G-structures and affine immersions
Paolo Piccione
Departamento de Matemática Instituto de Matemática e Estatística
Universidade de São Paulo
1o Congres(s)o Latino-Americano de Grupos de Lie en(m) Geometria
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 1 / 33
Outline.
1 G-structures
2 Principal spaces and fiber products
3 Principal fiber bundles
4 Connections
5 Inner torsion of aG-structure
6 Immersion theorems
7 Examples
Outline.
1 G-structures
2 Principal spaces and fiber products
3 Principal fiber bundles
4 Connections
5 Inner torsion of aG-structure
6 Immersion theorems
7 Examples
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 2 / 33
Outline.
1 G-structures
2 Principal spaces and fiber products
3 Principal fiber bundles
4 Connections
5 Inner torsion of aG-structure
6 Immersion theorems
7 Examples
Outline.
1 G-structures
2 Principal spaces and fiber products
3 Principal fiber bundles
4 Connections
5 Inner torsion of aG-structure
6 Immersion theorems
7 Examples
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 2 / 33
Outline.
1 G-structures
2 Principal spaces and fiber products
3 Principal fiber bundles
4 Connections
5 Inner torsion of aG-structure
6 Immersion theorems
7 Examples
Outline.
1 G-structures
2 Principal spaces and fiber products
3 Principal fiber bundles
4 Connections
5 Inner torsion of aG-structure
6 Immersion theorems
7 Examples
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 2 / 33
Outline.
1 G-structures
2 Principal spaces and fiber products
3 Principal fiber bundles
4 Connections
5 Inner torsion of aG-structure
6 Immersion theorems
7 Examples
Outline
1 G-structures
2 Principal spaces and fiber products
3 Principal fiber bundles
4 Connections
5 Inner torsion of aG-structure
6 Immersion theorems
7 Examples
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 3 / 33
G-structure on sets
X0set,Bij(X0)group of bijectionsf :X0→X0.
X another set, withBij(X0,X)6=∅.
Bij(X0)acts transitively onBij(X0,X)by composition on the right. G⊂Bij(X0)subgroup
Definition
AG-structureonX modeled onX0is a subsetP ofBij(X0,X)which is aG-orbit.
(a) p−1◦q :X0→X0is inG, for allp,q ∈P;
(b) p◦g :X0→X is inP, for allp∈P and allg∈G.
Given aG-structurePonX and aG-structureQonY, a map f :X →Y isG-structure preservingiff ◦p∈Qfor allp∈P.
G-structure on sets
X0set,Bij(X0)group of bijectionsf :X0→X0. X another set, withBij(X0,X)6=∅.
Bij(X0)acts transitively onBij(X0,X)by composition on the right. G⊂Bij(X0)subgroup
Definition
AG-structureonX modeled onX0is a subsetP ofBij(X0,X)which is aG-orbit.
(a) p−1◦q :X0→X0is inG, for allp,q ∈P;
(b) p◦g :X0→X is inP, for allp∈P and allg∈G.
Given aG-structurePonX and aG-structureQonY, a map f :X →Y isG-structure preservingiff ◦p∈Qfor allp∈P.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 4 / 33
G-structure on sets
X0set,Bij(X0)group of bijectionsf :X0→X0. X another set, withBij(X0,X)6=∅.
Bij(X0)acts transitively onBij(X0,X)by composition on the right.
G⊂Bij(X0)subgroup Definition
AG-structureonX modeled onX0is a subsetP ofBij(X0,X)which is aG-orbit.
(a) p−1◦q :X0→X0is inG, for allp,q ∈P;
(b) p◦g :X0→X is inP, for allp∈P and allg∈G.
Given aG-structurePonX and aG-structureQonY, a map f :X →Y isG-structure preservingiff ◦p∈Qfor allp∈P.
G-structure on sets
X0set,Bij(X0)group of bijectionsf :X0→X0. X another set, withBij(X0,X)6=∅.
Bij(X0)acts transitively onBij(X0,X)by composition on the right.
G⊂Bij(X0)subgroup
Definition
AG-structureonX modeled onX0is a subsetP ofBij(X0,X)which is aG-orbit.
(a) p−1◦q :X0→X0is inG, for allp,q ∈P;
(b) p◦g :X0→X is inP, for allp∈P and allg∈G.
Given aG-structurePonX and aG-structureQonY, a map f :X →Y isG-structure preservingiff ◦p∈Qfor allp∈P.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 4 / 33
G-structure on sets
X0set,Bij(X0)group of bijectionsf :X0→X0. X another set, withBij(X0,X)6=∅.
Bij(X0)acts transitively onBij(X0,X)by composition on the right.
G⊂Bij(X0)subgroup Definition
AG-structureonX modeled onX0is a subsetP ofBij(X0,X)which is aG-orbit.
(a) p−1◦q :X0→X0is inG, for allp,q ∈P;
(b) p◦g :X0→X is inP, for allp∈P and allg∈G.
Given aG-structurePonX and aG-structureQonY, a map f :X →Y isG-structure preservingiff ◦p∈Qfor allp∈P.
G-structure on sets
X0set,Bij(X0)group of bijectionsf :X0→X0. X another set, withBij(X0,X)6=∅.
Bij(X0)acts transitively onBij(X0,X)by composition on the right.
G⊂Bij(X0)subgroup Definition
AG-structureonX modeled onX0is a subsetP ofBij(X0,X)which is aG-orbit.
(a) p−1◦q :X0→X0is inG, for allp,q ∈P;
(b) p◦g :X0→X is inP, for allp∈P and allg∈G.
Given aG-structurePonX and aG-structureQonY, a map f :X →Y isG-structure preservingiff ◦p∈Qfor allp∈P.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 4 / 33
G-structure on sets
X0set,Bij(X0)group of bijectionsf :X0→X0. X another set, withBij(X0,X)6=∅.
Bij(X0)acts transitively onBij(X0,X)by composition on the right.
G⊂Bij(X0)subgroup Definition
AG-structureonX modeled onX0is a subsetP ofBij(X0,X)which is aG-orbit.
(a) p−1◦q :X0→X0is inG, for allp,q ∈P;
(b) p◦g :X0→X is inP, for allp∈P and allg∈G.
Given aG-structurePonX and aG-structureQonY, a map f :X →Y isG-structure preservingiff ◦p∈Qfor allp∈P.
G-structure on sets
X0set,Bij(X0)group of bijectionsf :X0→X0. X another set, withBij(X0,X)6=∅.
Bij(X0)acts transitively onBij(X0,X)by composition on the right.
G⊂Bij(X0)subgroup Definition
AG-structureonX modeled onX0is a subsetP ofBij(X0,X)which is aG-orbit.
(a) p−1◦q :X0→X0is inG, for allp,q ∈P;
(b) p◦g :X0→X is inP, for allp∈P and allg∈G.
Given aG-structurePonX and aG-structureQonY, a map f :X →Y isG-structure preservingiff ◦p∈Qfor allp∈P.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 4 / 33
Examples of G-structure
Example (1)
V n-dimensional vector space
Aframeis an isop:Rn→V,FR(V)⊂Bij(Rn,V)
GL(n)acts on the right transitively onFR(V), henceFR(V)is a GL(n)-structure onV.
Conversely, given aGL(n)-structureP⊂Bij(Rn,V)on a setV, there exists a unique vector space structure onV such thatP =FR(V). More generally,FRV0(V)is aGL(V0)-structure onV.
Example (2)
M0,M diffeomorphic differentiable manifolds
Diff(M0,M)⊂Bij(M0,M)is aDiff(M0)-structure onM.
Conversely, given aDiff(M0)-structureP ⊂Bij(M0,M)on a setM, there exists a unique differentiable structure onMsuch that P =Diff(M0,M).
Examples of G-structure
Example (1)
V n-dimensional vector space
Aframeis an isop:Rn→V,FR(V)⊂Bij(Rn,V)
GL(n)acts on the right transitively onFR(V), henceFR(V)is a GL(n)-structure onV.
Conversely, given aGL(n)-structureP⊂Bij(Rn,V)on a setV, there exists a unique vector space structure onV such thatP =FR(V). More generally,FRV0(V)is aGL(V0)-structure onV.
Example (2)
M0,M diffeomorphic differentiable manifolds
Diff(M0,M)⊂Bij(M0,M)is aDiff(M0)-structure onM.
Conversely, given aDiff(M0)-structureP ⊂Bij(M0,M)on a setM, there exists a unique differentiable structure onMsuch that P =Diff(M0,M).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 5 / 33
Examples of G-structure
Example (1)
V n-dimensional vector space
Aframeis an isop:Rn→V,FR(V)⊂Bij(Rn,V)
GL(n)acts on the right transitively onFR(V), henceFR(V)is a GL(n)-structure onV.
Conversely, given aGL(n)-structureP⊂Bij(Rn,V)on a setV, there exists a unique vector space structure onV such thatP =FR(V). More generally,FRV0(V)is aGL(V0)-structure onV.
Example (2)
M0,M diffeomorphic differentiable manifolds
Diff(M0,M)⊂Bij(M0,M)is aDiff(M0)-structure onM.
Conversely, given aDiff(M0)-structureP ⊂Bij(M0,M)on a setM, there exists a unique differentiable structure onMsuch that P =Diff(M0,M).
Examples of G-structure
Example (1)
V n-dimensional vector space
Aframeis an isop:Rn→V,FR(V)⊂Bij(Rn,V)
GL(n)acts on the right transitively onFR(V), henceFR(V)is a GL(n)-structure onV.
Conversely, given aGL(n)-structureP⊂Bij(Rn,V)on a setV, there exists a unique vector space structure onV such thatP =FR(V).
More generally,FRV0(V)is aGL(V0)-structure onV.
Example (2)
M0,M diffeomorphic differentiable manifolds
Diff(M0,M)⊂Bij(M0,M)is aDiff(M0)-structure onM.
Conversely, given aDiff(M0)-structureP ⊂Bij(M0,M)on a setM, there exists a unique differentiable structure onMsuch that P =Diff(M0,M).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 5 / 33
Examples of G-structure
Example (1)
V n-dimensional vector space
Aframeis an isop:Rn→V,FR(V)⊂Bij(Rn,V)
GL(n)acts on the right transitively onFR(V), henceFR(V)is a GL(n)-structure onV.
Conversely, given aGL(n)-structureP⊂Bij(Rn,V)on a setV, there exists a unique vector space structure onV such thatP =FR(V).
More generally,FRV0(V)is aGL(V0)-structure onV.
Example (2)
M0,M diffeomorphic differentiable manifolds
Diff(M0,M)⊂Bij(M0,M)is aDiff(M0)-structure onM.
Conversely, given aDiff(M0)-structureP ⊂Bij(M0,M)on a setM, there exists a unique differentiable structure onMsuch that P =Diff(M0,M).
Examples of G-structure
Example (1)
V n-dimensional vector space
Aframeis an isop:Rn→V,FR(V)⊂Bij(Rn,V)
GL(n)acts on the right transitively onFR(V), henceFR(V)is a GL(n)-structure onV.
Conversely, given aGL(n)-structureP⊂Bij(Rn,V)on a setV, there exists a unique vector space structure onV such thatP =FR(V).
More generally,FRV0(V)is aGL(V0)-structure onV.
Example (2)
M0,M diffeomorphic differentiable manifolds
Diff(M0,M)⊂Bij(M0,M)is aDiff(M0)-structure onM.
Conversely, given aDiff(M0)-structureP ⊂Bij(M0,M)on a setM, there exists a unique differentiable structure onMsuch that P =Diff(M0,M).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 5 / 33
Examples of G-structure
Example (1)
V n-dimensional vector space
Aframeis an isop:Rn→V,FR(V)⊂Bij(Rn,V)
GL(n)acts on the right transitively onFR(V), henceFR(V)is a GL(n)-structure onV.
Conversely, given aGL(n)-structureP⊂Bij(Rn,V)on a setV, there exists a unique vector space structure onV such thatP =FR(V).
More generally,FRV0(V)is aGL(V0)-structure onV.
Example (2)
M0,M diffeomorphic differentiable manifolds
Diff(M0,M)⊂Bij(M0,M)is aDiff(M0)-structure onM.
Conversely, given aDiff(M0)-structureP ⊂Bij(M0,M)on a setM, there exists a unique differentiable structure onMsuch that P =Diff(M0,M).
Examples of G-structure
Example (1)
V n-dimensional vector space
Aframeis an isop:Rn→V,FR(V)⊂Bij(Rn,V)
GL(n)acts on the right transitively onFR(V), henceFR(V)is a GL(n)-structure onV.
Conversely, given aGL(n)-structureP⊂Bij(Rn,V)on a setV, there exists a unique vector space structure onV such thatP =FR(V).
More generally,FRV0(V)is aGL(V0)-structure onV.
Example (2)
M0,M diffeomorphic differentiable manifolds
Diff(M0,M)⊂Bij(M0,M)is aDiff(M0)-structure onM.
Conversely, given aDiff(M0)-structureP ⊂Bij(M0,M)on a setM, there exists a unique differentiable structure onMsuch that P =Diff(M0,M).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 5 / 33
Strengthening G-structures
Given aG-structureP⊂Bij(X0,X)onX and a subgroupH⊂G.
AnH-structureQ⊂P is astrengtheningofP. Example
Giving anO(n)-structureQ ⊂FR(V)is the same as giving a positive definite inner product onV.
SO(n)-structureQ⊂FR(V)⇐⇒inner product + orientation onV Ifn=2m,U(m)-structureQ⊂FR(V)⇐⇒real positive definite inner product onV and an orthogonal complex structure onV SL(n)-structureQ⊂FR(V)⇐⇒a volume form onV
A 1-structureQ ⊂FR(V)is an identification ofV withRn.
Given aG-structureP⊂Bij(X0,X)and a subgroupH⊂G, there are [G:H]strengtheningH-structures ofP.
Strengthening G-structures
Given aG-structureP⊂Bij(X0,X)onX and a subgroupH⊂G.
AnH-structureQ⊂P is astrengtheningofP.
Example
Giving anO(n)-structureQ ⊂FR(V)is the same as giving a positive definite inner product onV.
SO(n)-structureQ⊂FR(V)⇐⇒inner product + orientation onV Ifn=2m,U(m)-structureQ⊂FR(V)⇐⇒real positive definite inner product onV and an orthogonal complex structure onV SL(n)-structureQ⊂FR(V)⇐⇒a volume form onV
A 1-structureQ ⊂FR(V)is an identification ofV withRn.
Given aG-structureP⊂Bij(X0,X)and a subgroupH⊂G, there are [G:H]strengtheningH-structures ofP.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 6 / 33
Strengthening G-structures
Given aG-structureP⊂Bij(X0,X)onX and a subgroupH⊂G.
AnH-structureQ⊂P is astrengtheningofP.
Example
Giving anO(n)-structureQ ⊂FR(V)is the same as giving a positive definite inner product onV.
SO(n)-structureQ⊂FR(V)⇐⇒inner product + orientation onV Ifn=2m,U(m)-structureQ⊂FR(V)⇐⇒real positive definite inner product onV and an orthogonal complex structure onV SL(n)-structureQ⊂FR(V)⇐⇒a volume form onV
A 1-structureQ ⊂FR(V)is an identification ofV withRn.
Given aG-structureP⊂Bij(X0,X)and a subgroupH⊂G, there are [G:H]strengtheningH-structures ofP.
Strengthening G-structures
Given aG-structureP⊂Bij(X0,X)onX and a subgroupH⊂G.
AnH-structureQ⊂P is astrengtheningofP.
Example
Giving anO(n)-structureQ ⊂FR(V)is the same as giving a positive definite inner product onV.
SO(n)-structureQ⊂FR(V)⇐⇒inner product + orientation onV
Ifn=2m,U(m)-structureQ⊂FR(V)⇐⇒real positive definite inner product onV and an orthogonal complex structure onV SL(n)-structureQ⊂FR(V)⇐⇒a volume form onV
A 1-structureQ ⊂FR(V)is an identification ofV withRn.
Given aG-structureP⊂Bij(X0,X)and a subgroupH⊂G, there are [G:H]strengtheningH-structures ofP.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 6 / 33
Strengthening G-structures
Given aG-structureP⊂Bij(X0,X)onX and a subgroupH⊂G.
AnH-structureQ⊂P is astrengtheningofP.
Example
Giving anO(n)-structureQ ⊂FR(V)is the same as giving a positive definite inner product onV.
SO(n)-structureQ⊂FR(V)⇐⇒inner product + orientation onV Ifn=2m,U(m)-structureQ⊂FR(V)⇐⇒real positive definite inner product onV and an orthogonal complex structure onV
SL(n)-structureQ⊂FR(V)⇐⇒a volume form onV A 1-structureQ ⊂FR(V)is an identification ofV withRn.
Given aG-structureP⊂Bij(X0,X)and a subgroupH⊂G, there are [G:H]strengtheningH-structures ofP.
Strengthening G-structures
Given aG-structureP⊂Bij(X0,X)onX and a subgroupH⊂G.
AnH-structureQ⊂P is astrengtheningofP.
Example
Giving anO(n)-structureQ ⊂FR(V)is the same as giving a positive definite inner product onV.
SO(n)-structureQ⊂FR(V)⇐⇒inner product + orientation onV Ifn=2m,U(m)-structureQ⊂FR(V)⇐⇒real positive definite inner product onV and an orthogonal complex structure onV SL(n)-structureQ⊂FR(V)⇐⇒a volume form onV
A 1-structureQ ⊂FR(V)is an identification ofV withRn.
Given aG-structureP⊂Bij(X0,X)and a subgroupH⊂G, there are [G:H]strengtheningH-structures ofP.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 6 / 33
Strengthening G-structures
Given aG-structureP⊂Bij(X0,X)onX and a subgroupH⊂G.
AnH-structureQ⊂P is astrengtheningofP.
Example
Giving anO(n)-structureQ ⊂FR(V)is the same as giving a positive definite inner product onV.
SO(n)-structureQ⊂FR(V)⇐⇒inner product + orientation onV Ifn=2m,U(m)-structureQ⊂FR(V)⇐⇒real positive definite inner product onV and an orthogonal complex structure onV SL(n)-structureQ⊂FR(V)⇐⇒a volume form onV
A 1-structureQ ⊂FR(V)is an identification ofV withRn.
Given aG-structureP⊂Bij(X0,X)and a subgroupH⊂G, there are [G:H]strengtheningH-structures ofP.
Strengthening G-structures
Given aG-structureP⊂Bij(X0,X)onX and a subgroupH⊂G.
AnH-structureQ⊂P is astrengtheningofP.
Example
Giving anO(n)-structureQ ⊂FR(V)is the same as giving a positive definite inner product onV.
SO(n)-structureQ⊂FR(V)⇐⇒inner product + orientation onV Ifn=2m,U(m)-structureQ⊂FR(V)⇐⇒real positive definite inner product onV and an orthogonal complex structure onV SL(n)-structureQ⊂FR(V)⇐⇒a volume form onV
A 1-structureQ ⊂FR(V)is an identification ofV withRn.
Given aG-structureP⊂Bij(X0,X)and a subgroupH⊂G, there are [G:H]strengtheningH-structures ofP.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 6 / 33
Outline
1 G-structures
2 Principal spaces and fiber products
3 Principal fiber bundles
4 Connections
5 Inner torsion of aG-structure
6 Immersion theorems
7 Examples
Principal spaces
Definition
A principal space consists of a setP6=∅, a groupG(structural group) and a free and transitive right action ofGonP.
Eachp∈Pgives abijectionβp:G→P, βp(g) =p·g Example
G=V vector space, a principal space with structural groupGis anaffine spaceparallel toV.
Any group is a principal space with structural groupG(right multiplication)
Given a subgroupH ⊂G, for allg ∈Gthe left cosetgH is a principal space with structural groupH.
FRV0(V)is a principal space with structural groupGL(V0).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 8 / 33
Principal spaces
Definition
A principal space consists of a setP6=∅, a groupG(structural group) and a free and transitive right action ofGonP.
Eachp∈Pgives abijectionβp:G→P, βp(g) =p·g
Example
G=V vector space, a principal space with structural groupGis anaffine spaceparallel toV.
Any group is a principal space with structural groupG(right multiplication)
Given a subgroupH ⊂G, for allg ∈Gthe left cosetgH is a principal space with structural groupH.
FRV0(V)is a principal space with structural groupGL(V0).
Principal spaces
Definition
A principal space consists of a setP6=∅, a groupG(structural group) and a free and transitive right action ofGonP.
Eachp∈Pgives abijectionβp:G→P, βp(g) =p·g Example
G=V vector space, a principal space with structural groupGis anaffine spaceparallel toV.
Any group is a principal space with structural groupG(right multiplication)
Given a subgroupH ⊂G, for allg ∈Gthe left cosetgH is a principal space with structural groupH.
FRV0(V)is a principal space with structural groupGL(V0).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 8 / 33
Principal spaces
Definition
A principal space consists of a setP6=∅, a groupG(structural group) and a free and transitive right action ofGonP.
Eachp∈Pgives abijectionβp:G→P, βp(g) =p·g Example
G=V vector space, a principal space with structural groupGis anaffine spaceparallel toV.
Any group is a principal space with structural groupG(right multiplication)
Given a subgroupH ⊂G, for allg ∈Gthe left cosetgH is a principal space with structural groupH.
FRV0(V)is a principal space with structural groupGL(V0).
Principal spaces
Definition
A principal space consists of a setP6=∅, a groupG(structural group) and a free and transitive right action ofGonP.
Eachp∈Pgives abijectionβp:G→P, βp(g) =p·g Example
G=V vector space, a principal space with structural groupGis anaffine spaceparallel toV.
Any group is a principal space with structural groupG(right multiplication)
Given a subgroupH ⊂G, for allg ∈Gthe left cosetgH is a principal space with structural groupH.
FRV0(V)is a principal space with structural groupGL(V0).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 8 / 33
Principal spaces
Definition
A principal space consists of a setP6=∅, a groupG(structural group) and a free and transitive right action ofGonP.
Eachp∈Pgives abijectionβp:G→P, βp(g) =p·g Example
G=V vector space, a principal space with structural groupGis anaffine spaceparallel toV.
Any group is a principal space with structural groupG(right multiplication)
Given a subgroupH ⊂G, for allg ∈Gthe left cosetgH is a principal space with structural groupH.
FR (V)is a principal space with structural groupGL(V ).
Fiber products
AG-spaceis a setNcarrying a leftG-action.
Given a principal spacePwith structural groupGand aG-spaceN, there is a leftG-action onP×N: g·(p,n) = (pg−1,gn).
Definition
Thefiber product P×GNis the quotient(P×N)/G.
For allp∈P, the mappˆ:N →P×GN, p(n) = [p,ˆ n] is a bijection. Example
Given a representationρ:G→GL(V0)(i.e., a left action ofGonV0by linear isomorphisms) and aG-principal spaceP, the set:
Pb ⊂FRV0(P×GV0)
consisting of all bijectionspb:V0→P×GV0is aGL(V0)-structure. Hence,P×GV0has the structure of a vector space.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 9 / 33
Fiber products
AG-spaceis a setNcarrying a leftG-action.
Given a principal spacePwith structural groupGand aG-spaceN, there is a leftG-action onP×N: g·(p,n) = (pg−1,gn).
Definition
Thefiber product P×GNis the quotient(P×N)/G.
For allp∈P, the mappˆ:N →P×GN, p(n) = [p,ˆ n] is a bijection. Example
Given a representationρ:G→GL(V0)(i.e., a left action ofGonV0by linear isomorphisms) and aG-principal spaceP, the set:
Pb ⊂FRV0(P×GV0)
consisting of all bijectionspb:V0→P×GV0is aGL(V0)-structure. Hence,P×GV0has the structure of a vector space.
Fiber products
AG-spaceis a setNcarrying a leftG-action.
Given a principal spacePwith structural groupGand aG-spaceN, there is a leftG-action onP×N: g·(p,n) = (pg−1,gn).
Definition
Thefiber product P×GNis the quotient(P×N)/G.
For allp∈P, the mappˆ:N →P×GN, p(n) = [p,ˆ n] is a bijection. Example
Given a representationρ:G→GL(V0)(i.e., a left action ofGonV0by linear isomorphisms) and aG-principal spaceP, the set:
Pb ⊂FRV0(P×GV0)
consisting of all bijectionspb:V0→P×GV0is aGL(V0)-structure. Hence,P×GV0has the structure of a vector space.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 9 / 33
Fiber products
AG-spaceis a setNcarrying a leftG-action.
Given a principal spacePwith structural groupGand aG-spaceN, there is a leftG-action onP×N: g·(p,n) = (pg−1,gn).
Definition
Thefiber product P×GNis the quotient(P×N)/G.
For allp∈P, the mappˆ:N →P×GN, p(n) = [p,ˆ n] is a bijection.
Example
Given a representationρ:G→GL(V0)(i.e., a left action ofGonV0by linear isomorphisms) and aG-principal spaceP, the set:
Pb ⊂FRV0(P×GV0)
consisting of all bijectionspb:V0→P×GV0is aGL(V0)-structure. Hence,P×GV0has the structure of a vector space.
Fiber products
AG-spaceis a setNcarrying a leftG-action.
Given a principal spacePwith structural groupGand aG-spaceN, there is a leftG-action onP×N: g·(p,n) = (pg−1,gn).
Definition
Thefiber product P×GNis the quotient(P×N)/G.
For allp∈P, the mappˆ:N →P×GN, p(n) = [p,ˆ n] is a bijection.
Example
Given a representationρ:G→GL(V0)(i.e., a left action ofGonV0by linear isomorphisms) and aG-principal spaceP, the set:
Pb ⊂FRV0(P×GV0)
consisting of all bijectionspb:V0→P×GV0is aGL(V0)-structure.
Hence,P×GV0has the structure of a vector space.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 9 / 33
Outline
1 G-structures
2 Principal spaces and fiber products
3 Principal fiber bundles
4 Connections
5 Inner torsion of aG-structure
6 Immersion theorems
7 Examples
Principal fiber bundles
a setP (total space)
a differentiable manifoldM(base space) a mapΠ :P →M(projection)
a Lie groupG(structural group)
a right action ofGonP that makes thefiber Px = Π−1(x)a principal space, for allx ∈m
a maximal atlas ofadmissiblelocal sections ofΠ. Lemma
There exists a unique differentiable structure on P that makes the action of G on P smooth,Πa smooth submersion, Px a smooth submanifold, every admissible local section s :U⊂M →P smooth,.
Verp=Ker dΠp
⊂TpP vertical space; canonical isomorphismdβp(1) :g−→∼= VerpP.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 11 / 33
Principal fiber bundles
a setP (total space)
a differentiable manifoldM(base space)
a mapΠ :P →M(projection) a Lie groupG(structural group)
a right action ofGonP that makes thefiber Px = Π−1(x)a principal space, for allx ∈m
a maximal atlas ofadmissiblelocal sections ofΠ. Lemma
There exists a unique differentiable structure on P that makes the action of G on P smooth,Πa smooth submersion, Px a smooth submanifold, every admissible local section s :U⊂M →P smooth,.
Verp=Ker dΠp
⊂TpP vertical space; canonical isomorphismdβp(1) :g−→∼= VerpP.
Principal fiber bundles
a setP (total space)
a differentiable manifoldM(base space) a mapΠ :P →M(projection)
a Lie groupG(structural group)
a right action ofGonP that makes thefiber Px = Π−1(x)a principal space, for allx ∈m
a maximal atlas ofadmissiblelocal sections ofΠ. Lemma
There exists a unique differentiable structure on P that makes the action of G on P smooth,Πa smooth submersion, Px a smooth submanifold, every admissible local section s :U⊂M →P smooth,.
Verp=Ker dΠp
⊂TpP vertical space; canonical isomorphismdβp(1) :g−→∼= VerpP.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 11 / 33
Principal fiber bundles
a setP (total space)
a differentiable manifoldM(base space) a mapΠ :P →M(projection)
a Lie groupG(structural group)
a right action ofGonP that makes thefiber Px = Π−1(x)a principal space, for allx ∈m
a maximal atlas ofadmissiblelocal sections ofΠ. Lemma
There exists a unique differentiable structure on P that makes the action of G on P smooth,Πa smooth submersion, Px a smooth submanifold, every admissible local section s :U⊂M →P smooth,.
Verp=Ker dΠp
⊂TpP vertical space; canonical isomorphismdβp(1) :g−→∼= VerpP.
Principal fiber bundles
a setP (total space)
a differentiable manifoldM(base space) a mapΠ :P →M(projection)
a Lie groupG(structural group)
a right action ofGonP that makes thefiber Px = Π−1(x)a principal space, for allx ∈m
a maximal atlas ofadmissiblelocal sections ofΠ. Lemma
There exists a unique differentiable structure on P that makes the action of G on P smooth,Πa smooth submersion, Px a smooth submanifold, every admissible local section s :U⊂M →P smooth,.
Verp=Ker dΠp
⊂TpP vertical space; canonical isomorphismdβp(1) :g−→∼= VerpP.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 11 / 33
Principal fiber bundles
a setP (total space)
a differentiable manifoldM(base space) a mapΠ :P →M(projection)
a Lie groupG(structural group)
a right action ofGonP that makes thefiber Px = Π−1(x)a principal space, for allx ∈m
a maximal atlas ofadmissiblelocal sections ofΠ.
Lemma
There exists a unique differentiable structure on P that makes the action of G on P smooth,Πa smooth submersion, Px a smooth submanifold, every admissible local section s :U⊂M →P smooth,.
Verp=Ker dΠp
⊂TpP vertical space; canonical isomorphismdβp(1) :g−→∼= VerpP.
Principal fiber bundles
a setP (total space)
a differentiable manifoldM(base space) a mapΠ :P →M(projection)
a Lie groupG(structural group)
a right action ofGonP that makes thefiber Px = Π−1(x)a principal space, for allx ∈m
a maximal atlas ofadmissiblelocal sections ofΠ.
Lemma
There exists a unique differentiable structure on P that makes the action of G on P smooth,Πa smooth submersion, Px a smooth submanifold, every admissible local section s :U⊂M →P smooth,.
Verp=Ker dΠp
⊂TpP vertical space; canonical isomorphismdβp(1) :g−→∼= VerpP.
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 11 / 33
Principal fiber bundles
a setP (total space)
a differentiable manifoldM(base space) a mapΠ :P →M(projection)
a Lie groupG(structural group)
a right action ofGonP that makes thefiber Px = Π−1(x)a principal space, for allx ∈m
a maximal atlas ofadmissiblelocal sections ofΠ.
Lemma
There exists a unique differentiable structure on P that makes the action of G on P smooth,Πa smooth submersion, Px a smooth submanifold, every admissible local section s :U⊂M →P smooth,.
Verp=Ker dΠp
⊂TpP vertical space;
Examples of principal fiber bundles
trivial principal bundle: Mmanifold,GLie group,P0a principal G-space,P =M×P0.
quotient of Lie groups:GLie group,H⊂Gclosed subgroup, Π :G→G/H projection; forx ∈G/H,Π−1(x)is a left coset ofH inG. His the structural group.
restriction:Π :P→M principal fiber bundle,U ⊂M open subset P|U= Π−1(U).
principal subbundles:Π :P →Mprincipal fiber bundle with structural groupG,H ⊂Ga Lie subgroup,Q ⊂Psatisfying:
I for allx ∈M,Qx =Px∩Qis a principal subspace ofPx with structural groupH;
I for allx ∈M, there exists a smooth local sections:U→Pwith x ∈Uands(U)⊂Q.
pull-backs:Π :P→M principal fiber bundle,f :M0 →Msmooth map,f∗P =S
y∈M0 {y} ×Pf(y) .
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 12 / 33
Examples of principal fiber bundles
trivial principal bundle: Mmanifold,GLie group,P0a principal G-space,P =M×P0.
quotient of Lie groups:GLie group,H⊂Gclosed subgroup, Π :G→G/H projection; forx ∈G/H,Π−1(x)is a left coset ofH inG. His the structural group.
restriction:Π :P→M principal fiber bundle,U ⊂M open subset P|U= Π−1(U).
principal subbundles:Π :P →Mprincipal fiber bundle with structural groupG,H ⊂Ga Lie subgroup,Q ⊂Psatisfying:
I for allx ∈M,Qx =Px∩Qis a principal subspace ofPx with structural groupH;
I for allx ∈M, there exists a smooth local sections:U→Pwith x ∈Uands(U)⊂Q.
pull-backs:Π :P→M principal fiber bundle,f :M0 →Msmooth map,f∗P =S
y∈M0 {y} ×Pf(y) .
Examples of principal fiber bundles
trivial principal bundle: Mmanifold,GLie group,P0a principal G-space,P =M×P0.
quotient of Lie groups:GLie group,H⊂Gclosed subgroup, Π :G→G/H projection; forx ∈G/H,Π−1(x)is a left coset ofH inG. His the structural group.
restriction:Π :P→M principal fiber bundle,U ⊂M open subset P|U= Π−1(U).
principal subbundles:Π :P →Mprincipal fiber bundle with structural groupG,H ⊂Ga Lie subgroup,Q ⊂Psatisfying:
I for allx ∈M,Qx =Px∩Qis a principal subspace ofPx with structural groupH;
I for allx ∈M, there exists a smooth local sections:U→Pwith x ∈Uands(U)⊂Q.
pull-backs:Π :P→M principal fiber bundle,f :M0 →Msmooth map,f∗P =S
y∈M0 {y} ×Pf(y) .
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 12 / 33
Examples of principal fiber bundles
trivial principal bundle: Mmanifold,GLie group,P0a principal G-space,P =M×P0.
quotient of Lie groups:GLie group,H⊂Gclosed subgroup, Π :G→G/H projection; forx ∈G/H,Π−1(x)is a left coset ofH inG. His the structural group.
restriction:Π :P→M principal fiber bundle,U ⊂M open subset P|U= Π−1(U).
principal subbundles:Π :P →Mprincipal fiber bundle with structural groupG,H ⊂Ga Lie subgroup,Q ⊂Psatisfying:
I for allx ∈M,Qx =Px∩Qis a principal subspace ofPx with structural groupH;
I for allx ∈M, there exists a smooth local sections:U→Pwith x ∈Uands(U)⊂Q.
pull-backs:Π :P→M principal fiber bundle,f :M0 →Msmooth map,f∗P =S
y∈M0 {y} ×Pf(y) .
Examples of principal fiber bundles
trivial principal bundle: Mmanifold,GLie group,P0a principal G-space,P =M×P0.
quotient of Lie groups:GLie group,H⊂Gclosed subgroup, Π :G→G/H projection; forx ∈G/H,Π−1(x)is a left coset ofH inG. His the structural group.
restriction:Π :P→M principal fiber bundle,U ⊂M open subset P|U= Π−1(U).
principal subbundles:Π :P →Mprincipal fiber bundle with structural groupG,H ⊂Ga Lie subgroup,Q ⊂Psatisfying:
I for allx ∈M,Qx =Px∩Qis a principal subspace ofPx with structural groupH;
I for allx ∈M, there exists a smooth local sections:U→Pwith x ∈Uands(U)⊂Q.
pull-backs:Π :P→M principal fiber bundle,f :M0 →Msmooth map,f∗P =S
y∈M0 {y} ×Pf(y) .
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 12 / 33
Examples of principal fiber bundles
trivial principal bundle: Mmanifold,GLie group,P0a principal G-space,P =M×P0.
quotient of Lie groups:GLie group,H⊂Gclosed subgroup, Π :G→G/H projection; forx ∈G/H,Π−1(x)is a left coset ofH inG. His the structural group.
restriction:Π :P→M principal fiber bundle,U ⊂M open subset P|U= Π−1(U).
principal subbundles:Π :P →Mprincipal fiber bundle with structural groupG,H ⊂Ga Lie subgroup,Q ⊂Psatisfying:
I for allx ∈M,Qx =Px∩Qis a principal subspace ofPx with structural groupH;
I for allx ∈M, there exists a smooth local sections:U→Pwith x ∈Uands(U)⊂Q.
pull-backs:Π :P→M principal fiber bundle,f :M0 →Msmooth map,f∗P =S
y∈M0 {y} ×Pf(y) .
Examples of principal fiber bundles
trivial principal bundle: Mmanifold,GLie group,P0a principal G-space,P =M×P0.
quotient of Lie groups:GLie group,H⊂Gclosed subgroup, Π :G→G/H projection; forx ∈G/H,Π−1(x)is a left coset ofH inG. His the structural group.
restriction:Π :P→M principal fiber bundle,U ⊂M open subset P|U= Π−1(U).
principal subbundles:Π :P →Mprincipal fiber bundle with structural groupG,H ⊂Ga Lie subgroup,Q ⊂Psatisfying:
I for allx ∈M,Qx =Px∩Qis a principal subspace ofPx with structural groupH;
I for allx ∈M, there exists a smooth local sections:U→Pwith x ∈Uands(U)⊂Q.
pull-backs:Π :P→M principal fiber bundle,f :M0 →Msmooth map,f∗P =S
y∈M0 {y} ×Pf(y) .
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 12 / 33
Associated and vector bundles
GLie group,Π :P→M aG-principal bundle,N a differentialG-space
Associated bundle:P×GN =S
x∈MPx×GN Definition
Avector bundleconsists of: a setE (total space)
a differentiable manifoldM(base manifold) a mapπ:E →M (projection)
a finite dimensional vector spaceE0(typical fiber)
a vector space structure on each fiberEx =π−1isomorphic toE0 a maximal atlas of admissible local sections of the fiber bundle FRE0(E) =S
x∈MFRE0(Ex).
FRE0(E)×GL(E0)E0∼=E, [p,e0]7→p(e0)∈E
Def.: AG-structure on Eis aG-principal subbundle ofFR(E).
Associated and vector bundles
GLie group,Π :P→M aG-principal bundle,N a differentialG-space Associated bundle:P×GN =S
x∈MPx×GN
Definition
Avector bundleconsists of: a setE (total space)
a differentiable manifoldM(base manifold) a mapπ:E →M (projection)
a finite dimensional vector spaceE0(typical fiber)
a vector space structure on each fiberEx =π−1isomorphic toE0 a maximal atlas of admissible local sections of the fiber bundle FRE0(E) =S
x∈MFRE0(Ex).
FRE0(E)×GL(E0)E0∼=E, [p,e0]7→p(e0)∈E
Def.: AG-structure on Eis aG-principal subbundle ofFR(E).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 13 / 33
Associated and vector bundles
GLie group,Π :P→M aG-principal bundle,N a differentialG-space Associated bundle:P×GN =S
x∈MPx×GN Definition
Avector bundleconsists of:
a setE (total space)
a differentiable manifoldM(base manifold) a mapπ:E →M (projection)
a finite dimensional vector spaceE0(typical fiber)
a vector space structure on each fiberEx =π−1isomorphic toE0 a maximal atlas of admissible local sections of the fiber bundle FRE0(E) =S
x∈MFRE0(Ex).
FRE0(E)×GL(E0)E0∼=E, [p,e0]7→p(e0)∈E
Def.: AG-structure on Eis aG-principal subbundle ofFR(E).
Associated and vector bundles
GLie group,Π :P→M aG-principal bundle,N a differentialG-space Associated bundle:P×GN =S
x∈MPx×GN Definition
Avector bundleconsists of:
a setE (total space)
a differentiable manifoldM(base manifold) a mapπ:E →M (projection)
a finite dimensional vector spaceE0(typical fiber)
a vector space structure on each fiberEx =π−1isomorphic toE0 a maximal atlas of admissible local sections of the fiber bundle FRE0(E) =S
x∈MFRE0(Ex).
FRE0(E)×GL(E0)E0∼=E, [p,e0]7→p(e0)∈E
Def.: AG-structure on Eis aG-principal subbundle ofFR(E).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 13 / 33
Associated and vector bundles
GLie group,Π :P→M aG-principal bundle,N a differentialG-space Associated bundle:P×GN =S
x∈MPx×GN Definition
Avector bundleconsists of:
a setE (total space)
a differentiable manifoldM(base manifold)
a mapπ:E →M (projection)
a finite dimensional vector spaceE0(typical fiber)
a vector space structure on each fiberEx =π−1isomorphic toE0 a maximal atlas of admissible local sections of the fiber bundle FRE0(E) =S
x∈MFRE0(Ex).
FRE0(E)×GL(E0)E0∼=E, [p,e0]7→p(e0)∈E
Def.: AG-structure on Eis aG-principal subbundle ofFR(E).
Associated and vector bundles
GLie group,Π :P→M aG-principal bundle,N a differentialG-space Associated bundle:P×GN =S
x∈MPx×GN Definition
Avector bundleconsists of:
a setE (total space)
a differentiable manifoldM(base manifold) a mapπ:E →M (projection)
a finite dimensional vector spaceE0(typical fiber)
a vector space structure on each fiberEx =π−1isomorphic toE0 a maximal atlas of admissible local sections of the fiber bundle FRE0(E) =S
x∈MFRE0(Ex).
FRE0(E)×GL(E0)E0∼=E, [p,e0]7→p(e0)∈E
Def.: AG-structure on Eis aG-principal subbundle ofFR(E).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 13 / 33
Associated and vector bundles
GLie group,Π :P→M aG-principal bundle,N a differentialG-space Associated bundle:P×GN =S
x∈MPx×GN Definition
Avector bundleconsists of:
a setE (total space)
a differentiable manifoldM(base manifold) a mapπ:E →M (projection)
a finite dimensional vector spaceE0(typical fiber)
a vector space structure on each fiberEx =π−1isomorphic toE0 a maximal atlas of admissible local sections of the fiber bundle FRE0(E) =S
x∈MFRE0(Ex).
FRE0(E)×GL(E0)E0∼=E, [p,e0]7→p(e0)∈E
Def.: AG-structure on Eis aG-principal subbundle ofFR(E).
Associated and vector bundles
GLie group,Π :P→M aG-principal bundle,N a differentialG-space Associated bundle:P×GN =S
x∈MPx×GN Definition
Avector bundleconsists of:
a setE (total space)
a differentiable manifoldM(base manifold) a mapπ:E →M (projection)
a finite dimensional vector spaceE0(typical fiber)
a vector space structure on each fiberEx =π−1isomorphic toE0
a maximal atlas of admissible local sections of the fiber bundle FRE0(E) =S
x∈MFRE0(Ex).
FRE0(E)×GL(E0)E0∼=E, [p,e0]7→p(e0)∈E
Def.: AG-structure on Eis aG-principal subbundle ofFR(E).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 13 / 33
Associated and vector bundles
GLie group,Π :P→M aG-principal bundle,N a differentialG-space Associated bundle:P×GN =S
x∈MPx×GN Definition
Avector bundleconsists of:
a setE (total space)
a differentiable manifoldM(base manifold) a mapπ:E →M (projection)
a finite dimensional vector spaceE0(typical fiber)
a vector space structure on each fiberEx =π−1isomorphic toE0 a maximal atlas of admissible local sections of the fiber bundle FRE0(E) =S
x∈MFRE0(Ex).
FRE0(E)×GL(E0)E0∼=E, [p,e0]7→p(e0)∈E
Def.: AG-structure on Eis aG-principal subbundle ofFR(E).
Associated and vector bundles
GLie group,Π :P→M aG-principal bundle,N a differentialG-space Associated bundle:P×GN =S
x∈MPx×GN Definition
Avector bundleconsists of:
a setE (total space)
a differentiable manifoldM(base manifold) a mapπ:E →M (projection)
a finite dimensional vector spaceE0(typical fiber)
a vector space structure on each fiberEx =π−1isomorphic toE0 a maximal atlas of admissible local sections of the fiber bundle FRE0(E) =S
x∈MFRE0(Ex).
FRE0(E)×GL(E0)E0∼=E, [p,e0]7→p(e0)∈E
Def.: AG-structure on Eis aG-principal subbundle ofFR(E).
Paolo Piccione (IME–USP) G-structures and affine immersions Unicamp, June 2006 13 / 33
Associated and vector bundles
GLie group,Π :P→M aG-principal bundle,N a differentialG-space Associated bundle:P×GN =S
x∈MPx×GN Definition
Avector bundleconsists of:
a setE (total space)
a differentiable manifoldM(base manifold) a mapπ:E →M (projection)
a finite dimensional vector spaceE0(typical fiber)
a vector space structure on each fiberEx =π−1isomorphic toE0 a maximal atlas of admissible local sections of the fiber bundle FRE0(E) =S
x∈MFRE0(Ex).
FRE0(E)×GL(E0)E0∼=E, [p,e0]7→p(e0)∈E
