Symplectic Geometry and Moment Maps Flashcards by Andrew Frohmader

What one definition and 4 theorems are the core of the course?

Def. A Hamiltonian T-Space is (M manifold with T action, w T-invariant 2-form, mu: M –> Lie(T)*) s.t.
1. (M, mu) is symplectic
2. mu is T-invariant
3. fundamental v.f. exact form…

Thm 1 (Symplectic reduction)

Thm 2 (Convexity)

Thm 3 (Localization)

Thm 4 ([Q, R] = 0)

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Define: symplectic vector space, w flat, pi sharp

f.d vector aspace over R with 2-form w s.t. ker(w) = 0

w flat (x) = w(x, _)

pi sharp = pi(z, _)

pg 4, 5

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Prove: (V,w) symplectic <=> w flat an iso

pg 4

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Prove: (V, w) symplectic <=> exist unique pi s.t. (w flat)^(-1) = pi sharp

pg 5

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Define: standard symplectic form, show symplectic

pg 6

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Discuss how to define a symplectic form on V + V*

pg 7

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Discuss how to define a symplectic form on a Hermitian vector space over C. Show symplectic

pg 8

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Define U^(w) and compare to U perp

pg 9

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What is (U^w)^w? Proof?

= U
pg 10

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Define: isotropic, co-isotropic, Lagrangian, symplectic

relationships?

iso: U < U^w
co-iso: U^w < U
Lagrangian: U^w = U
symplectic: U int U^w = {0}

pg 10

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Lagrangian subspaces in L(+)L*?

pg 11

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What is Linear Reduction Lemma? Proof?

(V,w) symplectic, U < V isotropic => W = U^w/U is symplectic.

pg 12

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Prove there always exist Lagrangian subspaces in a symplectic vector space. Dimension?

dim V even

L = max isotropic

pg 13-14

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Prove: L < V Lagrangian => there exists M < V Lagrangian s.t. V = L (+) M

pg 15

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Prove any symplectic vector space is iso to standard symplectic vector space

pg 15

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Relationship between Liouville volume form and symplectic form?

pg 16-17

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Define: symplectic manifold

Smooth manifold with non-degenerate closed 2-form pg 18

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Discuss orientation, dimension, and volume of symplectic manifold. Cohomology?

Even dimensional

Louiville volume form gives canonical orientation and symplectic volume (if M compact)

Pg 18-19

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Define: symplectomorphism, Sp(M, w), Chi(M, w)

symplectomorphism if diffeo and f*w2 = w1

Sp(M,w) = group of symplectomorphisms inside Diff+(M)

Chi(M, w) = Lie algebra of symplectic vector fields

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Discuss relationship between symplectomorphisms and symplectic vector fields

If M compact, then have bijective correspondence. Only obstruction to this is integrability of vector field

pg 20

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Define: Hamiltonian vector field of f, Chi_Ham(M, w)

Equivalent defs?

pi#(df) := v_f

Just turn differential of f, a 1-form, into a vector field by the uniquely defined map pi# : T*M –> TM

Chi_Ham(M,w) is Lie algebra of all Hamiltonian vector fields

pg 21

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Prove Chi(M,w) is a Lie subalgebra of Chi(M) and Chi_Ham(M,w) a Lie subalgebra of Chi(M,w)

Define: Poisson bracket, prove properties

Show map from C^inf(M) –> Chi_Ham(M, w) is a Lie algebra hom

That Chi(M,w) a Lie subalgebra is easy from Lie derivative commutes with bracket.

Check v_f symplectic (Cartan magic)
Prove Lie subalgebra
Show different expressions for Poisson bracket are equivalent
Show {,} is Lie bracket (Skew automatic, Jacobi just compute from defs)
Show C^inf(M) –> Chi_Ham(M, w) is a Lie algebra hom (already proved above)

pg 22-24

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Discuss symplectic structure on R^2n
2-form, volume form, Hamiltonian VFs, flows, Poisson structure

pg 25

Discuss symplectic structure on orientable 2-manifolds

No constraint on w - choose any non-degenerate 2-form and it is automatically closed by dimension

pg 26

Discuss symplectic structure on Cotangent bundles. What does it look like in local coordinates?

Prop. Exists unique one form on cotangent bundle s.t. if we pull it back to base manifold with 1-form alpha we get alpha. pg 26 - 30

Discuss/define Kahler manifolds

N = mfld/C with Hermitian structure M = N_R <-- viewed as real manifold w(u,v) = Im h(u,v) (N,h) is Kahler if dw = 0 => (M,w) symplectic pg 31 - 32

Show N = C^n is Kahler. What is w in basis zj = qj + ipj?

pg 32

Show N = CP^1 is Kahler.

Have SU(2) action on CP^1 by mobius transforms. Define hermitian structure on CP^1 by SU(2) invariance. Unlike defining structures on Lie groups, we don't have a free action here - need to make sure this is well-defined. (Working with symmetric space not Lie group) Check stabilizer of point preserves structure. Get SU(2) invariant symplectic structure on CP^1 pg 33 - 35

Are complex submanifolds of Kahler manifold Kahler? Proof?

Hermitian structure restricts nicely Differential commutes with pullback --> 2-form still closed. pg 35

Discuss time dependent vector fields

Lee

What is Moser Lemma/Trick? Proof?

M = mfld, w_t smooth family of symplectic forms on M. Assume: dw_t/dt is exact 2-form => [w_t] constant. Define: Moser vector field Assume v_t integrates to theta_t Then (M, w0) and (M, wt) symplectomorphic. PF. Again Cartan magic. pg 36 - 37

State and Prove Darboux Thm.

There exists a cover of any symplectic manifold by Darboux charts --> symplectic manifolds have no local invariants, in sharp contrast with curvature in Riemannian geometry. All patches look the same. pg 38-41

Discuss when two Kahler structures on N yield symplectomorphic manifolds.

Banyaga. 2 Kahler structures, only need to compare cohomology class of symplectic form. pg 41

Define: normal bundle Example of Lagrangian submanifold?

Lee pg 45

Discuss Tubular Neighborhood Thm using Riemannian metric

Lee pg 46

State and Prove a normal form theory for Lagrangian submanifolds

Weinstein Lagrangian Embedding Thm. (M,w) symplectic, L < M cpt Lagrangian submfd => exists tubular neighborhood symplectomorphic to a neighborhood of zero-section in T*L. pg 47 - 51

State and discuss (without proof) Co-isotropic embedding thm

Again gives symplectomorphism of tubular neighborhoods, but not a standard model - in absence of more structure there is not a convincing normal form for co-isotropic submanifolds

Discuss torus representations over R

pg 53-55

Discuss stabilizer subgroups for different S^1 actions on R^2.

pg 59

Discuss orbits of S^1 on R^2 and S^2. Oribits of U(n) on Hermitian matrices

pg 60-61

Define: fundamental vector fields compare to treatment in ch 20 of Lie on infinitesimal generator of group action. Relation to orbits?

action yields vector field in M associated to each element of g <-- fund vector field. Remember g = left-invariant vector fields on G. VF <=> flows... Orbits are submanifolds - tangent space at a point of orbit spanned by fundamental VFs pg 63-64

Discuss G-action on Normal bundle to an orbit. Gm-action? Example of S^1 acting on S^2?

pg 64-65

Discuss tubular neighborhoods of orbit

We can always choose tubular neighborhood using geodesic rays - diffeo between subset of M and NOrbit. If G compact, can average Riemannian metric over group action --> get G-invaraint metric --> geodesics now G invaraint so get our diffeo G-invaraint pg 66

Show normal bundle of orbit equivalent to kind of product. Proof?

pg 67-68

Prove: Connected components of M^G are submanifolds of M

pg 69-70

In what sense are group actions rigid?

pg 71

Discuss the relationship between zero-set of fundamental vector field and fixed points of 1-param subgroups. Structure of zero-set? What about for arbitrary vector fields?

zeros of vector field = fixed points of 1-param subgroup --> a smooth manifold (assuming group compact) Zero set need not be a subman if vector field doesn't come from group action pg 72-73

What is the kernel of the map from Lie(G) = g --> TmOm (the tangent space of the orbit of m at m)? What about the orbit map G --> Om? Relationship?

g_m ={elements of Lie algebra s.t. associated fundamental vector field vanishes at m} g_m = {0} <=> G_m finite dim O_m = dim g - dim g_m pg 74

Define: equivariant map, maps between stabilizer subgroups?

f(gm) = gf(m) G_m includes into G_n pg 75

Define: Adjoint action, examples?

pg 76

Define: coadjoint action, examples?

pg 77

Define: Hamiltonian G-Space

Triple: (Manifold, 2-form, moment map) s.t. 1. (M, w) symplectic 2. mu: M --> g* equivariant w.r.t. coadjoint action 3. Contracting w w.r.t fundamental v.f. = differential of natural pairing of moment map and element of g yielding fund v.f. <-- compare to def of Hamiltonian vector field pg 78

Discuss the kernel and image of differential of moment map

1. kernel at m = TmOn^w = subspace orthogonal to tangent space of orbit 2. Image at m = g_m^perp = elements of g* vanishing on g_m pg 79-81

Discuss properties of Hamiltonian G-spaces were G is abelian

pg 81

Discuss with proof: regular values of moment map and symplectic structures on regular level sets

x regular value <=> every point mapping to x has finite stabilizer pg 83