)
I Their attempts were justified by Beltrami-Klein modelwhich is a disk and lines were replaced by chordsand lengths were given by the logarithms of crossratios. See Beltrami-Klein model. Later other modelssuch as Poincare half space model and Poincaredisk model were developed. Poincare model (Inst.figuring).
I Here the group of rigid motions is the Lie groupSL(2,R).
I Higher-dimensional hyperbolic spaces were laterconstructed. Actually, an upper part of a hyperboloidin the Lorentzian space would be a model andPO(1,n) forms the group of rigid motions.Minkowsky model
Hyperbolic geometry.I Lobachevsky and Bolyai tried to build a geometry
that did not satisfy the fifth axiom of Euclid. (Seehyperbolic.cdy)
I Their attempts were justified by Beltrami-Klein modelwhich is a disk and lines were replaced by chordsand lengths were given by the logarithms of crossratios. See Beltrami-Klein model. Later other modelssuch as Poincare half space model and Poincaredisk model were developed. Poincare model (Inst.figuring).
I Here the group of rigid motions is the Lie groupSL(2,R).
I Higher-dimensional hyperbolic spaces were laterconstructed. Actually, an upper part of a hyperboloidin the Lorentzian space would be a model andPO(1,n) forms the group of rigid motions.Minkowsky model
Hyperbolic geometry.I Lobachevsky and Bolyai tried to build a geometry
that did not satisfy the fifth axiom of Euclid. (Seehyperbolic.cdy)
I Their attempts were justified by Beltrami-Klein modelwhich is a disk and lines were replaced by chordsand lengths were given by the logarithms of crossratios. See Beltrami-Klein model. Later other modelssuch as Poincare half space model and Poincaredisk model were developed. Poincare model (Inst.figuring).
I Here the group of rigid motions is the Lie groupSL(2,R).
I Higher-dimensional hyperbolic spaces were laterconstructed. Actually, an upper part of a hyperboloidin the Lorentzian space would be a model andPO(1,n) forms the group of rigid motions.Minkowsky model
Conformal geometryI Suppose that we use to study circles and spheres
only... We allow all transformations that preservescircles...
I This geometry looses a notion of lengths but has anotion of angles. There are no lines or geodesics butthere are circles.
I The Euclidean plane is compactifed by adding aunique point as an infinity. The group of motions isgenerated by inversions in circles. The group iscalled the Mobius transformation group. That is, thegroup of transformations of form
z → az + bcz + d
,az + bcz + d
I The space itself is considered as a complex sphere,i.e., the complex plane with the infinity added.
Conformal geometryI Suppose that we use to study circles and spheres
only... We allow all transformations that preservescircles...
I This geometry looses a notion of lengths but has anotion of angles. There are no lines or geodesics butthere are circles.
I The Euclidean plane is compactifed by adding aunique point as an infinity. The group of motions isgenerated by inversions in circles. The group iscalled the Mobius transformation group. That is, thegroup of transformations of form
z → az + bcz + d
,az + bcz + d
I The space itself is considered as a complex sphere,i.e., the complex plane with the infinity added.
Conformal geometryI Suppose that we use to study circles and spheres
only... We allow all transformations that preservescircles...
I This geometry looses a notion of lengths but has anotion of angles. There are no lines or geodesics butthere are circles.
I The Euclidean plane is compactifed by adding aunique point as an infinity. The group of motions isgenerated by inversions in circles. The group iscalled the Mobius transformation group. That is, thegroup of transformations of form
z → az + bcz + d
,az + bcz + d
I The space itself is considered as a complex sphere,i.e., the complex plane with the infinity added.
Conformal geometryI Suppose that we use to study circles and spheres
only... We allow all transformations that preservescircles...
I This geometry looses a notion of lengths but has anotion of angles. There are no lines or geodesics butthere are circles.
I The Euclidean plane is compactifed by adding aunique point as an infinity. The group of motions isgenerated by inversions in circles. The group iscalled the Mobius transformation group. That is, thegroup of transformations of form
z → az + bcz + d
,az + bcz + d
I The space itself is considered as a complex sphere,i.e., the complex plane with the infinity added.
Projective geometryI Projective geometry naturally arose in fine art
drawing perspectives during Renascence.Perspective drawings
I Desargues, Kepler first tried to add infinite pointscorresponding to each direction that a line in a planecan take. Thus, a plane with infinite point for eachdirection form a projective plane.
I Transformations are ones generated by change ofperspectives. In fact, when we are taking x-rays orother scans.
I The notions such as lengths, angles lose meaning.But notions of lines or geodesics are preserved. TheGreeks discovered that the cross ratios, i.e., ratios ofratios, are preserved.
I The infinite points are just like the ordinary pointsunder transformations.
Projective geometryI Projective geometry naturally arose in fine art
drawing perspectives during Renascence.Perspective drawings
I Desargues, Kepler first tried to add infinite pointscorresponding to each direction that a line in a planecan take. Thus, a plane with infinite point for eachdirection form a projective plane.
I Transformations are ones generated by change ofperspectives. In fact, when we are taking x-rays orother scans.
I The notions such as lengths, angles lose meaning.But notions of lines or geodesics are preserved. TheGreeks discovered that the cross ratios, i.e., ratios ofratios, are preserved.
I The infinite points are just like the ordinary pointsunder transformations.
Projective geometryI Projective geometry naturally arose in fine art
drawing perspectives during Renascence.Perspective drawings
I Desargues, Kepler first tried to add infinite pointscorresponding to each direction that a line in a planecan take. Thus, a plane with infinite point for eachdirection form a projective plane.
I Transformations are ones generated by change ofperspectives. In fact, when we are taking x-rays orother scans.
I The notions such as lengths, angles lose meaning.But notions of lines or geodesics are preserved. TheGreeks discovered that the cross ratios, i.e., ratios ofratios, are preserved.
I The infinite points are just like the ordinary pointsunder transformations.
Projective geometryI Projective geometry naturally arose in fine art
drawing perspectives during Renascence.Perspective drawings
I Desargues, Kepler first tried to add infinite pointscorresponding to each direction that a line in a planecan take. Thus, a plane with infinite point for eachdirection form a projective plane.
I Transformations are ones generated by change ofperspectives. In fact, when we are taking x-rays orother scans.
I The notions such as lengths, angles lose meaning.But notions of lines or geodesics are preserved. TheGreeks discovered that the cross ratios, i.e., ratios ofratios, are preserved.
I The infinite points are just like the ordinary pointsunder transformations.
Projective geometryI Projective geometry naturally arose in fine art
drawing perspectives during Renascence.Perspective drawings
I Desargues, Kepler first tried to add infinite pointscorresponding to each direction that a line in a planecan take. Thus, a plane with infinite point for eachdirection form a projective plane.
I Transformations are ones generated by change ofperspectives. In fact, when we are taking x-rays orother scans.
I The notions such as lengths, angles lose meaning.But notions of lines or geodesics are preserved. TheGreeks discovered that the cross ratios, i.e., ratios ofratios, are preserved.
I The infinite points are just like the ordinary pointsunder transformations.
I Klein finally formalized it by considering a projectiveplane as the space of lines through origin in theEuclidean 3-space. Here, a Euclidean plane isrecovered as a subspace of the lines not in thexy -plane and the infinite points are the lines in thexy -plane.
I Many interesting geometric theorems hold:Desargues, Pappus,... Moreover, the theorems comein pairs: duality of lines and points: PappusTheorem, Pascal Theorem
I Klein finally formalized it by considering a projectiveplane as the space of lines through origin in theEuclidean 3-space. Here, a Euclidean plane isrecovered as a subspace of the lines not in thexy -plane and the infinite points are the lines in thexy -plane.
I Many interesting geometric theorems hold:Desargues, Pappus,... Moreover, the theorems comein pairs: duality of lines and points: PappusTheorem, Pascal Theorem
I Many geometries are actually sub-geometries ofprojective (conformal) geometry. (The twocorrespond to maximal finite-dimensional Liealgebras acting locally on manifolds...)
I Hyperbolic geometry: Hn imbeds in RPn andPO(n,1) in PGL(n + 1,R) in canonical way.
I spherical, euclidean geometry.I The affine geometry G = SL(n,R) · Rn and X = Rn.I Six of eight 3-dimensional geometries: euclidean,
spherical, hyperbolic, nil, sol, SL(2,R) (B. Thiel)I S2 × R and H2 × R. (almost)I Some symmetric spaces and matrix groups...
I Recently, some graphics applications.... imagesuperposition,..
I Many geometries are actually sub-geometries ofprojective (conformal) geometry. (The twocorrespond to maximal finite-dimensional Liealgebras acting locally on manifolds...)
I Hyperbolic geometry: Hn imbeds in RPn andPO(n,1) in PGL(n + 1,R) in canonical way.
I spherical, euclidean geometry.I The affine geometry G = SL(n,R) · Rn and X = Rn.I Six of eight 3-dimensional geometries: euclidean,
spherical, hyperbolic, nil, sol, SL(2,R) (B. Thiel)I S2 × R and H2 × R. (almost)I Some symmetric spaces and matrix groups...
I Recently, some graphics applications.... imagesuperposition,..
I Many geometries are actually sub-geometries ofprojective (conformal) geometry. (The twocorrespond to maximal finite-dimensional Liealgebras acting locally on manifolds...)
I Hyperbolic geometry: Hn imbeds in RPn andPO(n,1) in PGL(n + 1,R) in canonical way.
I spherical, euclidean geometry.I The affine geometry G = SL(n,R) · Rn and X = Rn.I Six of eight 3-dimensional geometries: euclidean,
spherical, hyperbolic, nil, sol, SL(2,R) (B. Thiel)I S2 × R and H2 × R. (almost)I Some symmetric spaces and matrix groups...
I Recently, some graphics applications.... imagesuperposition,..
I Many geometries are actually sub-geometries ofprojective (conformal) geometry. (The twocorrespond to maximal finite-dimensional Liealgebras acting locally on manifolds...)
I Hyperbolic geometry: Hn imbeds in RPn andPO(n,1) in PGL(n + 1,R) in canonical way.
I spherical, euclidean geometry.I The affine geometry G = SL(n,R) · Rn and X = Rn.I Six of eight 3-dimensional geometries: euclidean,
spherical, hyperbolic, nil, sol, SL(2,R) (B. Thiel)I S2 × R and H2 × R. (almost)I Some symmetric spaces and matrix groups...
I Recently, some graphics applications.... imagesuperposition,..
I Many geometries are actually sub-geometries ofprojective (conformal) geometry. (The twocorrespond to maximal finite-dimensional Liealgebras acting locally on manifolds...)
I Hyperbolic geometry: Hn imbeds in RPn andPO(n,1) in PGL(n + 1,R) in canonical way.
I spherical, euclidean geometry.I The affine geometry G = SL(n,R) · Rn and X = Rn.I Six of eight 3-dimensional geometries: euclidean,
spherical, hyperbolic, nil, sol, SL(2,R) (B. Thiel)I S2 × R and H2 × R. (almost)I Some symmetric spaces and matrix groups...
I Recently, some graphics applications.... imagesuperposition,..
I Many geometries are actually sub-geometries ofprojective (conformal) geometry. (The twocorrespond to maximal finite-dimensional Liealgebras acting locally on manifolds...)
I Hyperbolic geometry: Hn imbeds in RPn andPO(n,1) in PGL(n + 1,R) in canonical way.
I spherical, euclidean geometry.I The affine geometry G = SL(n,R) · Rn and X = Rn.I Six of eight 3-dimensional geometries: euclidean,
spherical, hyperbolic, nil, sol, SL(2,R) (B. Thiel)I S2 × R and H2 × R. (almost)I Some symmetric spaces and matrix groups...
I Recently, some graphics applications.... imagesuperposition,..
I Many geometries are actually sub-geometries ofprojective (conformal) geometry. (The twocorrespond to maximal finite-dimensional Liealgebras acting locally on manifolds...)
I Hyperbolic geometry: Hn imbeds in RPn andPO(n,1) in PGL(n + 1,R) in canonical way.
I spherical, euclidean geometry.I The affine geometry G = SL(n,R) · Rn and X = Rn.I Six of eight 3-dimensional geometries: euclidean,
spherical, hyperbolic, nil, sol, SL(2,R) (B. Thiel)I S2 × R and H2 × R. (almost)I Some symmetric spaces and matrix groups...
I Recently, some graphics applications.... imagesuperposition,..
I Many geometries are actually sub-geometries ofprojective (conformal) geometry. (The twocorrespond to maximal finite-dimensional Liealgebras acting locally on manifolds...)
I Hyperbolic geometry: Hn imbeds in RPn andPO(n,1) in PGL(n + 1,R) in canonical way.
I spherical, euclidean geometry.I The affine geometry G = SL(n,R) · Rn and X = Rn.I Six of eight 3-dimensional geometries: euclidean,
spherical, hyperbolic, nil, sol, SL(2,R) (B. Thiel)I S2 × R and H2 × R. (almost)I Some symmetric spaces and matrix groups...
I Recently, some graphics applications.... imagesuperposition,..
Cartan, Ehresmann connections: Mostgeneral geometry possible
I Cartan proposed that for each pair (X ,G), we canimpart their properties to at each point of themanifolds so that they vary also. The flatness impliesthat the (X ,G)-geometry is actually recovered.
I Ehresmann introduced the most general notion ofconnections by generalizing Cartan connections.
I For example, for euclidean geometry, a Cartanconnection gives us Riemannian geometry.
I For projective geometry, a Cartan connectioncorresponds to a projectively flat torsion-free affineconnections and conversely.
I Wilson Stothers’ Geometry PagesI Reference: Projective and Cayley-Klein Geometries
(Springer Monographs in Mathematics) (Hardcover)by Arkadij L. Onishchik (Author), Rolf Sulanke(Author)
Cartan, Ehresmann connections: Mostgeneral geometry possible
I Cartan proposed that for each pair (X ,G), we canimpart their properties to at each point of themanifolds so that they vary also. The flatness impliesthat the (X ,G)-geometry is actually recovered.
I Ehresmann introduced the most general notion ofconnections by generalizing Cartan connections.
I For example, for euclidean geometry, a Cartanconnection gives us Riemannian geometry.
I For projective geometry, a Cartan connectioncorresponds to a projectively flat torsion-free affineconnections and conversely.
I Wilson Stothers’ Geometry PagesI Reference: Projective and Cayley-Klein Geometries
(Springer Monographs in Mathematics) (Hardcover)by Arkadij L. Onishchik (Author), Rolf Sulanke(Author)
Cartan, Ehresmann connections: Mostgeneral geometry possible
I Cartan proposed that for each pair (X ,G), we canimpart their properties to at each point of themanifolds so that they vary also. The flatness impliesthat the (X ,G)-geometry is actually recovered.
I Ehresmann introduced the most general notion ofconnections by generalizing Cartan connections.
I For example, for euclidean geometry, a Cartanconnection gives us Riemannian geometry.
I For projective geometry, a Cartan connectioncorresponds to a projectively flat torsion-free affineconnections and conversely.
I Wilson Stothers’ Geometry PagesI Reference: Projective and Cayley-Klein Geometries
(Springer Monographs in Mathematics) (Hardcover)by Arkadij L. Onishchik (Author), Rolf Sulanke(Author)
Cartan, Ehresmann connections: Mostgeneral geometry possible
I Cartan proposed that for each pair (X ,G), we canimpart their properties to at each point of themanifolds so that they vary also. The flatness impliesthat the (X ,G)-geometry is actually recovered.
I Ehresmann introduced the most general notion ofconnections by generalizing Cartan connections.
I For example, for euclidean geometry, a Cartanconnection gives us Riemannian geometry.
I For projective geometry, a Cartan connectioncorresponds to a projectively flat torsion-free affineconnections and conversely.
I Wilson Stothers’ Geometry PagesI Reference: Projective and Cayley-Klein Geometries
(Springer Monographs in Mathematics) (Hardcover)by Arkadij L. Onishchik (Author), Rolf Sulanke(Author)
Cartan, Ehresmann connections: Mostgeneral geometry possible
I Cartan proposed that for each pair (X ,G), we canimpart their properties to at each point of themanifolds so that they vary also. The flatness impliesthat the (X ,G)-geometry is actually recovered.
I Ehresmann introduced the most general notion ofconnections by generalizing Cartan connections.
I For example, for euclidean geometry, a Cartanconnection gives us Riemannian geometry.
I For projective geometry, a Cartan connectioncorresponds to a projectively flat torsion-free affineconnections and conversely.
I Wilson Stothers’ Geometry PagesI Reference: Projective and Cayley-Klein Geometries
(Springer Monographs in Mathematics) (Hardcover)by Arkadij L. Onishchik (Author), Rolf Sulanke(Author)
Cartan, Ehresmann connections: Mostgeneral geometry possible
I Cartan proposed that for each pair (X ,G), we canimpart their properties to at each point of themanifolds so that they vary also. The flatness impliesthat the (X ,G)-geometry is actually recovered.
I Ehresmann introduced the most general notion ofconnections by generalizing Cartan connections.
I For example, for euclidean geometry, a Cartanconnection gives us Riemannian geometry.
I For projective geometry, a Cartan connectioncorresponds to a projectively flat torsion-free affineconnections and conversely.
I Wilson Stothers’ Geometry PagesI Reference: Projective and Cayley-Klein Geometries
(Springer Monographs in Mathematics) (Hardcover)by Arkadij L. Onishchik (Author), Rolf Sulanke(Author)
I Manifolds come up in many areas of technology andsciences.
I Studying the topological structures of manifolds iscomplicated by the fact that there is no uniform wayto describe many topologically important featuresand provides useful coordinates.
I We would like to find some good descriptions andperhaps even classify collections of manifolds.
I Of course these are for pure-mathematical uses fornow....
I Manifolds come up in many areas of technology andsciences.
I Studying the topological structures of manifolds iscomplicated by the fact that there is no uniform wayto describe many topologically important featuresand provides useful coordinates.
I We would like to find some good descriptions andperhaps even classify collections of manifolds.
I Of course these are for pure-mathematical uses fornow....
I Manifolds come up in many areas of technology andsciences.
I Studying the topological structures of manifolds iscomplicated by the fact that there is no uniform wayto describe many topologically important featuresand provides useful coordinates.
I We would like to find some good descriptions andperhaps even classify collections of manifolds.
I Of course these are for pure-mathematical uses fornow....
I Manifolds come up in many areas of technology andsciences.
I Studying the topological structures of manifolds iscomplicated by the fact that there is no uniform wayto describe many topologically important featuresand provides useful coordinates.
I We would like to find some good descriptions andperhaps even classify collections of manifolds.
I Of course these are for pure-mathematical uses fornow....
ExamplesI Closed surfaces have either a spherical, euclidean,
or hyperbolic structures depending on genus. Wecan classify these to form deformation spaces suchas Teichmuller spaces.
I A hyperbolic surface equals H2/Γ for the image Γ ofthe representation π → PSL(2,R).
I A Teichmuller space can be identified with acomponent of the space Hom(π,PSL(2,R))/ ∼ ofconjugacy classes of representations. Thecomponent consists of discrete faithfulrepresentations.
I For closed 3-manifolds, it is recently proved that theydecompose into pieces admitting one of eightgeometrical structures including hyperbolic,euclidean, spherical ones... Thus, these manifoldsare now being classified... (Proof of thegeometrization conjecture by Perelman).
I For higher-dimensional manifolds, there are othertypes of geometric structures...
ExamplesI Closed surfaces have either a spherical, euclidean,
or hyperbolic structures depending on genus. Wecan classify these to form deformation spaces suchas Teichmuller spaces.
I A hyperbolic surface equals H2/Γ for the image Γ ofthe representation π → PSL(2,R).
I A Teichmuller space can be identified with acomponent of the space Hom(π,PSL(2,R))/ ∼ ofconjugacy classes of representations. Thecomponent consists of discrete faithfulrepresentations.
I For closed 3-manifolds, it is recently proved that theydecompose into pieces admitting one of eightgeometrical structures including hyperbolic,euclidean, spherical ones... Thus, these manifoldsare now being classified... (Proof of thegeometrization conjecture by Perelman).
I For higher-dimensional manifolds, there are othertypes of geometric structures...
ExamplesI Closed surfaces have either a spherical, euclidean,
or hyperbolic structures depending on genus. Wecan classify these to form deformation spaces suchas Teichmuller spaces.
I A hyperbolic surface equals H2/Γ for the image Γ ofthe representation π → PSL(2,R).
I A Teichmuller space can be identified with acomponent of the space Hom(π,PSL(2,R))/ ∼ ofconjugacy classes of representations. Thecomponent consists of discrete faithfulrepresentations.
I For closed 3-manifolds, it is recently proved that theydecompose into pieces admitting one of eightgeometrical structures including hyperbolic,euclidean, spherical ones... Thus, these manifoldsare now being classified... (Proof of thegeometrization conjecture by Perelman).
I For higher-dimensional manifolds, there are othertypes of geometric structures...
ExamplesI Closed surfaces have either a spherical, euclidean,
or hyperbolic structures depending on genus. Wecan classify these to form deformation spaces suchas Teichmuller spaces.
I A hyperbolic surface equals H2/Γ for the image Γ ofthe representation π → PSL(2,R).
I A Teichmuller space can be identified with acomponent of the space Hom(π,PSL(2,R))/ ∼ ofconjugacy classes of representations. Thecomponent consists of discrete faithfulrepresentations.
I For closed 3-manifolds, it is recently proved that theydecompose into pieces admitting one of eightgeometrical structures including hyperbolic,euclidean, spherical ones... Thus, these manifoldsare now being classified... (Proof of thegeometrization conjecture by Perelman).
I For higher-dimensional manifolds, there are othertypes of geometric structures...
ExamplesI Closed surfaces have either a spherical, euclidean,
or hyperbolic structures depending on genus. Wecan classify these to form deformation spaces suchas Teichmuller spaces.
I A hyperbolic surface equals H2/Γ for the image Γ ofthe representation π → PSL(2,R).
I A Teichmuller space can be identified with acomponent of the space Hom(π,PSL(2,R))/ ∼ ofconjugacy classes of representations. Thecomponent consists of discrete faithfulrepresentations.
I For closed 3-manifolds, it is recently proved that theydecompose into pieces admitting one of eightgeometrical structures including hyperbolic,euclidean, spherical ones... Thus, these manifoldsare now being classified... (Proof of thegeometrization conjecture by Perelman).
I For higher-dimensional manifolds, there are othertypes of geometric structures...
I Cartan first considered projective structures onsurfaces as projectively flat torsion-free connectionson surfaces.
I Chern worked on general type of projectivestructures in the differential geometry point of view.
I The study of affine manifolds precede that ofprojective manifolds.
I There were extensive work on affine manifolds, andaffine Lie groups by Chern, Auslander, Goldman,Fried, Smillie, Nagano, Yagi, Shima, Carriere,Margulis,...
I There are outstanding questions such as the Chernconjecture, Auslander conjecture,...
I Cartan first considered projective structures onsurfaces as projectively flat torsion-free connectionson surfaces.
I Chern worked on general type of projectivestructures in the differential geometry point of view.
I The study of affine manifolds precede that ofprojective manifolds.
I There were extensive work on affine manifolds, andaffine Lie groups by Chern, Auslander, Goldman,Fried, Smillie, Nagano, Yagi, Shima, Carriere,Margulis,...
I There are outstanding questions such as the Chernconjecture, Auslander conjecture,...
I Cartan first considered projective structures onsurfaces as projectively flat torsion-free connectionson surfaces.
I Chern worked on general type of projectivestructures in the differential geometry point of view.
I The study of affine manifolds precede that ofprojective manifolds.
I There were extensive work on affine manifolds, andaffine Lie groups by Chern, Auslander, Goldman,Fried, Smillie, Nagano, Yagi, Shima, Carriere,Margulis,...
I There are outstanding questions such as the Chernconjecture, Auslander conjecture,...
I Cartan first considered projective structures onsurfaces as projectively flat torsion-free connectionson surfaces.
I Chern worked on general type of projectivestructures in the differential geometry point of view.
I The study of affine manifolds precede that ofprojective manifolds.
I There were extensive work on affine manifolds, andaffine Lie groups by Chern, Auslander, Goldman,Fried, Smillie, Nagano, Yagi, Shima, Carriere,Margulis,...
I There are outstanding questions such as the Chernconjecture, Auslander conjecture,...
I Cartan first considered projective structures onsurfaces as projectively flat torsion-free connectionson surfaces.
I Chern worked on general type of projectivestructures in the differential geometry point of view.
I The study of affine manifolds precede that ofprojective manifolds.
I There were extensive work on affine manifolds, andaffine Lie groups by Chern, Auslander, Goldman,Fried, Smillie, Nagano, Yagi, Shima, Carriere,Margulis,...
I There are outstanding questions such as the Chernconjecture, Auslander conjecture,...
Projective manifolds: how many?I By the Klein-Beltrami model of hyperbolic space, we
can consider Hn as a unit ball B in an affinesubspace of RPn and PO(n + 1,R) as a subgroup ofPGL(n + 1,R) acting on B. Thus, a (complete)hyperbolic manifold has a projective structure. (Infact any closed surface has one.)
I In fact, 3-manifolds with six types of geometricstructures have real projective structures. (In fact,up to coverings of order two, 3-manifolds with eightgeometric structures have real projective structures.)
I A question arises whether these projective structurescan be deformed to purely projective structures.
I Deformed projective manifolds from closedhyperbolic ones are convex in the sense that anypath can be homotoped to a geodesic path. (Koszul’sopenness)
Projective manifolds: how many?I By the Klein-Beltrami model of hyperbolic space, we
can consider Hn as a unit ball B in an affinesubspace of RPn and PO(n + 1,R) as a subgroup ofPGL(n + 1,R) acting on B. Thus, a (complete)hyperbolic manifold has a projective structure. (Infact any closed surface has one.)
I In fact, 3-manifolds with six types of geometricstructures have real projective structures. (In fact,up to coverings of order two, 3-manifolds with eightgeometric structures have real projective structures.)
I A question arises whether these projective structurescan be deformed to purely projective structures.
I Deformed projective manifolds from closedhyperbolic ones are convex in the sense that anypath can be homotoped to a geodesic path. (Koszul’sopenness)
Projective manifolds: how many?I By the Klein-Beltrami model of hyperbolic space, we
can consider Hn as a unit ball B in an affinesubspace of RPn and PO(n + 1,R) as a subgroup ofPGL(n + 1,R) acting on B. Thus, a (complete)hyperbolic manifold has a projective structure. (Infact any closed surface has one.)
I In fact, 3-manifolds with six types of geometricstructures have real projective structures. (In fact,up to coverings of order two, 3-manifolds with eightgeometric structures have real projective structures.)
I A question arises whether these projective structurescan be deformed to purely projective structures.
I Deformed projective manifolds from closedhyperbolic ones are convex in the sense that anypath can be homotoped to a geodesic path. (Koszul’sopenness)
Projective manifolds: how many?I By the Klein-Beltrami model of hyperbolic space, we
can consider Hn as a unit ball B in an affinesubspace of RPn and PO(n + 1,R) as a subgroup ofPGL(n + 1,R) acting on B. Thus, a (complete)hyperbolic manifold has a projective structure. (Infact any closed surface has one.)
I In fact, 3-manifolds with six types of geometricstructures have real projective structures. (In fact,up to coverings of order two, 3-manifolds with eightgeometric structures have real projective structures.)
I A question arises whether these projective structurescan be deformed to purely projective structures.
I Deformed projective manifolds from closedhyperbolic ones are convex in the sense that anypath can be homotoped to a geodesic path. (Koszul’sopenness)
I In 1960s, Benzecri started working on strictly convexdomain Ω where a projective transformation group Γacted with a compact quotient. That is, M = Ω/Γ is acompact manifold (orbifold). (Related to convexcones and group transformations (Kuiper,Koszul,Vinberg,...)). He showed that the boundary iseither C1 or is an ellipsoid.
I Kac-Vinberg found the first example for n = 2 for atriangle reflection group associated with Kac-Moodyalgebra.
I Recently, Benoist found many interesting propertiessuch as the fact that Ω is strictly convex if and only ifthe group Γ is Gromov-hyperbolic.
I In 1960s, Benzecri started working on strictly convexdomain Ω where a projective transformation group Γacted with a compact quotient. That is, M = Ω/Γ is acompact manifold (orbifold). (Related to convexcones and group transformations (Kuiper,Koszul,Vinberg,...)). He showed that the boundary iseither C1 or is an ellipsoid.
I Kac-Vinberg found the first example for n = 2 for atriangle reflection group associated with Kac-Moodyalgebra.
I Recently, Benoist found many interesting propertiessuch as the fact that Ω is strictly convex if and only ifthe group Γ is Gromov-hyperbolic.
I In 1960s, Benzecri started working on strictly convexdomain Ω where a projective transformation group Γacted with a compact quotient. That is, M = Ω/Γ is acompact manifold (orbifold). (Related to convexcones and group transformations (Kuiper,Koszul,Vinberg,...)). He showed that the boundary iseither C1 or is an ellipsoid.
I Kac-Vinberg found the first example for n = 2 for atriangle reflection group associated with Kac-Moodyalgebra.
I Recently, Benoist found many interesting propertiessuch as the fact that Ω is strictly convex if and only ifthe group Γ is Gromov-hyperbolic.
I For closed surfaces, in 80s, Goldman found ageneral dimension counting method for thenonsingular part of the surface-group representationspace into the Lie group G which is dim G× (2g − 2)if g ≥ 2.
I Since the deformation space D(RP2,PGL(3,R))(S) for aclosed surface S is locally homeomorphic toHom(π1(S),PGL(3,R))/PGL(3,R), we know thedimension of the deformation space to be 8(2g − 2).
I For closed surfaces, in 80s, Goldman found ageneral dimension counting method for thenonsingular part of the surface-group representationspace into the Lie group G which is dim G× (2g − 2)if g ≥ 2.
I Since the deformation space D(RP2,PGL(3,R))(S) for aclosed surface S is locally homeomorphic toHom(π1(S),PGL(3,R))/PGL(3,R), we know thedimension of the deformation space to be 8(2g − 2).
I Goldman found that the the deformation space ofconvex projective structures is a cell of dimension16g − 16. Deformations (The real pro. str. on hyp.mflds.
I Choi showed that if g ≥ 2, then a projective surfacealways decomposes into convex projective surfacesand annuli along disjoint closed geodesics. (Someare not convex)
I The annuli were classified by Nagano, Yagi, andGoldman earlier.
I Using this, we have a complete classification ofprojective structures on closed surfaces (evenconstructive one).
I For 2-orbifolds, Goldman and Choi completed theclassification. developing images
I Goldman found that the the deformation space ofconvex projective structures is a cell of dimension16g − 16. Deformations (The real pro. str. on hyp.mflds.
I Choi showed that if g ≥ 2, then a projective surfacealways decomposes into convex projective surfacesand annuli along disjoint closed geodesics. (Someare not convex)
I The annuli were classified by Nagano, Yagi, andGoldman earlier.
I Using this, we have a complete classification ofprojective structures on closed surfaces (evenconstructive one).
I For 2-orbifolds, Goldman and Choi completed theclassification. developing images
I Goldman found that the the deformation space ofconvex projective structures is a cell of dimension16g − 16. Deformations (The real pro. str. on hyp.mflds.
I Choi showed that if g ≥ 2, then a projective surfacealways decomposes into convex projective surfacesand annuli along disjoint closed geodesics. (Someare not convex)
I The annuli were classified by Nagano, Yagi, andGoldman earlier.
I Using this, we have a complete classification ofprojective structures on closed surfaces (evenconstructive one).
I For 2-orbifolds, Goldman and Choi completed theclassification. developing images
I Goldman found that the the deformation space ofconvex projective structures is a cell of dimension16g − 16. Deformations (The real pro. str. on hyp.mflds.
I Choi showed that if g ≥ 2, then a projective surfacealways decomposes into convex projective surfacesand annuli along disjoint closed geodesics. (Someare not convex)
I The annuli were classified by Nagano, Yagi, andGoldman earlier.
I Using this, we have a complete classification ofprojective structures on closed surfaces (evenconstructive one).
I For 2-orbifolds, Goldman and Choi completed theclassification. developing images
I Goldman found that the the deformation space ofconvex projective structures is a cell of dimension16g − 16. Deformations (The real pro. str. on hyp.mflds.
I Choi showed that if g ≥ 2, then a projective surfacealways decomposes into convex projective surfacesand annuli along disjoint closed geodesics. (Someare not convex)
I The annuli were classified by Nagano, Yagi, andGoldman earlier.
I Using this, we have a complete classification ofprojective structures on closed surfaces (evenconstructive one).
I For 2-orbifolds, Goldman and Choi completed theclassification. developing images
I Hitchin used Higgs field on principal G-bundles oversurfaces to obtain parametrizations of flatG-connections over surfaces. (G is a real split formof a reductive group.) (90s)
I Hitchin used Higgs field on principal G-bundles oversurfaces to obtain parametrizations of flatG-connections over surfaces. (G is a real split formof a reductive group.) (90s)
I A convex projective surface is of form Ω/Γ. Hence,there is a representation π1(Σ)→ Γ determined onlyup to conjugation by PGL(3,R). This gives us a map
hol : D(Σ)→ Hom(π1(Σ),PGL(3,R))/ ∼ .
This map is known to be a local-homeomorphism(Ehresmann, Thurston) and is injective (Goldman)
I The map is in fact a homeomorphism ontoHitchin-Teichmüller component (Goldman, Choi) Themain idea for proof is to show that the image of themap is closed.
I As a consequence, we know that theHitchin-Teichmuller component for PGL(3,R))consists of discrete faithful representations only.
I A convex projective surface is of form Ω/Γ. Hence,there is a representation π1(Σ)→ Γ determined onlyup to conjugation by PGL(3,R). This gives us a map
hol : D(Σ)→ Hom(π1(Σ),PGL(3,R))/ ∼ .
This map is known to be a local-homeomorphism(Ehresmann, Thurston) and is injective (Goldman)
I The map is in fact a homeomorphism ontoHitchin-Teichmüller component (Goldman, Choi) Themain idea for proof is to show that the image of themap is closed.
I As a consequence, we know that theHitchin-Teichmuller component for PGL(3,R))consists of discrete faithful representations only.
I A convex projective surface is of form Ω/Γ. Hence,there is a representation π1(Σ)→ Γ determined onlyup to conjugation by PGL(3,R). This gives us a map
hol : D(Σ)→ Hom(π1(Σ),PGL(3,R))/ ∼ .
This map is known to be a local-homeomorphism(Ehresmann, Thurston) and is injective (Goldman)
I The map is in fact a homeomorphism ontoHitchin-Teichmüller component (Goldman, Choi) Themain idea for proof is to show that the image of themap is closed.
I As a consequence, we know that theHitchin-Teichmuller component for PGL(3,R))consists of discrete faithful representations only.
I Labourie recently generalized this to n ≥ 2,3: Thatthe component of representations that can bedeformed to the Fuschian representation acts on ahyperconvex curve in RPn−1.
I Corollary: Every Hitchin representation is a discretefaithful and “purely loxodromic”. The mapping classgroup acts properly on H(n).
I Burger, Iozzi, Labourie, Wienhard generalized thisresult to maximal representations into other Liegroups.
I Labourie recently generalized this to n ≥ 2,3: Thatthe component of representations that can bedeformed to the Fuschian representation acts on ahyperconvex curve in RPn−1.
I Corollary: Every Hitchin representation is a discretefaithful and “purely loxodromic”. The mapping classgroup acts properly on H(n).
I Burger, Iozzi, Labourie, Wienhard generalized thisresult to maximal representations into other Liegroups.
I Labourie recently generalized this to n ≥ 2,3: Thatthe component of representations that can bedeformed to the Fuschian representation acts on ahyperconvex curve in RPn−1.
I Corollary: Every Hitchin representation is a discretefaithful and “purely loxodromic”. The mapping classgroup acts properly on H(n).
I Burger, Iozzi, Labourie, Wienhard generalized thisresult to maximal representations into other Liegroups.
Projective surfaces and affine differentialgeometry
I John Loftin (simultaneously Labourie) showed usingthe work of Calabi, Cheng-Yau, and C.P. Wang onaffine spheres: Let Mn be a closed projectively flatmanifold.
I Then M is convex if and only if M admits an affinesphere structure in Rn+1.
I For n = 2, the affine sphere structure is equivalent tothe conformal structure on M with a holomorphiccubic-differential.
I In particular, this shows that the deformation spaceD(Σ) of convex projective structures on Σ admits acomplex structure, which is preserved under themoduli group actions. (Is it Kahler?)
I Loftin also worked out Mumford typecompactifications of the moduli space M(Σ) ofconvex projective structures.
Projective surfaces and affine differentialgeometry
I John Loftin (simultaneously Labourie) showed usingthe work of Calabi, Cheng-Yau, and C.P. Wang onaffine spheres: Let Mn be a closed projectively flatmanifold.
I Then M is convex if and only if M admits an affinesphere structure in Rn+1.
I For n = 2, the affine sphere structure is equivalent tothe conformal structure on M with a holomorphiccubic-differential.
I In particular, this shows that the deformation spaceD(Σ) of convex projective structures on Σ admits acomplex structure, which is preserved under themoduli group actions. (Is it Kahler?)
I Loftin also worked out Mumford typecompactifications of the moduli space M(Σ) ofconvex projective structures.
Projective surfaces and affine differentialgeometry
I John Loftin (simultaneously Labourie) showed usingthe work of Calabi, Cheng-Yau, and C.P. Wang onaffine spheres: Let Mn be a closed projectively flatmanifold.
I Then M is convex if and only if M admits an affinesphere structure in Rn+1.
I For n = 2, the affine sphere structure is equivalent tothe conformal structure on M with a holomorphiccubic-differential.
I In particular, this shows that the deformation spaceD(Σ) of convex projective structures on Σ admits acomplex structure, which is preserved under themoduli group actions. (Is it Kahler?)
I Loftin also worked out Mumford typecompactifications of the moduli space M(Σ) ofconvex projective structures.
Projective surfaces and affine differentialgeometry
I John Loftin (simultaneously Labourie) showed usingthe work of Calabi, Cheng-Yau, and C.P. Wang onaffine spheres: Let Mn be a closed projectively flatmanifold.
I Then M is convex if and only if M admits an affinesphere structure in Rn+1.
I For n = 2, the affine sphere structure is equivalent tothe conformal structure on M with a holomorphiccubic-differential.
I In particular, this shows that the deformation spaceD(Σ) of convex projective structures on Σ admits acomplex structure, which is preserved under themoduli group actions. (Is it Kahler?)
I Loftin also worked out Mumford typecompactifications of the moduli space M(Σ) ofconvex projective structures.
Projective surfaces and affine differentialgeometry
I John Loftin (simultaneously Labourie) showed usingthe work of Calabi, Cheng-Yau, and C.P. Wang onaffine spheres: Let Mn be a closed projectively flatmanifold.
I Then M is convex if and only if M admits an affinesphere structure in Rn+1.
I For n = 2, the affine sphere structure is equivalent tothe conformal structure on M with a holomorphiccubic-differential.
I In particular, this shows that the deformation spaceD(Σ) of convex projective structures on Σ admits acomplex structure, which is preserved under themoduli group actions. (Is it Kahler?)
I Loftin also worked out Mumford typecompactifications of the moduli space M(Σ) ofconvex projective structures.
I We saw that many 3-manifolds has projectivestructures.
I Cooper showed that RP3#RP3 does not admit a realprojective structures. (answers Bill Goldman’squestion). Other examples?
I We wish to understand the deformation spaces forhigher-dimensional manifolds and so on. (Thedeformation theory is still very difficult for conformalstructures as well.)
I There is a well-known construction of Apanasovusing bending: deforming along a closed totallygeodesic hypersurface.
I Johnson and Millson found deformations ofhigher-dimensional hyperbolic manifolds which arelocally singular. (also bending constructions)
I We saw that many 3-manifolds has projectivestructures.
I Cooper showed that RP3#RP3 does not admit a realprojective structures. (answers Bill Goldman’squestion). Other examples?
I We wish to understand the deformation spaces forhigher-dimensional manifolds and so on. (Thedeformation theory is still very difficult for conformalstructures as well.)
I There is a well-known construction of Apanasovusing bending: deforming along a closed totallygeodesic hypersurface.
I Johnson and Millson found deformations ofhigher-dimensional hyperbolic manifolds which arelocally singular. (also bending constructions)
I We saw that many 3-manifolds has projectivestructures.
I Cooper showed that RP3#RP3 does not admit a realprojective structures. (answers Bill Goldman’squestion). Other examples?
I We wish to understand the deformation spaces forhigher-dimensional manifolds and so on. (Thedeformation theory is still very difficult for conformalstructures as well.)
I There is a well-known construction of Apanasovusing bending: deforming along a closed totallygeodesic hypersurface.
I Johnson and Millson found deformations ofhigher-dimensional hyperbolic manifolds which arelocally singular. (also bending constructions)
I We saw that many 3-manifolds has projectivestructures.
I Cooper showed that RP3#RP3 does not admit a realprojective structures. (answers Bill Goldman’squestion). Other examples?
I We wish to understand the deformation spaces forhigher-dimensional manifolds and so on. (Thedeformation theory is still very difficult for conformalstructures as well.)
I There is a well-known construction of Apanasovusing bending: deforming along a closed totallygeodesic hypersurface.
I Johnson and Millson found deformations ofhigher-dimensional hyperbolic manifolds which arelocally singular. (also bending constructions)
I We saw that many 3-manifolds has projectivestructures.
I Cooper showed that RP3#RP3 does not admit a realprojective structures. (answers Bill Goldman’squestion). Other examples?
I We wish to understand the deformation spaces forhigher-dimensional manifolds and so on. (Thedeformation theory is still very difficult for conformalstructures as well.)
I There is a well-known construction of Apanasovusing bending: deforming along a closed totallygeodesic hypersurface.
I Johnson and Millson found deformations ofhigher-dimensional hyperbolic manifolds which arelocally singular. (also bending constructions)
I Cooper,Long and Thistlethwaite found a numerical(some exact algebraic) evidences that out of the first1000 closed hyperbolic 3-manifolds in theHodgson-Weeks census, a handful admit non-trivialdeformations of their SO+(3,1)-representations intoSL(4,R); each resulting representation variety thengives rise to a family of real projective structures onthe manifold.
I Benoist and Choi found some class of 3-dimensionalreflection orbifold that has positive dimensionaldeformation spaces. But we haven’t been able tostudy the wider class of reflection orbifolds. (prisms,pyramids, icosahedron,...)
I We have been working algebraically and numericallyfor simple polytopes.
I So far, there are no Gauge theoretic or affine sphereapproach to 3-dimensional projective manifolds.
I Cooper,Long and Thistlethwaite found a numerical(some exact algebraic) evidences that out of the first1000 closed hyperbolic 3-manifolds in theHodgson-Weeks census, a handful admit non-trivialdeformations of their SO+(3,1)-representations intoSL(4,R); each resulting representation variety thengives rise to a family of real projective structures onthe manifold.
I Benoist and Choi found some class of 3-dimensionalreflection orbifold that has positive dimensionaldeformation spaces. But we haven’t been able tostudy the wider class of reflection orbifolds. (prisms,pyramids, icosahedron,...)
I We have been working algebraically and numericallyfor simple polytopes.
I So far, there are no Gauge theoretic or affine sphereapproach to 3-dimensional projective manifolds.
I Cooper,Long and Thistlethwaite found a numerical(some exact algebraic) evidences that out of the first1000 closed hyperbolic 3-manifolds in theHodgson-Weeks census, a handful admit non-trivialdeformations of their SO+(3,1)-representations intoSL(4,R); each resulting representation variety thengives rise to a family of real projective structures onthe manifold.
I Benoist and Choi found some class of 3-dimensionalreflection orbifold that has positive dimensionaldeformation spaces. But we haven’t been able tostudy the wider class of reflection orbifolds. (prisms,pyramids, icosahedron,...)
I We have been working algebraically and numericallyfor simple polytopes.
I So far, there are no Gauge theoretic or affine sphereapproach to 3-dimensional projective manifolds.
I Cooper,Long and Thistlethwaite found a numerical(some exact algebraic) evidences that out of the first1000 closed hyperbolic 3-manifolds in theHodgson-Weeks census, a handful admit non-trivialdeformations of their SO+(3,1)-representations intoSL(4,R); each resulting representation variety thengives rise to a family of real projective structures onthe manifold.
I Benoist and Choi found some class of 3-dimensionalreflection orbifold that has positive dimensionaldeformation spaces. But we haven’t been able tostudy the wider class of reflection orbifolds. (prisms,pyramids, icosahedron,...)
I We have been working algebraically and numericallyfor simple polytopes.
I So far, there are no Gauge theoretic or affine sphereapproach to 3-dimensional projective manifolds.