Di¤erential Geometry of Singular Spaces and Reduction of Symmetries

J ¾edrzej ´Sniatycki

Department of Mathematics and Statistics University of Calgary

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 1 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Reduction of symmetries in mechanics was an important theme in Jerry Marsden’sresearch. In 1974, Marsden and Weinstein formulated a reduction scheme for free and proper action of the symmetry group. In a 1981 paper, Arms, Marsden and Moncrief showed that the reduction of the zero level of a proper action of a Lie group exhibits properties of a strati…ed space. In 1998, Koon and Marsden proposed Poisson reduction of non-holonomic systems. In a 2007 paper, Yoshimura and Marsden investigated reduction of symmetries of Dirac structures.

Introduction

I would like to than the organizers of this event for inviting me and giving me an opportunity to give this series of lectures.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 2 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 In 1974, Marsden and Weinstein formulated a reduction scheme for free and proper action of the symmetry group. In a 1981 paper, Arms, Marsden and Moncrief showed that the reduction of the zero level of a proper action of a Lie group exhibits properties of a strati…ed space. In 1998, Koon and Marsden proposed Poisson reduction of non-holonomic systems. In a 2007 paper, Yoshimura and Marsden investigated reduction of symmetries of Dirac structures.

Introduction

I would like to than the organizers of this event for inviting me and giving me an opportunity to give this series of lectures. Reduction of symmetries in mechanics was an important theme in Jerry Marsden’sresearch.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 2 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 In a 1981 paper, Arms, Marsden and Moncrief showed that the reduction of the zero level of a proper action of a Lie group exhibits properties of a strati…ed space. In 1998, Koon and Marsden proposed Poisson reduction of non-holonomic systems. In a 2007 paper, Yoshimura and Marsden investigated reduction of symmetries of Dirac structures.

Introduction

I would like to than the organizers of this event for inviting me and giving me an opportunity to give this series of lectures. Reduction of symmetries in mechanics was an important theme in Jerry Marsden’sresearch. In 1974, Marsden and Weinstein formulated a reduction scheme for free and proper action of the symmetry group.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 2 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 In 1998, Koon and Marsden proposed Poisson reduction of non-holonomic systems. In a 2007 paper, Yoshimura and Marsden investigated reduction of symmetries of Dirac structures.

Introduction

I would like to than the organizers of this event for inviting me and giving me an opportunity to give this series of lectures. Reduction of symmetries in mechanics was an important theme in Jerry Marsden’sresearch. In 1974, Marsden and Weinstein formulated a reduction scheme for free and proper action of the symmetry group. In a 1981 paper, Arms, Marsden and Moncrief showed that the reduction of the zero level of a proper action of a Lie group exhibits properties of a strati…ed space.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 2 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 In a 2007 paper, Yoshimura and Marsden investigated reduction of symmetries of Dirac structures.

Introduction

I would like to than the organizers of this event for inviting me and giving me an opportunity to give this series of lectures. Reduction of symmetries in mechanics was an important theme in Jerry Marsden’sresearch. In 1974, Marsden and Weinstein formulated a reduction scheme for free and proper action of the symmetry group. In a 1981 paper, Arms, Marsden and Moncrief showed that the reduction of the zero level of a proper action of a Lie group exhibits properties of a strati…ed space. In 1998, Koon and Marsden proposed Poisson reduction of non-holonomic systems.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 2 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Introduction

I would like to than the organizers of this event for inviting me and giving me an opportunity to give this series of lectures. Reduction of symmetries in mechanics was an important theme in Jerry Marsden’sresearch. In 1974, Marsden and Weinstein formulated a reduction scheme for free and proper action of the symmetry group. In a 1981 paper, Arms, Marsden and Moncrief showed that the reduction of the zero level of a proper action of a Lie group exhibits properties of a strati…ed space. In 1998, Koon and Marsden proposed Poisson reduction of non-holonomic systems. In a 2007 paper, Yoshimura and Marsden investigated reduction of symmetries of Dirac structures.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 2 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 If the action of the symmetry group G on the phase space P of a system is free and proper, then the space of orbits is a manifold and the the space of orbits P/G is a quotient manifold of P. If the action on P of a Lie group G is proper, the orbit space P/G is strati…ed (Bierstone 1980). Strati…ed spaces were introduced by Whitney (1955), who called them manifold collections. The term strati…ed spaces (strati…cations) was coined by Thom (1955). The following photograph of soap bubbles illustrates the structure of a strati…ed space.

Orbit spaces

In reduction of symmetries of a Hamiltonian system, we study the interplay between the structure of the space of orbits of the symmetry group of the system and its symplectic structure.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 3 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 If the action on P of a Lie group G is proper, the orbit space P/G is strati…ed (Bierstone 1980). Strati…ed spaces were introduced by Whitney (1955), who called them manifold collections. The term strati…ed spaces (strati…cations) was coined by Thom (1955). The following photograph of soap bubbles illustrates the structure of a strati…ed space.

Orbit spaces

In reduction of symmetries of a Hamiltonian system, we study the interplay between the structure of the space of orbits of the symmetry group of the system and its symplectic structure. If the action of the symmetry group G on the phase space P of a system is free and proper, then the space of orbits is a manifold and the the space of orbits P/G is a quotient manifold of P.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 3 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Strati…ed spaces were introduced by Whitney (1955), who called them manifold collections. The term strati…ed spaces (strati…cations) was coined by Thom (1955). The following photograph of soap bubbles illustrates the structure of a strati…ed space.

Orbit spaces

In reduction of symmetries of a Hamiltonian system, we study the interplay between the structure of the space of orbits of the symmetry group of the system and its symplectic structure. If the action of the symmetry group G on the phase space P of a system is free and proper, then the space of orbits is a manifold and the the space of orbits P/G is a quotient manifold of P. If the action on P of a Lie group G is proper, the orbit space P/G is strati…ed (Bierstone 1980).

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 3 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 The following photograph of soap bubbles illustrates the structure of a strati…ed space.

Orbit spaces

In reduction of symmetries of a Hamiltonian system, we study the interplay between the structure of the space of orbits of the symmetry group of the system and its symplectic structure. If the action of the symmetry group G on the phase space P of a system is free and proper, then the space of orbits is a manifold and the the space of orbits P/G is a quotient manifold of P. If the action on P of a Lie group G is proper, the orbit space P/G is strati…ed (Bierstone 1980). Strati…ed spaces were introduced by Whitney (1955), who called them manifold collections. The term strati…ed spaces (strati…cations) was coined by Thom (1955).

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 3 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Orbit spaces

In reduction of symmetries of a Hamiltonian system, we study the interplay between the structure of the space of orbits of the symmetry group of the system and its symplectic structure. If the action of the symmetry group G on the phase space P of a system is free and proper, then the space of orbits is a manifold and the the space of orbits P/G is a quotient manifold of P. If the action on P of a Lie group G is proper, the orbit space P/G is strati…ed (Bierstone 1980). Strati…ed spaces were introduced by Whitney (1955), who called them manifold collections. The term strati…ed spaces (strati…cations) was coined by Thom (1955). The following photograph of soap bubbles illustrates the structure of a strati…ed space.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 3 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 I was fascinated by Cushman’sresults and tried to understand his theory. His explanations were very helpful. Nevertheless, for a long time I did not understand was he was doing. Finally, I realized that Cushman was using the language of di¤erential geometry in the sense of Sikorski. In his 1972 book, Sikorski introduced the notion of a di¤erential structure on a topological space that is given by a class of continuous functions which are deemed to be smooth.

Singular reduction

In 1983, Cushman initiated his technique of singular reduction with a study of a Hamiltonian action of a compact Lie group. Using the 1975 paper of G. Schwarz, in many cases, Cushman was able to describe explicitly the strati…cation structure of the reduced space in terms of algebraic invariants of the action.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 4 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Finally, I realized that Cushman was using the language of di¤erential geometry in the sense of Sikorski. In his 1972 book, Sikorski introduced the notion of a di¤erential structure on a topological space that is given by a class of continuous functions which are deemed to be smooth.

Singular reduction

In 1983, Cushman initiated his technique of singular reduction with a study of a Hamiltonian action of a compact Lie group. Using the 1975 paper of G. Schwarz, in many cases, Cushman was able to describe explicitly the strati…cation structure of the reduced space in terms of algebraic invariants of the action. I was fascinated by Cushman’sresults and tried to understand his theory. His explanations were very helpful. Nevertheless, for a long time I did not understand was he was doing.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 4 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 In his 1972 book, Sikorski introduced the notion of a di¤erential structure on a topological space that is given by a class of continuous functions which are deemed to be smooth.

Singular reduction

In 1983, Cushman initiated his technique of singular reduction with a study of a Hamiltonian action of a compact Lie group. Using the 1975 paper of G. Schwarz, in many cases, Cushman was able to describe explicitly the strati…cation structure of the reduced space in terms of algebraic invariants of the action. I was fascinated by Cushman’sresults and tried to understand his theory. His explanations were very helpful. Nevertheless, for a long time I did not understand was he was doing. Finally, I realized that Cushman was using the language of di¤erential geometry in the sense of Sikorski.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 4 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Singular reduction

In 1983, Cushman initiated his technique of singular reduction with a study of a Hamiltonian action of a compact Lie group. Using the 1975 paper of G. Schwarz, in many cases, Cushman was able to describe explicitly the strati…cation structure of the reduced space in terms of algebraic invariants of the action. I was fascinated by Cushman’sresults and tried to understand his theory. His explanations were very helpful. Nevertheless, for a long time I did not understand was he was doing. Finally, I realized that Cushman was using the language of di¤erential geometry in the sense of Sikorski. In his 1972 book, Sikorski introduced the notion of a di¤erential structure on a topological space that is given by a class of continuous functions which are deemed to be smooth.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 4 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 1 The collection of sets

1 ∞ f ((a, b)) f C (S), a, b R f j 2 2 g is a subbasis for the topology of P/G. ∞ n ∞ 2 For every n N, F C (R ) and f1, ..., fn C (S), 2 2 2 ∞ F (f1, ..., fn) C (S). 2 3 If h : S R has the property that for every point x S, there exists an open! neighbourhood U of x in S and a function f2 C ∞(S) such that 2 h U = f U , j j then h C ∞(S). 2 A di¤erential space is a topological space S endowed with a di¤erential structure C ∞(S).

Di¤erential structures

A di¤erential structure on a topological space S is a family C ∞(S) of functions on S such that

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 5 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 ∞ n ∞ 2 For every n N, F C (R ) and f1, ..., fn C (S), 2 2 2 ∞ F (f1, ..., fn) C (S). 2 3 If h : S R has the property that for every point x S, there exists an open! neighbourhood U of x in S and a function f2 C ∞(S) such that 2 h U = f U , j j then h C ∞(S). 2 A di¤erential space is a topological space S endowed with a di¤erential structure C ∞(S).

Di¤erential structures

A di¤erential structure on a topological space S is a family C ∞(S) of functions on S such that 1 The collection of sets

1 ∞ f ((a, b)) f C (S), a, b R f j 2 2 g is a subbasis for the topology of P/G.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 5 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 3 If h : S R has the property that for every point x S, there exists an open! neighbourhood U of x in S and a function f2 C ∞(S) such that 2 h U = f U , j j then h C ∞(S). 2 A di¤erential space is a topological space S endowed with a di¤erential structure C ∞(S).

Di¤erential structures

A di¤erential structure on a topological space S is a family C ∞(S) of functions on S such that 1 The collection of sets

1 ∞ f ((a, b)) f C (S), a, b R f j 2 2 g is a subbasis for the topology of P/G. ∞ n ∞ 2 For every n N, F C (R ) and f1, ..., fn C (S), 2 2 2 ∞ F (f1, ..., fn) C (S). 2

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 5 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 A di¤erential space is a topological space S endowed with a di¤erential structure C ∞(S).

Di¤erential structures

A di¤erential structure on a topological space S is a family C ∞(S) of functions on S such that 1 The collection of sets

1 ∞ f ((a, b)) f C (S), a, b R f j 2 2 g is a subbasis for the topology of P/G. ∞ n ∞ 2 For every n N, F C (R ) and f1, ..., fn C (S), 2 2 2 ∞ F (f1, ..., fn) C (S). 2 3 If h : S R has the property that for every point x S, there exists an open! neighbourhood U of x in S and a function f2 C ∞(S) such that 2 h U = f U , j j then h C ∞(S). 2

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 5 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Di¤erential structures

A di¤erential structure on a topological space S is a family C ∞(S) of functions on S such that 1 The collection of sets

1 ∞ f ((a, b)) f C (S), a, b R f j 2 2 g is a subbasis for the topology of P/G. ∞ n ∞ 2 For every n N, F C (R ) and f1, ..., fn C (S), 2 2 2 ∞ F (f1, ..., fn) C (S). 2 3 If h : S R has the property that for every point x S, there exists an open! neighbourhood U of x in S and a function f2 C ∞(S) such that 2 h U = f U , j j then h C ∞(S). 2 A di¤erential space is a topological space S endowed with a di¤erential structure C ∞(S). Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 5 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 ∞ ∞ A map ϕ : S T is smooth if ϕf C (S) for every f C (T ). A map ϕ : S ! T is a di¤eomorphism2 if ϕ is smooth, invertible2 and 1 ! ϕ is smooth. If T is a di¤erential space, and S T , then S is a di¤erential space with the di¤erential structure generated by restrictions to S of functions in C ∞(T ). A di¤erential space S is a manifold if every point of S has a neighbourhood di¤eomorphic to an open subset of Rn. A di¤erential space S is subcartesian if it is Hausdor¤ and every point of S has a neighbourhood di¤eomorphic to a subset of Rn.

Category of di¤erential spaces

Let S and T be di¤erential spaces with di¤erential structures C ∞(S) and C ∞(T ), respectively.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 6 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 A map ϕ : S T is a di¤eomorphism if ϕ is smooth, invertible and 1 ! ϕ is smooth. If T is a di¤erential space, and S T , then S is a di¤erential space with the di¤erential structure generated by restrictions to S of functions in C ∞(T ). A di¤erential space S is a manifold if every point of S has a neighbourhood di¤eomorphic to an open subset of Rn. A di¤erential space S is subcartesian if it is Hausdor¤ and every point of S has a neighbourhood di¤eomorphic to a subset of Rn.

Category of di¤erential spaces

Let S and T be di¤erential spaces with di¤erential structures C ∞(S) and C ∞(T ), respectively. A map ϕ : S T is smooth if ϕ f C ∞(S) for every f C ∞(T ). !  2 2

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 6 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 If T is a di¤erential space, and S T , then S is a di¤erential space with the di¤erential structure generated by restrictions to S of functions in C ∞(T ). A di¤erential space S is a manifold if every point of S has a neighbourhood di¤eomorphic to an open subset of Rn. A di¤erential space S is subcartesian if it is Hausdor¤ and every point of S has a neighbourhood di¤eomorphic to a subset of Rn.

Category of di¤erential spaces

Let S and T be di¤erential spaces with di¤erential structures C ∞(S) and C ∞(T ), respectively. ∞ ∞ A map ϕ : S T is smooth if ϕf C (S) for every f C (T ). A map ϕ : S ! T is a di¤eomorphism2 if ϕ is smooth, invertible2 and 1 ! ϕ is smooth.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 6 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 A di¤erential space S is a manifold if every point of S has a neighbourhood di¤eomorphic to an open subset of Rn. A di¤erential space S is subcartesian if it is Hausdor¤ and every point of S has a neighbourhood di¤eomorphic to a subset of Rn.

Category of di¤erential spaces

Let S and T be di¤erential spaces with di¤erential structures C ∞(S) and C ∞(T ), respectively. ∞ ∞ A map ϕ : S T is smooth if ϕf C (S) for every f C (T ). A map ϕ : S ! T is a di¤eomorphism2 if ϕ is smooth, invertible2 and 1 ! ϕ is smooth. If T is a di¤erential space, and S T , then S is a di¤erential space with the di¤erential structure generated by restrictions to S of functions in C ∞(T ).

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 6 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 A di¤erential space S is subcartesian if it is Hausdor¤ and every point of S has a neighbourhood di¤eomorphic to a subset of Rn.

Category of di¤erential spaces

Let S and T be di¤erential spaces with di¤erential structures C ∞(S) and C ∞(T ), respectively. ∞ ∞ A map ϕ : S T is smooth if ϕf C (S) for every f C (T ). A map ϕ : S ! T is a di¤eomorphism2 if ϕ is smooth, invertible2 and 1 ! ϕ is smooth. If T is a di¤erential space, and S T , then S is a di¤erential space with the di¤erential structure generated by restrictions to S of functions in C ∞(T ). A di¤erential space S is a manifold if every point of S has a neighbourhood di¤eomorphic to an open subset of Rn.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 6 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Category of di¤erential spaces

Let S and T be di¤erential spaces with di¤erential structures C ∞(S) and C ∞(T ), respectively. ∞ ∞ A map ϕ : S T is smooth if ϕf C (S) for every f C (T ). A map ϕ : S ! T is a di¤eomorphism2 if ϕ is smooth, invertible2 and 1 ! ϕ is smooth. If T is a di¤erential space, and S T , then S is a di¤erential space with the di¤erential structure generated by restrictions to S of functions in C ∞(T ). A di¤erential space S is a manifold if every point of S has a neighbourhood di¤eomorphic to an open subset of Rn. A di¤erential space S is subcartesian if it is Hausdor¤ and every point of S has a neighbourhood di¤eomorphic to a subset of Rn.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 6 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Properness of Φ means that for every convergent sequence (pn) in P and a sequence (gn) in G such that the sequence (gnpn) is convergent

in P, there exists a convergent subsequence (gnk ) of G and

lim gnpn = ( lim gnk )( lim pn). n ∞ k ∞ n ∞ ! ! ! Properness of the action implies that all isotropy groups

Gp = g G gp = p f 2 j g are compact.

Proper action

Let Φ : G P P : (g, p) Φg (p) = gp  ! 7! be a proper action of a connected Lie group G on a manifold P.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 7 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Properness of the action implies that all isotropy groups

Gp = g G gp = p f 2 j g are compact.

Proper action

Let Φ : G P P : (g, p) Φg (p) = gp  ! 7! be a proper action of a connected Lie group G on a manifold P.

Properness of Φ means that for every convergent sequence (pn) in P and a sequence (gn) in G such that the sequence (gnpn) is convergent

in P, there exists a convergent subsequence (gnk ) of G and

lim gnpn = ( lim gnk )( lim pn). n ∞ k ∞ n ∞ ! ! !

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 7 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Proper action

Let Φ : G P P : (g, p) Φg (p) = gp  ! 7! be a proper action of a connected Lie group G on a manifold P.

Properness of Φ means that for every convergent sequence (pn) in P and a sequence (gn) in G such that the sequence (gnpn) is convergent

in P, there exists a convergent subsequence (gnk ) of G and

lim gnpn = ( lim gnk )( lim pn). n ∞ k ∞ n ∞ ! ! ! Properness of the action implies that all isotropy groups

Gp = g G gp = p f 2 j g are compact.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 7 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Theorem The orbit space P/G endowed with the di¤erential structure C ∞(P/G ) is a subcartesian di¤erential space.

The di¤erential structure of the orbit space P/G is

∞ 0 ∞ G C (P/G ) = f C (P/G ) ρf C (P) , f 2 j 2 g where C ∞(P)G is the space of smooth G-invariant functions on P.

Proof of this theorem involves all the steps which entered in Cushman’ssingular reduction theory. Our aim is to decode the structure of P/G from the data encoded in its di¤erential structure C ∞(P/G ).

Space of orbits of a proper action

We denote the space of G orbits in P by P/G and the canonical projection by ρ : P P/G : p Gp. ! 7!

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 8 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Theorem The orbit space P/G endowed with the di¤erential structure C ∞(P/G ) is a subcartesian di¤erential space.

Proof of this theorem involves all the steps which entered in Cushman’ssingular reduction theory. Our aim is to decode the structure of P/G from the data encoded in its di¤erential structure C ∞(P/G ).

Space of orbits of a proper action

We denote the space of G orbits in P by P/G and the canonical projection by ρ : P P/G : p Gp. ! 7! The di¤erential structure of the orbit space P/G is

∞ 0 ∞ G C (P/G ) = f C (P/G ) ρf C (P) , f 2 j 2 g where C ∞(P)G is the space of smooth G-invariant functions on P.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 8 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Proof of this theorem involves all the steps which entered in Cushman’ssingular reduction theory. Our aim is to decode the structure of P/G from the data encoded in its di¤erential structure C ∞(P/G ).

Space of orbits of a proper action

We denote the space of G orbits in P by P/G and the canonical projection by ρ : P P/G : p Gp. ! 7! The di¤erential structure of the orbit space P/G is

∞ 0 ∞ G C (P/G ) = f C (P/G ) ρf C (P) , f 2 j 2 g where C ∞(P)G is the space of smooth G-invariant functions on P. Theorem The orbit space P/G endowed with the di¤erential structure C ∞(P/G ) is a subcartesian di¤erential space.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 8 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Our aim is to decode the structure of P/G from the data encoded in its di¤erential structure C ∞(P/G ).

Space of orbits of a proper action

We denote the space of G orbits in P by P/G and the canonical projection by ρ : P P/G : p Gp. ! 7! The di¤erential structure of the orbit space P/G is

∞ 0 ∞ G C (P/G ) = f C (P/G ) ρf C (P) , f 2 j 2 g where C ∞(P)G is the space of smooth G-invariant functions on P. Theorem The orbit space P/G endowed with the di¤erential structure C ∞(P/G ) is a subcartesian di¤erential space.

Proof of this theorem involves all the steps which entered in Cushman’ssingular reduction theory.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 8 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Space of orbits of a proper action

We denote the space of G orbits in P by P/G and the canonical projection by ρ : P P/G : p Gp. ! 7! The di¤erential structure of the orbit space P/G is

∞ 0 ∞ G C (P/G ) = f C (P/G ) ρf C (P) , f 2 j 2 g where C ∞(P)G is the space of smooth G-invariant functions on P. Theorem The orbit space P/G endowed with the di¤erential structure C ∞(P/G ) is a subcartesian di¤erential space.

Proof of this theorem involves all the steps which entered in Cushman’ssingular reduction theory. Our aim is to decode the structure of P/G from the data encoded in its di¤erential structure C ∞(P/G ). Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 8 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Theorem Let S be a subcartesian space, and let X be a derivation of C ∞(S). For every x S, there exists a unique maximal integral curve c : I S of X such that2 c(0) = x. !

Let I be an interval in R. A smooth map c : I S is an integral curve of a derivation X if ! d f (c(t)) = (X (f ))(c(t)) dt for every t I . 2

Di¤erential equations on subcartesian spaces

A derivation of C ∞(S) is a map X : C ∞(S) C ∞(S) satis…ying Leibniz’srule ! X (f1f2) = X (f1)f2 + f1X (f2).

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 9 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Theorem Let S be a subcartesian space, and let X be a derivation of C ∞(S). For every x S, there exists a unique maximal integral curve c : I S of X such that2 c(0) = x. !

Di¤erential equations on subcartesian spaces

A derivation of C ∞(S) is a map X : C ∞(S) C ∞(S) satis…ying Leibniz’srule ! X (f1f2) = X (f1)f2 + f1X (f2). Let I be an interval in R. A smooth map c : I S is an integral curve of a derivation X if ! d f (c(t)) = (X (f ))(c(t)) dt for every t I . 2

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 9 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Di¤erential equations on subcartesian spaces

A derivation of C ∞(S) is a map X : C ∞(S) C ∞(S) satis…ying Leibniz’srule ! X (f1f2) = X (f1)f2 + f1X (f2). Let I be an interval in R. A smooth map c : I S is an integral curve of a derivation X if ! d f (c(t)) = (X (f ))(c(t)) dt for every t I . 2 Theorem Let S be a subcartesian space, and let X be a derivation of C ∞(S). For every x S, there exists a unique maximal integral curve c : I S of X such that2 c(0) = x. !

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 9 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 De…nition We say that X is a vector …eld on S if translations along integral curves of X give rise to local one-parameter groups exp tX of local di¤eomorphisms of S. Let F be a family of vector …elds on a subcartesian space S.

For x0 S, the orbit of through x0 is 2 F ∞ n

Ox0 = (exp ti Xi )(xi 1) S ti Ji , f 2 j 2 g n[=1 X1[,..,Xn J1,...,[Jn i[=1

where the vector …elds X1, ..., Xn are in F and, for each i = 1, ..., n,

the interval Ji Ixi 1 is either [0, τi ] or [τi , 0] with  xi = (exp τi Xi )(xi 1).

Orbits of families of vector …elds

Let X be a derivation of the di¤erential structure C ∞(S) of a subcartesian space S.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 10 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Let F be a family of vector …elds on a subcartesian space S.

For x0 S, the orbit of through x0 is 2 F ∞ n

Ox0 = (exp ti Xi )(xi 1) S ti Ji , f 2 j 2 g n[=1 X1[,..,Xn J1,...,[Jn i[=1

where the vector …elds X1, ..., Xn are in F and, for each i = 1, ..., n,

the interval Ji Ixi 1 is either [0, τi ] or [τi , 0] with  xi = (exp τi Xi )(xi 1).

Orbits of families of vector …elds

Let X be a derivation of the di¤erential structure C ∞(S) of a subcartesian space S. De…nition We say that X is a vector …eld on S if translations along integral curves of X give rise to local one-parameter groups exp tX of local di¤eomorphisms of S.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 10 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 For x0 S, the orbit of through x0 is 2 F ∞ n

Ox0 = (exp ti Xi )(xi 1) S ti Ji , f 2 j 2 g n[=1 X1[,..,Xn J1,...,[Jn i[=1

where the vector …elds X1, ..., Xn are in F and, for each i = 1, ..., n,

the interval Ji Ixi 1 is either [0, τi ] or [τi , 0] with  xi = (exp τi Xi )(xi 1).

Orbits of families of vector …elds

Let X be a derivation of the di¤erential structure C ∞(S) of a subcartesian space S. De…nition We say that X is a vector …eld on S if translations along integral curves of X give rise to local one-parameter groups exp tX of local di¤eomorphisms of S. Let F be a family of vector …elds on a subcartesian space S.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 10 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Orbits of families of vector …elds

Let X be a derivation of the di¤erential structure C ∞(S) of a subcartesian space S. De…nition We say that X is a vector …eld on S if translations along integral curves of X give rise to local one-parameter groups exp tX of local di¤eomorphisms of S. Let F be a family of vector …elds on a subcartesian space S.

For x0 S, the orbit of through x0 is 2 F ∞ n

Ox0 = (exp ti Xi )(xi 1) S ti Ji , f 2 j 2 g n[=1 X1[,..,Xn J1,...,[Jn i[=1

where the vector …elds X1, ..., Xn are in F and, for each i = 1, ..., n,

the interval Ji Ixi 1 is either [0, τi ] or [τi , 0] with  xi = (exp τi Xi )(xi 1). Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 10 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Let X(S) be the family of all vector …elds on S. By the theorem above, every subcartesian space S is partitioned by orbits of the family X(S) of all vector …elds on S. picture of soap bubbles Note, that we have got a structure theorem for a very general space. The category of subcartesian spaces contains all subsets of Rn and spaces that locally look like subsets Rn. According to our theorem, each of these spaces has a partition by immersed manifolds that are orbits of the family of all vector …elds. Moreover, this partition is minimal in the sense that there is no local one-parameter group of local di¤eomorphisms that acts transversally to the manifolds of the partition.

Generalized Sussmann’stheorem

Theorem Orbits of families of vector …elds on a subcartesian space S are manifolds immersed in S.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 11 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 picture of soap bubbles Note, that we have got a structure theorem for a very general space. The category of subcartesian spaces contains all subsets of Rn and spaces that locally look like subsets Rn. According to our theorem, each of these spaces has a partition by immersed manifolds that are orbits of the family of all vector …elds. Moreover, this partition is minimal in the sense that there is no local one-parameter group of local di¤eomorphisms that acts transversally to the manifolds of the partition.

Generalized Sussmann’stheorem

Theorem Orbits of families of vector …elds on a subcartesian space S are manifolds immersed in S.

Let X(S) be the family of all vector …elds on S. By the theorem above, every subcartesian space S is partitioned by orbits of the family X(S) of all vector …elds on S.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 11 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Note, that we have got a structure theorem for a very general space. The category of subcartesian spaces contains all subsets of Rn and spaces that locally look like subsets Rn. According to our theorem, each of these spaces has a partition by immersed manifolds that are orbits of the family of all vector …elds. Moreover, this partition is minimal in the sense that there is no local one-parameter group of local di¤eomorphisms that acts transversally to the manifolds of the partition.

Generalized Sussmann’stheorem

Theorem Orbits of families of vector …elds on a subcartesian space S are manifolds immersed in S.

Let X(S) be the family of all vector …elds on S. By the theorem above, every subcartesian space S is partitioned by orbits of the family X(S) of all vector …elds on S. picture of soap bubbles

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 11 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 The category of subcartesian spaces contains all subsets of Rn and spaces that locally look like subsets Rn. According to our theorem, each of these spaces has a partition by immersed manifolds that are orbits of the family of all vector …elds. Moreover, this partition is minimal in the sense that there is no local one-parameter group of local di¤eomorphisms that acts transversally to the manifolds of the partition.

Generalized Sussmann’stheorem

Theorem Orbits of families of vector …elds on a subcartesian space S are manifolds immersed in S.

Let X(S) be the family of all vector …elds on S. By the theorem above, every subcartesian space S is partitioned by orbits of the family X(S) of all vector …elds on S. picture of soap bubbles Note, that we have got a structure theorem for a very general space.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 11 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 According to our theorem, each of these spaces has a partition by immersed manifolds that are orbits of the family of all vector …elds. Moreover, this partition is minimal in the sense that there is no local one-parameter group of local di¤eomorphisms that acts transversally to the manifolds of the partition.

Generalized Sussmann’stheorem

Theorem Orbits of families of vector …elds on a subcartesian space S are manifolds immersed in S.

Let X(S) be the family of all vector …elds on S. By the theorem above, every subcartesian space S is partitioned by orbits of the family X(S) of all vector …elds on S. picture of soap bubbles Note, that we have got a structure theorem for a very general space. The category of subcartesian spaces contains all subsets of Rn and spaces that locally look like subsets Rn.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 11 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Moreover, this partition is minimal in the sense that there is no local one-parameter group of local di¤eomorphisms that acts transversally to the manifolds of the partition.

Generalized Sussmann’stheorem

Theorem Orbits of families of vector …elds on a subcartesian space S are manifolds immersed in S.

Let X(S) be the family of all vector …elds on S. By the theorem above, every subcartesian space S is partitioned by orbits of the family X(S) of all vector …elds on S. picture of soap bubbles Note, that we have got a structure theorem for a very general space. The category of subcartesian spaces contains all subsets of Rn and spaces that locally look like subsets Rn. According to our theorem, each of these spaces has a partition by immersed manifolds that are orbits of the family of all vector …elds.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 11 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Generalized Sussmann’stheorem

Theorem Orbits of families of vector …elds on a subcartesian space S are manifolds immersed in S.

Let X(S) be the family of all vector …elds on S. By the theorem above, every subcartesian space S is partitioned by orbits of the family X(S) of all vector …elds on S. picture of soap bubbles Note, that we have got a structure theorem for a very general space. The category of subcartesian spaces contains all subsets of Rn and spaces that locally look like subsets Rn. According to our theorem, each of these spaces has a partition by immersed manifolds that are orbits of the family of all vector …elds. Moreover, this partition is minimal in the sense that there is no local one-parameter group of local di¤eomorphisms that acts transversally to the manifolds of the partition. Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 11 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Theorem Strata of the orbit type strati…cation of P/G are orbits of the family X(P/G ) of all vector …elds on P/G.

If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space. The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries. For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P. The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure. The last step is to determine the geometric structure on P/G on the basis of the algebraic structure of C ∞(P/G ).

Application to reduction of symmetry

Let P/G be the space of orbits of a proper action of a connected Lie group G on a manifold P.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 12 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space. The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries. For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P. The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure. The last step is to determine the geometric structure on P/G on the basis of the algebraic structure of C ∞(P/G ).

Application to reduction of symmetry

Let P/G be the space of orbits of a proper action of a connected Lie group G on a manifold P. Theorem Strata of the orbit type strati…cation of P/G are orbits of the family X(P/G ) of all vector …elds on P/G.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 12 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries. For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P. The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure. The last step is to determine the geometric structure on P/G on the basis of the algebraic structure of C ∞(P/G ).

Application to reduction of symmetry

Let P/G be the space of orbits of a proper action of a connected Lie group G on a manifold P. Theorem Strata of the orbit type strati…cation of P/G are orbits of the family X(P/G ) of all vector …elds on P/G.

If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 12 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P. The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure. The last step is to determine the geometric structure on P/G on the basis of the algebraic structure of C ∞(P/G ).

Application to reduction of symmetry

Let P/G be the space of orbits of a proper action of a connected Lie group G on a manifold P. Theorem Strata of the orbit type strati…cation of P/G are orbits of the family X(P/G ) of all vector …elds on P/G.

If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space. The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 12 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure. The last step is to determine the geometric structure on P/G on the basis of the algebraic structure of C ∞(P/G ).

Application to reduction of symmetry

Let P/G be the space of orbits of a proper action of a connected Lie group G on a manifold P. Theorem Strata of the orbit type strati…cation of P/G are orbits of the family X(P/G ) of all vector …elds on P/G.

If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space. The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries. For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 12 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 The last step is to determine the geometric structure on P/G on the basis of the algebraic structure of C ∞(P/G ).

Application to reduction of symmetry

Let P/G be the space of orbits of a proper action of a connected Lie group G on a manifold P. Theorem Strata of the orbit type strati…cation of P/G are orbits of the family X(P/G ) of all vector …elds on P/G.

If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space. The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries. For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P. The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure. Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 12 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Application to reduction of symmetry

Let P/G be the space of orbits of a proper action of a connected Lie group G on a manifold P. Theorem Strata of the orbit type strati…cation of P/G are orbits of the family X(P/G ) of all vector …elds on P/G.

If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space. The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries. For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P. The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure. Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 12 J. ´SniatyckiThe (University last step of Calgary) is to determineSingular the Reduction geometric structure on P/G on the/ 22 basis of the algebraic structure of C ∞(P/G ). Corollary

The open neighboourhood GSp /G of the orbit Gp in P/G is di¤eomorphic to Sp /Gp .

1 Sp is invariant under the action of the iostropy group Gp of p. 2 GSp = gq g G, q Sp is an open G-invariant neighbourhood of p in P. f j 2 2 g 3 Sp (Gq) = Gp q for every q Sp . \ 2 By a Theorem of Palais (1961), the properness of the action of G on P ensures that for every point p P, there exists a slice Sp through p. 2

Palais’Slice Theorem

A slice for the action of G on P through a point p P is a 2 submanifold Sp containing p such that:

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 13 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary

The open neighboourhood GSp /G of the orbit Gp in P/G is di¤eomorphic to Sp /Gp .

2 GSp = gq g G, q Sp is an open G-invariant neighbourhood of p in P. f j 2 2 g 3 Sp (Gq) = Gp q for every q Sp . \ 2 By a Theorem of Palais (1961), the properness of the action of G on P ensures that for every point p P, there exists a slice Sp through p. 2

Palais’Slice Theorem

A slice for the action of G on P through a point p P is a 2 submanifold Sp containing p such that:

1 Sp is invariant under the action of the iostropy group Gp of p.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 13 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary

The open neighboourhood GSp /G of the orbit Gp in P/G is di¤eomorphic to Sp /Gp .

3 Sp (Gq) = Gp q for every q Sp . \ 2 By a Theorem of Palais (1961), the properness of the action of G on P ensures that for every point p P, there exists a slice Sp through p. 2

Palais’Slice Theorem

A slice for the action of G on P through a point p P is a 2 submanifold Sp containing p such that:

1 Sp is invariant under the action of the iostropy group Gp of p. 2 GSp = gq g G, q Sp is an open G-invariant neighbourhood of p in P. f j 2 2 g

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 13 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary

The open neighboourhood GSp /G of the orbit Gp in P/G is di¤eomorphic to Sp /Gp .

By a Theorem of Palais (1961), the properness of the action of G on P ensures that for every point p P, there exists a slice Sp through p. 2

Palais’Slice Theorem

A slice for the action of G on P through a point p P is a 2 submanifold Sp containing p such that:

1 Sp is invariant under the action of the iostropy group Gp of p. 2 GSp = gq g G, q Sp is an open G-invariant neighbourhood of p in P. f j 2 2 g 3 Sp (Gq) = Gp q for every q Sp . \ 2

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 13 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary

The open neighboourhood GSp /G of the orbit Gp in P/G is di¤eomorphic to Sp /Gp .

Palais’Slice Theorem

A slice for the action of G on P through a point p P is a 2 submanifold Sp containing p such that:

1 Sp is invariant under the action of the iostropy group Gp of p. 2 GSp = gq g G, q Sp is an open G-invariant neighbourhood of p in P. f j 2 2 g 3 Sp (Gq) = Gp q for every q Sp . \ 2 By a Theorem of Palais (1961), the properness of the action of G on P ensures that for every point p P, there exists a slice Sp through p. 2

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 13 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Palais’Slice Theorem

A slice for the action of G on P through a point p P is a 2 submanifold Sp containing p such that:

1 Sp is invariant under the action of the iostropy group Gp of p. 2 GSp = gq g G, q Sp is an open G-invariant neighbourhood of p in P. f j 2 2 g 3 Sp (Gq) = Gp q for every q Sp . \ 2 By a Theorem of Palais (1961), the properness of the action of G on P ensures that for every point p P, there exists a slice Sp through p. 2 Corollary

The open neighboourhood GSp /G of the orbit Gp in P/G is di¤eomorphic to Sp /Gp .

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 13 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary

Sp /Gp is di¤eomorphic to the orbit space Up /Gp of the linear action Ψp of Gp on Ep Up . 

By Bochner’sLinearization Lemma (1945), one can choose Sp so that the action of Gp on Sp is equivalent to the the restriction of Ψp to an open Gp -invariant neighbourhood Up of 0 in a subspace Ep of Tp P. In other words, there exists a di¤eomorphism ψ : Up Sp such that p ! ψ T Φg = Φg ψ p   p for all g Gp . 2

Bochner’sLinearization Lemma

Since p is a …xed point of Gp , the derived action of G restricted to Gp preserves Tp P, and induces a linear action

Ψp : Gp Tp P Tp P : (g, u) T Φg (u).  ! 7!

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 14 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary

Sp /Gp is di¤eomorphic to the orbit space Up /Gp of the linear action Ψp of Gp on Ep Up . 

In other words, there exists a di¤eomorphism ψ : Up Sp such that p ! ψ T Φg = Φg ψ p   p for all g Gp . 2

Bochner’sLinearization Lemma

Since p is a …xed point of Gp , the derived action of G restricted to Gp preserves Tp P, and induces a linear action

Ψp : Gp Tp P Tp P : (g, u) T Φg (u).  ! 7! By Bochner’sLinearization Lemma (1945), one can choose Sp so that the action of Gp on Sp is equivalent to the the restriction of Ψp to an open Gp -invariant neighbourhood Up of 0 in a subspace Ep of Tp P.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 14 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary

Sp /Gp is di¤eomorphic to the orbit space Up /Gp of the linear action Ψp of Gp on Ep Up . 

Bochner’sLinearization Lemma

Since p is a …xed point of Gp , the derived action of G restricted to Gp preserves Tp P, and induces a linear action

Ψp : Gp Tp P Tp P : (g, u) T Φg (u).  ! 7! By Bochner’sLinearization Lemma (1945), one can choose Sp so that the action of Gp on Sp is equivalent to the the restriction of Ψp to an open Gp -invariant neighbourhood Up of 0 in a subspace Ep of Tp P. In other words, there exists a di¤eomorphism ψ : Up Sp such that p ! ψ T Φg = Φg ψ p   p for all g Gp . 2

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 14 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Bochner’sLinearization Lemma

Since p is a …xed point of Gp , the derived action of G restricted to Gp preserves Tp P, and induces a linear action

Ψp : Gp Tp P Tp P : (g, u) T Φg (u).  ! 7! By Bochner’sLinearization Lemma (1945), one can choose Sp so that the action of Gp on Sp is equivalent to the the restriction of Ψp to an open Gp -invariant neighbourhood Up of 0 in a subspace Ep of Tp P. In other words, there exists a di¤eomorphism ψ : Up Sp such that p ! ψ T Φg = Φg ψ p   p for all g Gp . 2 Corollary

Sp /Gp is di¤eomorphic to the orbit space Up /Gp of the linear action Ψp of Gp on Ep Up .  Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 14 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 In other words, Up /Gp is a di¤erential subspace of the space Ep /Gp consisting of orbits of the linear action of Gp on Ep . By a theorem of G. Schwarz (1975), invariant smooth functions of a linear action of a compact group are smooth functions of algebraic invariants. By Weyl’sNullstellensatz (1946), the ring of algebraic invariants of a linear action of a compact group is …nitely generated.

Schwarz’sTheorem and Weyl’sNullstellensatz

The di¤erential structure of Up /Gp is generated by the restrictions to Up of smooth functions on Ep .

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 15 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 By a theorem of G. Schwarz (1975), invariant smooth functions of a linear action of a compact group are smooth functions of algebraic invariants. By Weyl’sNullstellensatz (1946), the ring of algebraic invariants of a linear action of a compact group is …nitely generated.

Schwarz’sTheorem and Weyl’sNullstellensatz

The di¤erential structure of Up /Gp is generated by the restrictions to Up of smooth functions on Ep .

In other words, Up /Gp is a di¤erential subspace of the space Ep /Gp consisting of orbits of the linear action of Gp on Ep .

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 15 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 By Weyl’sNullstellensatz (1946), the ring of algebraic invariants of a linear action of a compact group is …nitely generated.

Schwarz’sTheorem and Weyl’sNullstellensatz

The di¤erential structure of Up /Gp is generated by the restrictions to Up of smooth functions on Ep .

In other words, Up /Gp is a di¤erential subspace of the space Ep /Gp consisting of orbits of the linear action of Gp on Ep . By a theorem of G. Schwarz (1975), invariant smooth functions of a linear action of a compact group are smooth functions of algebraic invariants.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 15 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Schwarz’sTheorem and Weyl’sNullstellensatz

The di¤erential structure of Up /Gp is generated by the restrictions to Up of smooth functions on Ep .

In other words, Up /Gp is a di¤erential subspace of the space Ep /Gp consisting of orbits of the linear action of Gp on Ep . By a theorem of G. Schwarz (1975), invariant smooth functions of a linear action of a compact group are smooth functions of algebraic invariants. By Weyl’sNullstellensatz (1946), the ring of algebraic invariants of a linear action of a compact group is …nitely generated.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 15 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary n Ep /Gp is di¤eomorphic to Σ R . 

The corresponding Hilbert map is n σp : Ep R : v (σ1(v), ..., σn(v)). ! 7! n The map σp : Ep R induces a map ! σ˜ p : Ep /Gp Σ = σp (Ep ), ! which is an isomorphism onto its range. n Since the Hilbert map σp : Tp P R is algebraic, the ! Tarski-Seidenberg Theorem ensures that its range Σ is a semialgebraic subset of Rn. In particular, Σ is a closed subset of Rn and the di¤erential structure of Σ is given by restrictions of smooth functions on Rn to Σ.

Hilbert basis and Tarski-Seidenberg Theorem

A Hilbert basis of the action of Gp on Ep is a minimal set σ1, ..., σn of homogeneous generators of the ring of algebraic invariants.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 16 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary n Ep /Gp is di¤eomorphic to Σ R . 

n The map σp : Ep R induces a map ! σ˜ p : Ep /Gp Σ = σp (Ep ), ! which is an isomorphism onto its range. n Since the Hilbert map σp : Tp P R is algebraic, the ! Tarski-Seidenberg Theorem ensures that its range Σ is a semialgebraic subset of Rn. In particular, Σ is a closed subset of Rn and the di¤erential structure of Σ is given by restrictions of smooth functions on Rn to Σ.

Hilbert basis and Tarski-Seidenberg Theorem

A Hilbert basis of the action of Gp on Ep is a minimal set σ1, ..., σn of homogeneous generators of the ring of algebraic invariants. The corresponding Hilbert map is n σp : Ep R : v (σ1(v), ..., σn(v)). ! 7!

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 16 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary n Ep /Gp is di¤eomorphic to Σ R . 

n Since the Hilbert map σp : Tp P R is algebraic, the ! Tarski-Seidenberg Theorem ensures that its range Σ is a semialgebraic subset of Rn. In particular, Σ is a closed subset of Rn and the di¤erential structure of Σ is given by restrictions of smooth functions on Rn to Σ.

Hilbert basis and Tarski-Seidenberg Theorem

A Hilbert basis of the action of Gp on Ep is a minimal set σ1, ..., σn of homogeneous generators of the ring of algebraic invariants. The corresponding Hilbert map is n σp : Ep R : v (σ1(v), ..., σn(v)). ! 7! n The map σp : Ep R induces a map ! σ˜ p : Ep /Gp Σ = σp (Ep ), ! which is an isomorphism onto its range.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 16 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary n Ep /Gp is di¤eomorphic to Σ R . 

In particular, Σ is a closed subset of Rn and the di¤erential structure of Σ is given by restrictions of smooth functions on Rn to Σ.

Hilbert basis and Tarski-Seidenberg Theorem

A Hilbert basis of the action of Gp on Ep is a minimal set σ1, ..., σn of homogeneous generators of the ring of algebraic invariants. The corresponding Hilbert map is n σp : Ep R : v (σ1(v), ..., σn(v)). ! 7! n The map σp : Ep R induces a map ! σ˜ p : Ep /Gp Σ = σp (Ep ), ! which is an isomorphism onto its range. n Since the Hilbert map σp : Tp P R is algebraic, the ! Tarski-Seidenberg Theorem ensures that its range Σ is a semialgebraic subset of Rn.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 16 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Corollary n Ep /Gp is di¤eomorphic to Σ R . 

Hilbert basis and Tarski-Seidenberg Theorem

A Hilbert basis of the action of Gp on Ep is a minimal set σ1, ..., σn of homogeneous generators of the ring of algebraic invariants. The corresponding Hilbert map is n σp : Ep R : v (σ1(v), ..., σn(v)). ! 7! n The map σp : Ep R induces a map ! σ˜ p : Ep /Gp Σ = σp (Ep ), ! which is an isomorphism onto its range. n Since the Hilbert map σp : Tp P R is algebraic, the ! Tarski-Seidenberg Theorem ensures that its range Σ is a semialgebraic subset of Rn. In particular, Σ is a closed subset of Rn and the di¤erential structure of Σ is given by restrictions of smooth functions on Rn to Σ.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 16 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Hilbert basis and Tarski-Seidenberg Theorem

A Hilbert basis of the action of Gp on Ep is a minimal set σ1, ..., σn of homogeneous generators of the ring of algebraic invariants. The corresponding Hilbert map is n σp : Ep R : v (σ1(v), ..., σn(v)). ! 7! n The map σp : Ep R induces a map ! σ˜ p : Ep /Gp Σ = σp (Ep ), ! which is an isomorphism onto its range. n Since the Hilbert map σp : Tp P R is algebraic, the ! Tarski-Seidenberg Theorem ensures that its range Σ is a semialgebraic subset of Rn. In particular, Σ is a closed subset of Rn and the di¤erential structure of Σ is given by restrictions of smooth functions on Rn to Σ. Corollary n Ep /Gp is di¤eomorphic to Σ R .  Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 16 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 1 GSp /G is di¤eomorphic to Sp /Gp , where Sp is a slice at p and Gp is the isotropy group of p. 2 Sp /Gp is di¤eomorphic to Up /Gp , where Up is an invariant open subset of a vector subspace Ep of Tp P and the action of Gp on Ep is linear. n 3 Ep /Gp is di¤eomorphic to Σ R .  n Hence, GSp /G is di¤eomorphic to an open subset of R . It is easy to show that the topology of the orbit space is Hausdor¤. Therefore, P/G is subcartesian.

Summary

We have obtained a sequence of di¤eomorphisms of neighbourhoods of an arbitrary orbit Gp P/G. 2

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 17 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 2 Sp /Gp is di¤eomorphic to Up /Gp , where Up is an invariant open subset of a vector subspace Ep of Tp P and the action of Gp on Ep is linear. n 3 Ep /Gp is di¤eomorphic to Σ R .  n Hence, GSp /G is di¤eomorphic to an open subset of R . It is easy to show that the topology of the orbit space is Hausdor¤. Therefore, P/G is subcartesian.

Summary

We have obtained a sequence of di¤eomorphisms of neighbourhoods of an arbitrary orbit Gp P/G. 2 1 GSp /G is di¤eomorphic to Sp /Gp , where Sp is a slice at p and Gp is the isotropy group of p.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 17 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 n 3 Ep /Gp is di¤eomorphic to Σ R .  n Hence, GSp /G is di¤eomorphic to an open subset of R . It is easy to show that the topology of the orbit space is Hausdor¤. Therefore, P/G is subcartesian.

Summary

We have obtained a sequence of di¤eomorphisms of neighbourhoods of an arbitrary orbit Gp P/G. 2 1 GSp /G is di¤eomorphic to Sp /Gp , where Sp is a slice at p and Gp is the isotropy group of p. 2 Sp /Gp is di¤eomorphic to Up /Gp , where Up is an invariant open subset of a vector subspace Ep of Tp P and the action of Gp on Ep is linear.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 17 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 n Hence, GSp /G is di¤eomorphic to an open subset of R . It is easy to show that the topology of the orbit space is Hausdor¤. Therefore, P/G is subcartesian.

Summary

We have obtained a sequence of di¤eomorphisms of neighbourhoods of an arbitrary orbit Gp P/G. 2 1 GSp /G is di¤eomorphic to Sp /Gp , where Sp is a slice at p and Gp is the isotropy group of p. 2 Sp /Gp is di¤eomorphic to Up /Gp , where Up is an invariant open subset of a vector subspace Ep of Tp P and the action of Gp on Ep is linear. n 3 Ep /Gp is di¤eomorphic to Σ R . 

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 17 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 It is easy to show that the topology of the orbit space is Hausdor¤. Therefore, P/G is subcartesian.

Summary

We have obtained a sequence of di¤eomorphisms of neighbourhoods of an arbitrary orbit Gp P/G. 2 1 GSp /G is di¤eomorphic to Sp /Gp , where Sp is a slice at p and Gp is the isotropy group of p. 2 Sp /Gp is di¤eomorphic to Up /Gp , where Up is an invariant open subset of a vector subspace Ep of Tp P and the action of Gp on Ep is linear. n 3 Ep /Gp is di¤eomorphic to Σ R .  n Hence, GSp /G is di¤eomorphic to an open subset of R .

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 17 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Therefore, P/G is subcartesian.

Summary

We have obtained a sequence of di¤eomorphisms of neighbourhoods of an arbitrary orbit Gp P/G. 2 1 GSp /G is di¤eomorphic to Sp /Gp , where Sp is a slice at p and Gp is the isotropy group of p. 2 Sp /Gp is di¤eomorphic to Up /Gp , where Up is an invariant open subset of a vector subspace Ep of Tp P and the action of Gp on Ep is linear. n 3 Ep /Gp is di¤eomorphic to Σ R .  n Hence, GSp /G is di¤eomorphic to an open subset of R . It is easy to show that the topology of the orbit space is Hausdor¤.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 17 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Summary

We have obtained a sequence of di¤eomorphisms of neighbourhoods of an arbitrary orbit Gp P/G. 2 1 GSp /G is di¤eomorphic to Sp /Gp , where Sp is a slice at p and Gp is the isotropy group of p. 2 Sp /Gp is di¤eomorphic to Up /Gp , where Up is an invariant open subset of a vector subspace Ep of Tp P and the action of Gp on Ep is linear. n 3 Ep /Gp is di¤eomorphic to Σ R .  n Hence, GSp /G is di¤eomorphic to an open subset of R . It is easy to show that the topology of the orbit space is Hausdor¤. Therefore, P/G is subcartesian.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 17 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries. For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P. The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure. The next step is to determine the geometric structure on P/G on the basis of the algebraic structure of C ∞(P/G ).

Reduction of symmetries

If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 18 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P. The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure. The next step is to determine the geometric structure on P/G on the basis of the algebraic structure of C ∞(P/G ).

Reduction of symmetries

If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space. The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 18 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure. The next step is to determine the geometric structure on P/G on the basis of the algebraic structure of C ∞(P/G ).

Reduction of symmetries

If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space. The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries. For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 18 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 The next step is to determine the geometric structure on P/G on the basis of the algebraic structure of C ∞(P/G ).

Reduction of symmetries

If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space. The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries. For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P. The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 18 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Reduction of symmetries

If P has a geometric structure invariant under the action of G, then this structure induces an additional structure on the orbit space. The process of determination of the structure of the orbit space P/G induced by an invariant geometric structure on P is called reduction of symmetries. For a proper action of G on P, it is convenient to encode the geometric structure on P as an algebraic structure on the ring C ∞(P)G of smooth functions on P. The di¤erential structure C ∞(P/G ) inherits an isomorphic algebraic structure. The next step is to determine the geometric structure on P/G on the basis of the algebraic structure of C ∞(P/G ).

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 18 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 The symplectic structure on P de…nes a Poisson algebra structure on C ∞(P). Since ω is G-invariant, the induced Poisson structure on C ∞(P) is G-invariant. Hence, C ∞(P)G is a Poisson subalgebra of C ∞(P). Therefore, C ∞(P/G ) has the structure of a Poisson algebra. ∞ For each h C (P/G ), the Poisson derivation Xh of h is given by 2 ∞ Xh(f ) = f , h for all f C (P/G ). f g 2 Poisson derivations of C ∞(P/G ) are vector …elds on P/G. Orbits of the family of all Poisson vector …elds are symplectic manifolds. Thus, the orbit space P/G is strati…ed. Each stratum is a Poisson manifold singularly foliated by symplectic manifolds.

Symplectic reduction

Suppose that P is endowed with an invariant symplectic form ω.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 19 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Since ω is G-invariant, the induced Poisson structure on C ∞(P) is G-invariant. Hence, C ∞(P)G is a Poisson subalgebra of C ∞(P). Therefore, C ∞(P/G ) has the structure of a Poisson algebra. ∞ For each h C (P/G ), the Poisson derivation Xh of h is given by 2 ∞ Xh(f ) = f , h for all f C (P/G ). f g 2 Poisson derivations of C ∞(P/G ) are vector …elds on P/G. Orbits of the family of all Poisson vector …elds are symplectic manifolds. Thus, the orbit space P/G is strati…ed. Each stratum is a Poisson manifold singularly foliated by symplectic manifolds.

Symplectic reduction

Suppose that P is endowed with an invariant symplectic form ω. The symplectic structure on P de…nes a Poisson algebra structure on C ∞(P).

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 19 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Hence, C ∞(P)G is a Poisson subalgebra of C ∞(P). Therefore, C ∞(P/G ) has the structure of a Poisson algebra. ∞ For each h C (P/G ), the Poisson derivation Xh of h is given by 2 ∞ Xh(f ) = f , h for all f C (P/G ). f g 2 Poisson derivations of C ∞(P/G ) are vector …elds on P/G. Orbits of the family of all Poisson vector …elds are symplectic manifolds. Thus, the orbit space P/G is strati…ed. Each stratum is a Poisson manifold singularly foliated by symplectic manifolds.

Symplectic reduction

Suppose that P is endowed with an invariant symplectic form ω. The symplectic structure on P de…nes a Poisson algebra structure on C ∞(P). Since ω is G-invariant, the induced Poisson structure on C ∞(P) is G-invariant.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 19 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Therefore, C ∞(P/G ) has the structure of a Poisson algebra. ∞ For each h C (P/G ), the Poisson derivation Xh of h is given by 2 ∞ Xh(f ) = f , h for all f C (P/G ). f g 2 Poisson derivations of C ∞(P/G ) are vector …elds on P/G. Orbits of the family of all Poisson vector …elds are symplectic manifolds. Thus, the orbit space P/G is strati…ed. Each stratum is a Poisson manifold singularly foliated by symplectic manifolds.

Symplectic reduction

Suppose that P is endowed with an invariant symplectic form ω. The symplectic structure on P de…nes a Poisson algebra structure on C ∞(P). Since ω is G-invariant, the induced Poisson structure on C ∞(P) is G-invariant. Hence, C ∞(P)G is a Poisson subalgebra of C ∞(P).

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 19 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 ∞ For each h C (P/G ), the Poisson derivation Xh of h is given by 2 ∞ Xh(f ) = f , h for all f C (P/G ). f g 2 Poisson derivations of C ∞(P/G ) are vector …elds on P/G. Orbits of the family of all Poisson vector …elds are symplectic manifolds. Thus, the orbit space P/G is strati…ed. Each stratum is a Poisson manifold singularly foliated by symplectic manifolds.

Symplectic reduction

Suppose that P is endowed with an invariant symplectic form ω. The symplectic structure on P de…nes a Poisson algebra structure on C ∞(P). Since ω is G-invariant, the induced Poisson structure on C ∞(P) is G-invariant. Hence, C ∞(P)G is a Poisson subalgebra of C ∞(P). Therefore, C ∞(P/G ) has the structure of a Poisson algebra.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 19 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Poisson derivations of C ∞(P/G ) are vector …elds on P/G. Orbits of the family of all Poisson vector …elds are symplectic manifolds. Thus, the orbit space P/G is strati…ed. Each stratum is a Poisson manifold singularly foliated by symplectic manifolds.

Symplectic reduction

Suppose that P is endowed with an invariant symplectic form ω. The symplectic structure on P de…nes a Poisson algebra structure on C ∞(P). Since ω is G-invariant, the induced Poisson structure on C ∞(P) is G-invariant. Hence, C ∞(P)G is a Poisson subalgebra of C ∞(P). Therefore, C ∞(P/G ) has the structure of a Poisson algebra. ∞ For each h C (P/G ), the Poisson derivation Xh of h is given by 2 ∞ Xh(f ) = f , h for all f C (P/G ). f g 2

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 19 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Orbits of the family of all Poisson vector …elds are symplectic manifolds. Thus, the orbit space P/G is strati…ed. Each stratum is a Poisson manifold singularly foliated by symplectic manifolds.

Symplectic reduction

Suppose that P is endowed with an invariant symplectic form ω. The symplectic structure on P de…nes a Poisson algebra structure on C ∞(P). Since ω is G-invariant, the induced Poisson structure on C ∞(P) is G-invariant. Hence, C ∞(P)G is a Poisson subalgebra of C ∞(P). Therefore, C ∞(P/G ) has the structure of a Poisson algebra. ∞ For each h C (P/G ), the Poisson derivation Xh of h is given by 2 ∞ Xh(f ) = f , h for all f C (P/G ). f g 2 Poisson derivations of C ∞(P/G ) are vector …elds on P/G.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 19 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Thus, the orbit space P/G is strati…ed. Each stratum is a Poisson manifold singularly foliated by symplectic manifolds.

Symplectic reduction

Suppose that P is endowed with an invariant symplectic form ω. The symplectic structure on P de…nes a Poisson algebra structure on C ∞(P). Since ω is G-invariant, the induced Poisson structure on C ∞(P) is G-invariant. Hence, C ∞(P)G is a Poisson subalgebra of C ∞(P). Therefore, C ∞(P/G ) has the structure of a Poisson algebra. ∞ For each h C (P/G ), the Poisson derivation Xh of h is given by 2 ∞ Xh(f ) = f , h for all f C (P/G ). f g 2 Poisson derivations of C ∞(P/G ) are vector …elds on P/G. Orbits of the family of all Poisson vector …elds are symplectic manifolds.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 19 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Symplectic reduction

Suppose that P is endowed with an invariant symplectic form ω. The symplectic structure on P de…nes a Poisson algebra structure on C ∞(P). Since ω is G-invariant, the induced Poisson structure on C ∞(P) is G-invariant. Hence, C ∞(P)G is a Poisson subalgebra of C ∞(P). Therefore, C ∞(P/G ) has the structure of a Poisson algebra. ∞ For each h C (P/G ), the Poisson derivation Xh of h is given by 2 ∞ Xh(f ) = f , h for all f C (P/G ). f g 2 Poisson derivations of C ∞(P/G ) are vector …elds on P/G. Orbits of the family of all Poisson vector …elds are symplectic manifolds. Thus, the orbit space P/G is strati…ed. Each stratum is a Poisson manifold singularly foliated by symplectic manifolds. Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 19 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 If ω is symplectic, for every f C ∞(P) there exists a unique vector 2 …eld Xf on P such that

Xf ω = df . Xf is called the Hamiltonian vector …eld of f . If f describes the energy of a mechanical system, integral curves of Xf give time evolution of the system. ∞ The Posson bracket of f1 and f2 in C (P) is

f1, f2 = ω(Xf , Xf ). f g 1 2 Poisson bracket is bilinear, skew-symmetric, acts as a derivation and satis…es the Jacobi identity. C ∞(P) with the Poisson bracket , is called the Poisson algebra of f

g (P, ω).

Details of symplectic reduction

A 2-form ω on P is symplectic if dω = 0 and ω is nondegenerate.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 20 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Xf is called the Hamiltonian vector …eld of f . If f describes the energy of a mechanical system, integral curves of Xf give time evolution of the system. ∞ The Posson bracket of f1 and f2 in C (P) is

f1, f2 = ω(Xf , Xf ). f g 1 2 Poisson bracket is bilinear, skew-symmetric, acts as a derivation and satis…es the Jacobi identity. C ∞(P) with the Poisson bracket , is called the Poisson algebra of f

g (P, ω).

Details of symplectic reduction

A 2-form ω on P is symplectic if dω = 0 and ω is nondegenerate. If ω is symplectic, for every f C ∞(P) there exists a unique vector 2 …eld Xf on P such that

Xf ω = df .

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 20 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 If f describes the energy of a mechanical system, integral curves of Xf give time evolution of the system. ∞ The Posson bracket of f1 and f2 in C (P) is

f1, f2 = ω(Xf , Xf ). f g 1 2 Poisson bracket is bilinear, skew-symmetric, acts as a derivation and satis…es the Jacobi identity. C ∞(P) with the Poisson bracket , is called the Poisson algebra of f

g (P, ω).

Details of symplectic reduction

A 2-form ω on P is symplectic if dω = 0 and ω is nondegenerate. If ω is symplectic, for every f C ∞(P) there exists a unique vector 2 …eld Xf on P such that

Xf ω = df . Xf is called the Hamiltonian vector …eld of f .

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 20 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 ∞ The Posson bracket of f1 and f2 in C (P) is

f1, f2 = ω(Xf , Xf ). f g 1 2 Poisson bracket is bilinear, skew-symmetric, acts as a derivation and satis…es the Jacobi identity. C ∞(P) with the Poisson bracket , is called the Poisson algebra of f

g (P, ω).

Details of symplectic reduction

A 2-form ω on P is symplectic if dω = 0 and ω is nondegenerate. If ω is symplectic, for every f C ∞(P) there exists a unique vector 2 …eld Xf on P such that

Xf ω = df . Xf is called the Hamiltonian vector …eld of f . If f describes the energy of a mechanical system, integral curves of Xf give time evolution of the system.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 20 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Poisson bracket is bilinear, skew-symmetric, acts as a derivation and satis…es the Jacobi identity. C ∞(P) with the Poisson bracket , is called the Poisson algebra of f

g (P, ω).

Details of symplectic reduction

A 2-form ω on P is symplectic if dω = 0 and ω is nondegenerate. If ω is symplectic, for every f C ∞(P) there exists a unique vector 2 …eld Xf on P such that

Xf ω = df . Xf is called the Hamiltonian vector …eld of f . If f describes the energy of a mechanical system, integral curves of Xf give time evolution of the system. ∞ The Posson bracket of f1 and f2 in C (P) is

f1, f2 = ω(Xf , Xf ). f g 1 2

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 20 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 C ∞(P) with the Poisson bracket , is called the Poisson algebra of f

g (P, ω).

Details of symplectic reduction

A 2-form ω on P is symplectic if dω = 0 and ω is nondegenerate. If ω is symplectic, for every f C ∞(P) there exists a unique vector 2 …eld Xf on P such that

Xf ω = df . Xf is called the Hamiltonian vector …eld of f . If f describes the energy of a mechanical system, integral curves of Xf give time evolution of the system. ∞ The Posson bracket of f1 and f2 in C (P) is

f1, f2 = ω(Xf , Xf ). f g 1 2 Poisson bracket is bilinear, skew-symmetric, acts as a derivation and satis…es the Jacobi identity.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 20 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Details of symplectic reduction

A 2-form ω on P is symplectic if dω = 0 and ω is nondegenerate. If ω is symplectic, for every f C ∞(P) there exists a unique vector 2 …eld Xf on P such that

Xf ω = df . Xf is called the Hamiltonian vector …eld of f . If f describes the energy of a mechanical system, integral curves of Xf give time evolution of the system. ∞ The Posson bracket of f1 and f2 in C (P) is

f1, f2 = ω(Xf , Xf ). f g 1 2 Poisson bracket is bilinear, skew-symmetric, acts as a derivation and satis…es the Jacobi identity. C ∞(P) with the Poisson bracket , is called the Poisson algebra of f

g (P, ω). Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 20 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 A action Φ : G P P : (g, p) Φg (p) = gp is Hamiltonian if  ! 7! there exists an Ad -equivariant map J : P g such that for each G !  ξ g, the action of exp tξ on P is given by translations along integral 2 curves of the Hamiltonian vector …eld X J ξ . h j i Symplectic reduction deals with the structure of of the orbit space P/G of a proper Hamiltonian action of G on (P, ω). Hamiltonian action of G on (P, ω) preserves ω. Hence, Hamiltonian action of G on (P, ω) preserves preserves the Poisson bracket.

Hamiltonian action

Let G be a connected Lie group with a Lie algebra g and its dual g.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 21 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Symplectic reduction deals with the structure of of the orbit space P/G of a proper Hamiltonian action of G on (P, ω). Hamiltonian action of G on (P, ω) preserves ω. Hence, Hamiltonian action of G on (P, ω) preserves preserves the Poisson bracket.

Hamiltonian action

Let G be a connected Lie group with a Lie algebra g and its dual g. A action Φ : G P P : (g, p) Φg (p) = gp is Hamiltonian if  ! 7! there exists an Ad -equivariant map J : P g such that for each G !  ξ g, the action of exp tξ on P is given by translations along integral 2 curves of the Hamiltonian vector …eld X J ξ . h j i

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 21 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Hamiltonian action of G on (P, ω) preserves ω. Hence, Hamiltonian action of G on (P, ω) preserves preserves the Poisson bracket.

Hamiltonian action

Let G be a connected Lie group with a Lie algebra g and its dual g. A action Φ : G P P : (g, p) Φg (p) = gp is Hamiltonian if  ! 7! there exists an Ad -equivariant map J : P g such that for each G !  ξ g, the action of exp tξ on P is given by translations along integral 2 curves of the Hamiltonian vector …eld X J ξ . h j i Symplectic reduction deals with the structure of of the orbit space P/G of a proper Hamiltonian action of G on (P, ω).

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 21 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Hence, Hamiltonian action of G on (P, ω) preserves preserves the Poisson bracket.

Hamiltonian action

Let G be a connected Lie group with a Lie algebra g and its dual g. A action Φ : G P P : (g, p) Φg (p) = gp is Hamiltonian if  ! 7! there exists an Ad -equivariant map J : P g such that for each G !  ξ g, the action of exp tξ on P is given by translations along integral 2 curves of the Hamiltonian vector …eld X J ξ . h j i Symplectic reduction deals with the structure of of the orbit space P/G of a proper Hamiltonian action of G on (P, ω). Hamiltonian action of G on (P, ω) preserves ω.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 21 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Hamiltonian action

Let G be a connected Lie group with a Lie algebra g and its dual g. A action Φ : G P P : (g, p) Φg (p) = gp is Hamiltonian if  ! 7! there exists an Ad -equivariant map J : P g such that for each G !  ξ g, the action of exp tξ on P is given by translations along integral 2 curves of the Hamiltonian vector …eld X J ξ . h j i Symplectic reduction deals with the structure of of the orbit space P/G of a proper Hamiltonian action of G on (P, ω). Hamiltonian action of G on (P, ω) preserves ω. Hence, Hamiltonian action of G on (P, ω) preserves preserves the Poisson bracket.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 21 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Theorem Each stratum of the orbit type strati…cation of P/G is a Poisson manifold singularly foliated by symplectic manifolds that are orbits of the family of all Poisson vector …elds on P/G.

Hence, the di¤erential structure C ∞(P/G ) is a Poisson algebra. ∞ For each f C (P/G ), the derivation Yf de…ned by 2 ∞ Yf (h) = Yf , h h C (P/G ) f g 8 2 is a vector …eld on P/G, called the Poisson vector …eld of f . We know that orbits of the family X(P/G ) of all vector …elds on P/G are strata of the orbit type strati…cation of P/G.

Poisson reduction

The space C ∞(P)G is a Poisson subalgebra of C ∞(P).

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 22 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Theorem Each stratum of the orbit type strati…cation of P/G is a Poisson manifold singularly foliated by symplectic manifolds that are orbits of the family of all Poisson vector …elds on P/G.

∞ For each f C (P/G ), the derivation Yf de…ned by 2 ∞ Yf (h) = Yf , h h C (P/G ) f g 8 2 is a vector …eld on P/G, called the Poisson vector …eld of f . We know that orbits of the family X(P/G ) of all vector …elds on P/G are strata of the orbit type strati…cation of P/G.

Poisson reduction

The space C ∞(P)G is a Poisson subalgebra of C ∞(P). Hence, the di¤erential structure C ∞(P/G ) is a Poisson algebra.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 22 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Theorem Each stratum of the orbit type strati…cation of P/G is a Poisson manifold singularly foliated by symplectic manifolds that are orbits of the family of all Poisson vector …elds on P/G.

We know that orbits of the family X(P/G ) of all vector …elds on P/G are strata of the orbit type strati…cation of P/G.

Poisson reduction

The space C ∞(P)G is a Poisson subalgebra of C ∞(P). Hence, the di¤erential structure C ∞(P/G ) is a Poisson algebra. ∞ For each f C (P/G ), the derivation Yf de…ned by 2 ∞ Yf (h) = Yf , h h C (P/G ) f g 8 2 is a vector …eld on P/G, called the Poisson vector …eld of f .

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 22 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Theorem Each stratum of the orbit type strati…cation of P/G is a Poisson manifold singularly foliated by symplectic manifolds that are orbits of the family of all Poisson vector …elds on P/G.

Poisson reduction

The space C ∞(P)G is a Poisson subalgebra of C ∞(P). Hence, the di¤erential structure C ∞(P/G ) is a Poisson algebra. ∞ For each f C (P/G ), the derivation Yf de…ned by 2 ∞ Yf (h) = Yf , h h C (P/G ) f g 8 2 is a vector …eld on P/G, called the Poisson vector …eld of f . We know that orbits of the family X(P/G ) of all vector …elds on P/G are strata of the orbit type strati…cation of P/G.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 22 J. ´Sniatycki (University of Calgary) Singular Reduction / 22 Poisson reduction

The space C ∞(P)G is a Poisson subalgebra of C ∞(P). Hence, the di¤erential structure C ∞(P/G ) is a Poisson algebra. ∞ For each f C (P/G ), the derivation Yf de…ned by 2 ∞ Yf (h) = Yf , h h C (P/G ) f g 8 2 is a vector …eld on P/G, called the Poisson vector …eld of f . We know that orbits of the family X(P/G ) of all vector …elds on P/G are strata of the orbit type strati…cation of P/G. Theorem Each stratum of the orbit type strati…cation of P/G is a Poisson manifold singularly foliated by symplectic manifolds that are orbits of the family of all Poisson vector …elds on P/G.

Mechanics and Geometry in Canada Fields Instutute Toronto 16 July 2012 22 J. ´Sniatycki (University of Calgary) Singular Reduction / 22