Why I like homogeneous manifolds - School of Mathematicsjmf/CV/Seminars/EDGE2012.pdfBasic...

Why I like homogeneous manifolds

Jose Figueroa-O’Farrill

inburghometry

1 March 2012

Jose Figueroa-O’Farrill Why I like homogeneous manifolds 1 / 36

Basic terminology I“manifold”: smooth, connected, finite-dimensional

“Lie group”: finite-dimensional with identity 1G acts on M (on the left) via G×M→M, sending(γ,m) 7→ γ ·mactions are effective: γ ·m = m, ∀m =⇒ γ = 1M is homogeneous (under G) if either

1 G acts transitively: i.e., there is only one orbit; or2 for every m ∈M, G→M sending γ 7→ γ ·m is surjective3 given m,m ′ ∈M, ∃γ ∈ G with γ ·m = m ′

γ defined up to right multiplication by the stabiliser of m:H = {γ ∈ G|γ ·m = m}, a closed subgroup of GM ∼= G/H, hence M is a coset manifoldH→ G

is a principal H-bundle

Basic terminology I“manifold”: smooth, connected, finite-dimensional“Lie group”: finite-dimensional with identity 1

G acts on M (on the left) via G×M→M, sending(γ,m) 7→ γ ·mactions are effective: γ ·m = m, ∀m =⇒ γ = 1M is homogeneous (under G) if either

Basic terminology I“manifold”: smooth, connected, finite-dimensional“Lie group”: finite-dimensional with identity 1G acts on M (on the left) via G×M→M, sending(γ,m) 7→ γ ·m

actions are effective: γ ·m = m, ∀m =⇒ γ = 1M is homogeneous (under G) if either

Basic terminology I“manifold”: smooth, connected, finite-dimensional“Lie group”: finite-dimensional with identity 1G acts on M (on the left) via G×M→M, sending(γ,m) 7→ γ ·mactions are effective: γ ·m = m, ∀m =⇒ γ = 1

M is homogeneous (under G) if either

Basic terminology I“manifold”: smooth, connected, finite-dimensional“Lie group”: finite-dimensional with identity 1G acts on M (on the left) via G×M→M, sending(γ,m) 7→ γ ·mactions are effective: γ ·m = m, ∀m =⇒ γ = 1M is homogeneous (under G) if either

1 G acts transitively: i.e., there is only one orbit; or

2 for every m ∈M, G→M sending γ 7→ γ ·m is surjective3 given m,m ′ ∈M, ∃γ ∈ G with γ ·m = m ′

1 G acts transitively: i.e., there is only one orbit; or2 for every m ∈M, G→M sending γ 7→ γ ·m is surjective

3 given m,m ′ ∈M, ∃γ ∈ G with γ ·m = m ′

γ defined up to right multiplication by the stabiliser of m:H = {γ ∈ G|γ ·m = m}, a closed subgroup of G

M ∼= G/H, hence M is a coset manifoldH→ G

γ defined up to right multiplication by the stabiliser of m:H = {γ ∈ G|γ ·m = m}, a closed subgroup of GM ∼= G/H, hence M is a coset manifold

H→ G

Basic terminology IIthe action of G on M defines G→ DiffM

the differential g→X (M)

evaluating at m ∈M: exact sequence of H-modules

0 −−−−→ h −−−−→ g −−−−→ TmM −−−−→ 0

linear isotropy representation of H on TmM is defined forγ ∈ H as (dγ·)m : TmM→ TmM

it agrees with the representation on g/h induced by theadjoint representation restricted to h

G/H reductive: the sequence splits (as H-modules); i.e.,g = h⊕m with m an Ad(H)-modulethere is a one-to-one correspondence{

Ad(H)-invarianttensors on m

H-invarianttensors on TmM

G-invarianttensor fields on M

Basic terminology IIthe action of G on M defines G→ DiffMthe differential g→X (M)

0 −−−−→ h −−−−→ g −−−−→ TmM −−−−→ 0

G/H reductive: the sequence splits (as H-modules); i.e.,g = h⊕m with m an Ad(H)-module

there is a one-to-one correspondence{Ad(H)-invariant

tensors on m

0 −−−−→ h −−−−→ g −−−−→ TmM −−−−→ 0

Many of our favourite manifolds are homogeneous:

ExamplesLie groups (H = {1})

affine space An under the group of translationssphere Sn = SO(n+ 1)/SO(n)

hyperbolic space Hn ∼= SO(n, 1)/SO(n)

projective space CPn = U(n+ 1)/U(n)×U(1)real grassmannians SO(p+ q)/SO(p)× SO(q)

complex grassmannians U(p+ q)/U(p)×U(q)more exotic grassmannians SU(n)/SO(n),...our (spatial) universe!

ExamplesLie groups (H = {1})affine space An under the group of translations

sphere Sn = SO(n+ 1)/SO(n)

ExamplesLie groups (H = {1})affine space An under the group of translationssphere Sn = SO(n+ 1)/SO(n)

projective space CPn = U(n+ 1)/U(n)×U(1)

real grassmannians SO(p+ q)/SO(p)× SO(q)

complex grassmannians U(p+ q)/U(p)×U(q)

more exotic grassmannians SU(n)/SO(n),...our (spatial) universe!

complex grassmannians U(p+ q)/U(p)×U(q)more exotic grassmannians SU(n)/SO(n),...

our (spatial) universe!

The principle of mediocrity

The cosmological principleAt sufficiently large scales, the Universe looks the same to allobservers.

Mathematical rephrasingAs far as cosmology is concerned, the spatial universe is a3-dimensional homogeneous manifold.

The big questionsWhat (topological) manifold is it? Is it compact?Simply-connected?We know it’s expanding: but will it do so forever? or will iteventually contract?

Homogeneous (pseudo)riemannian manifolds(M,g) with g ∈ C∞(S2T∗M) everywhere nondegenerate

g has constant signature (p,q):

(n, 0): riemannian(n− 1, 1): lorentzian

(M,g) homogeneous under G if G acts transitively byisometries: γ∗g = g for all γ ∈ Gg acts via Killing vector fields: LXg = 0gm is H-invariant inner product on TmMin the reductive case and assuming π1(M) = {1} (or Hconnected), (M,g) is described algebraically by

1 g = h⊕m2 h-invariant inner product 〈−,−〉 on m

riemannian: H compact, so G/H reductive

Homogeneous (pseudo)riemannian manifolds(M,g) with g ∈ C∞(S2T∗M) everywhere nondegenerateg has constant signature (p,q):

(n, 0): riemannian

(n− 1, 1): lorentzian(M,g) homogeneous under G if G acts transitively byisometries: γ∗g = g for all γ ∈ Gg acts via Killing vector fields: LXg = 0gm is H-invariant inner product on TmMin the reductive case and assuming π1(M) = {1} (or Hconnected), (M,g) is described algebraically by

(M,g) homogeneous under G if G acts transitively byisometries: γ∗g = g for all γ ∈ G

g acts via Killing vector fields: LXg = 0gm is H-invariant inner product on TmMin the reductive case and assuming π1(M) = {1} (or Hconnected), (M,g) is described algebraically by

(M,g) homogeneous under G if G acts transitively byisometries: γ∗g = g for all γ ∈ Gg acts via Killing vector fields: LXg = 0

gm is H-invariant inner product on TmMin the reductive case and assuming π1(M) = {1} (or Hconnected), (M,g) is described algebraically by

(M,g) homogeneous under G if G acts transitively byisometries: γ∗g = g for all γ ∈ Gg acts via Killing vector fields: LXg = 0gm is H-invariant inner product on TmM

in the reductive case and assuming π1(M) = {1} (or Hconnected), (M,g) is described algebraically by

1 g = h⊕m

2 h-invariant inner product 〈−,−〉 on m

Reductive homogeneous manifolds

Theorem (Ambrose–Singer, 1958; Kostant, 1960)A simply-connected (pseudo)riemannian manifold (M,g) isreductive homogeneous if and only if there exists a linearconnection ∇ with torsion T and curvature R satisfying

1 ∇g = 02 ∇T = 03 ∇R = 0

if T = 0 then (M,g) is a symmetric spaceCartan (1926)

if R = 0 then (M,g) is either a Lie group with a bi-invariantmetric or the round 7-sphere

Cartan–Schouten (1926); Wolf (1971-2)

Reductive homogeneous manifoldsTheorem (Ambrose–Singer, 1958; Kostant, 1960)A simply-connected (pseudo)riemannian manifold (M,g) isreductive homogeneous if and only if there exists a linearconnection ∇ with torsion T and curvature R satisfying

1 ∇g = 02 ∇T = 03 ∇R = 0

1 ∇g = 0

2 ∇T = 03 ∇R = 0

1 ∇g = 02 ∇T = 0

3 ∇R = 0

1 ∇g = 02 ∇T = 03 ∇R = 0

Many of our favourite manifolds have homogeneous metrics:

Examplesleft-invariant metrics on Lie groups

euclidean metric on affine space An

round metric on sphere Sn = SO(n+ 1)/SO(n)

Poincare metric on hyperbolic space Hn ∼= SO(n, 1)/SO(n)

Fubini–Study metric on complex projective spaceCPn = U(n+ 1)/U(n)×U(1)

Examplesleft-invariant metrics on Lie groupseuclidean metric on affine space An

The name of the game in differential geometry

... is finding “nice” metrics; e.g.,

constant sectional curvatureconstant scalar curvatureEinstein: Ric = g

extremal, etcThese are curvature conditions =⇒ (hard) PDEs on gBut on a homogeneous manifold, for X, Y ∈ m Killing

∇XY|m = −12 [X, Y]m +U(X, Y)

where U : S2m→ m is defined by