Symmetric spaces and Cartan's classification...the Killing Form Decomposition of Symmetric Spaces...

Symmetricspaces and

Cartan’sclassification

Henry Twiss

History

SymmetricSpaces

Examples ofSymmetricSpaces

Curvature andLocallySymmetricSpaces

EffectiveOrthogonalSymmetric LieAlgebras andthe Killing Form

Decompositionof SymmetricSpaces

Rank andClassification

References

Symmetric spaces and Cartan’s classification

Henry Twiss

University of Minnesota

April 2020

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Outline

1 History

2 Symmetric Spaces

3 Examples of Symmetric Spaces

4 Curvature and Locally Symmetric Spaces

5 Effective Orthogonal Symmetric Lie Algebras and the KillingForm

6 Decomposition of Symmetric Spaces

7 Rank and Classification

8 References

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2 Symmetric Spaces

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History

Riemann showed that locally there is only one constantcurvature geometry. The most natural geometries to studynext are symmetric spaces. Elie Cartan alone classifiedsymmetric spaces.In the following we will introduce symmetric spaces, give afew prototypical examples, and discuss Cartan’s classification.We will assume throughout that every Lie algebra is a real Liealgebra unless otherwise specified.

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History

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History

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History

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History

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Introducing Symmetric Spaces

Definition 2.1

A Riemannian manifold M is a symmetric space if for eachp ∈ M, there exists an isometry sp ∈ Iso(M)p such that

s∗,p : TpM → TpM

is the negative of the identity map. The map sp is called asymmetry at p.

Geodesics are preserved by isometries, so sp ◦ γ is a geodesicfor all geodesics γ. Since

(sp ◦ γ)(t) = γ(−t),

symmetric spaces as spaces where at any point there exists asymmetry reversing geodesics through that point. Thisobservation tells us more.

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Definition 2.1

(sp ◦ γ)(t) = γ(−t),

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Definition 2.1

(sp ◦ γ)(t) = γ(−t),

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Definition 2.1

(sp ◦ γ)(t) = γ(−t),

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Definition 2.1

(sp ◦ γ)(t) = γ(−t),

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The Homogeneous Description

1 M is geodesically complete: domains of geodesicsγ : [0, s) are extended by reflecting using symmetries sγ(t)

for t ∈ (s/2, s).

2 Iso(M)◦ acts transitively on M: connect p and q by ageodesic. Letting m be the midpoint of this geodesic,sm(p) = q.

Definition 2.2

A Riemannian manifold M is a homogeneous space if Iso(M)◦

acts transitively on M.

In fact, the second property can be strengthened.

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for t ∈ (s/2, s).

Definition 2.2

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for t ∈ (s/2, s).

Definition 2.2

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for t ∈ (s/2, s).

Definition 2.2

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for t ∈ (s/2, s).

Definition 2.2

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for t ∈ (s/2, s).

Definition 2.2

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for t ∈ (s/2, s).

Definition 2.2

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Theorem 2.1

A symmetric space M is precisely a homogeneous space with asymmetry sp at some p ∈ M.

Proof. We are left to show that a homogeneous space with asymmetry is symmetric. Let g ∈ Iso(M)◦ be an isometrytaking p to q. By the chain rule,

sq := g ◦ sp ◦ g−1

defines a symmetry at q.

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Theorem 2.1

sq := g ◦ sp ◦ g−1

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Theorem 2.1

sq := g ◦ sp ◦ g−1

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Theorem 2.1

sq := g ◦ sp ◦ g−1

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The Isotropy Description

Theorem 2.2

Fixing a basepoint p ∈ M,

M ∼= Iso(M)◦/Iso(M)p.

Iso(M)◦/Iso(M)p is not necessarily a Lie group despiteIso(M)◦ being a connected Lie group. Indeed, Iso(M)p neednot be a normal subgroup.

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Theorem 2.2

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Theorem 2.2

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8 References

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Euclidean Space

Endow Rn with the Euclidean metric. The symmetry sp atp ∈ Rn is defined by

sp(p + v) := p − v .

Any line (i.e., geodesic) through p is of the form p + tv forsome v ∈ Rn, so sp reverse geodesics through p.Geometrically:

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Euclidean Space

sp(p + v) := p − v .

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Euclidean Space

sp(p + v) := p − v .

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Euclidean Space

sp(p + v) := p − v .

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Euclidean Space

sp(p + v) := p − v .

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The Sphere

Endow the unit sphere Sn in Rn+1 with the metric inducedfrom the standard inner product. For p ∈ Sn, sp is thereflection about the line tp for t ∈ R in Rn+1. Precisely,

sp(q) := 2〈q, p〉p − q.

The symmetry sp reverse the direction of great circles throughp. Geometrically (for S2):

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The Sphere

sp(q) := 2〈q, p〉p − q.

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The Sphere

sp(q) := 2〈q, p〉p − q.

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The Sphere

sp(q) := 2〈q, p〉p − q.

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The Sphere

sp(q) := 2〈q, p〉p − q.

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Real Hyperbolic Space

To define Hn, give Rn+1 the Lorentzian scalar product definedby

〈p, q〉 :=

)− pn+1qn+1,

and define Hn to be

Hn := {p ∈ Rn+1 | 〈p, p〉 = −1, pn+1 > 0}.

The induced scalar product on TpHn for p ∈ Hn makes Hn

into a Riemannian manifold. For any p ∈ Hn, the restriction of

sp(q) := 2〈q, p〉p − q

to Hn is the symmetry through p.

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〈p, q〉 :=

)− pn+1qn+1,

and define Hn to be

Hn := {p ∈ Rn+1 | 〈p, p〉 = −1, pn+1 > 0}.

sp(q) := 2〈q, p〉p − q

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〈p, q〉 :=

)− pn+1qn+1,

and define Hn to be

Hn := {p ∈ Rn+1 | 〈p, p〉 = −1, pn+1 > 0}.

sp(q) := 2〈q, p〉p − q

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〈p, q〉 :=

)− pn+1qn+1,

and define Hn to be

Hn := {p ∈ Rn+1 | 〈p, p〉 = −1, pn+1 > 0}.

sp(q) := 2〈q, p〉p − q

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The Orthogonal Group

We first show O(n) is a homogeneous space. O(n) is a regularsubmanifold of GL(n,R) by the regular level set theorem. TheRiemannian structure on O(n) is induced from the scalarproduct on Rn. In particular,

〈A,B〉 := trace(ATB).

If G ∈ O(n), then

〈GA,GB〉 = (GA)TGB = ATGTGB = ATB = 〈A,B〉.

Similarly, 〈AG ,BG 〉 = 〈A,B〉. Therefore O(n) is ahomogeneous space since O(n) acts transitively on itself.

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If G ∈ O(n), then

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If G ∈ O(n), then

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If G ∈ O(n), then

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If G ∈ O(n), then

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If G ∈ O(n), then

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If G ∈ O(n), then

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By Theorem 2.1 it suffices to exhibit a symmetry at the originI . Consider

sI : O(n)→ O(n) A 7→ AT .

sI is a isometry preserving the identity. It’s well-known (usingcurves)

TIO(n) = {A ∈ GL(n,R) | AT = −A}.

ComputingsI∗,I : TIO(n)→ TIO(n)

using curves, sI∗,I is the negative of the identity map. Thus sIis a symmetry at I .

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sI : O(n)→ O(n) A 7→ AT .

TIO(n) = {A ∈ GL(n,R) | AT = −A}.

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sI : O(n)→ O(n) A 7→ AT .

TIO(n) = {A ∈ GL(n,R) | AT = −A}.

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sI : O(n)→ O(n) A 7→ AT .

TIO(n) = {A ∈ GL(n,R) | AT = −A}.

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sI : O(n)→ O(n) A 7→ AT .

TIO(n) = {A ∈ GL(n,R) | AT = −A}.

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sI : O(n)→ O(n) A 7→ AT .

TIO(n) = {A ∈ GL(n,R) | AT = −A}.

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Compact Lie Group

Any compact Lie group is a symmetric space. If G is acompact compact Lie group it exhibits a biinvariant metric. Gacts transitively on itself, implying G is homogeneous.Consider

se : G → G g 7→ g−1.

sI is a diffeomorphism preserving the identity, and se∗,epreserves the metric. If g ∈ G is arbitrary, notese ◦ `g = rg−1 ◦ se . By the chain rule

se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .

So se is an isometry, and hence a symmetry at e proving G isa symmetric space.This generalizes our discussion about O(n) .

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Compact Lie Group

se : G → G g 7→ g−1.

se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .

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Compact Lie Group

se : G → G g 7→ g−1.

se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .

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Compact Lie Group

se : G → G g 7→ g−1.

se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .

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Compact Lie Group

se : G → G g 7→ g−1.

se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .

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Compact Lie Group

se : G → G g 7→ g−1.

se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .

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Compact Lie Group

se : G → G g 7→ g−1.

se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .

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Compact Lie Group

se : G → G g 7→ g−1.

se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .

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Compact Lie Group

se : G → G g 7→ g−1.

se∗,g ◦ `g∗,e = rg−1∗,e◦ se∗,e .

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8 References

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Parallel Curvature Tensor

Symmetric space have parallel curvature tensor.

Theorem 4.1

If M is a symmetric space with curvature tensor R, then thecurvature tensor is parallel, i.e., ∇R = 0.

Proof sketch. We prove ∇R is locally parallel. Note that if Tis a covariant k-tensor in a vector space which is invariantunder −id, then T = (−1)kT . If k is odd then necessarilyT = 0. If M is symmetric, each point p ∈ M admits asymmetry sp. We then check (∇R)p is invariant under sp∗,pand has odd rank.

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Theorem 4.1

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Theorem 4.1

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Theorem 4.1

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Theorem 4.1

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Theorem 4.1

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Theorem 4.1

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Locally Symmetric Spaces

Definition 4.1

A Riemannian manifold M is a locally symmetric space if ithas parallel curvature tensor.

A theorem of Cartan justifies this definition:

Theorem 4.2

If M is a locally symmetric space then for each p ∈ M, there isa symmetry at p defined in a neighborhood of p. Moreover, ifM is simply connected and complete, M is a symmetric space.

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Definition 4.1

Theorem 4.2

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Definition 4.1

Theorem 4.2

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Definition 4.1

Theorem 4.2

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Proof sketch. For ε > 0, the exponential

expp : B(0, ε)→ B(p, ε)

is a diffeomorphism. Under exp−1p we define sp(x) := −x .

This induces a diffeomorphism

sp : B(p, ε)→ B(p, ε),

and sp∗,p = −id on TpM automatically. We use the parallelcurvature tensor to prove sp is an isometry. The secondstatement is proved using an analytic continuation.

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expp : B(0, ε)→ B(p, ε)

sp : B(p, ε)→ B(p, ε),

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expp : B(0, ε)→ B(p, ε)

sp : B(p, ε)→ B(p, ε),

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expp : B(0, ε)→ B(p, ε)

sp : B(p, ε)→ B(p, ε),

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expp : B(0, ε)→ B(p, ε)

sp : B(p, ε)→ B(p, ε),

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Orthogonal Symmetric Lie Algebras

Definition 5.1

An orthogonal symmetric Lie algebra (g, s) is a Lie algebra gand an involution automorphism s of g such that theeigenspace u of s corresponding to 1 (i.e., the set of fixedpoints of s) is a compact Lie subalgebra. An orthogonalsymmetric Lie algebra is effective if u and Z (g) intersecttrivially.

As a prototypical example, let g = R and s = −id so thatu = {0}. Then (g, s) is an effective orthogonal symmetric Liealgebra.

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Definition 5.1

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Definition 5.1

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Definition 5.1

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The Killing Form

Definition 5.2

If g is a Lie algebra, we define the Killing form B of g over afield F to be the bilinear form

B : g⊗ g→ F x ⊗ y 7→ trace(Ad(x) ◦Ad(y)).

Usually F = R. The Killing form is symmetric and satisfiesother nice property. The sign of the Killing form plays animportant role in Cartan’s classification.

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The Killing Form

Definition 5.2

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The Killing Form

Definition 5.2

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The Killing Form

Definition 5.2

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Definition 5.3

Let g be and orthogonal symmetric Lie algebra.

• We say g is of compact type if B is negative definite.

• We say g is of noncompact type if B is positive definite.

• We say g is of flat type if B is identically zero.

Lie algebras of compact type are compact, and Lie algebras ofnoncompact type are noncompact.

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Definition 5.3

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Definition 5.3

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Definition 5.3

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Definition 5.3

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The Natural Decomposition

Definition 5.3 is useful when g is effective.

Theorem 5.1

Let g be an effective orthogonal symmetric Lie algebra. Theng admits the mutually orthogonal decomposition

g = g0 ⊕ g+ ⊕ g−

where g0 is of flat type, g+ is of compact type, and g− is ofnoncompact type.

The astonishing fact is that (simply connected) symmetricspaces decompose this way.

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Theorem 5.1

g = g0 ⊕ g+ ⊕ g−

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Theorem 5.1

g = g0 ⊕ g+ ⊕ g−

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Theorem 5.1

g = g0 ⊕ g+ ⊕ g−

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Riemannian Symmetric Pairs

Definition 5.4

Let G be a connected Lie group with K ≤ G a closedsubgroup. We say (G ,K ) is a Riemannian symmetric pair ifthe following two properties are satisfied:

1 AdG (K ) ≤ GL(g) is compact.

2 There exists an involution σ : G → G such that(Gσ)◦ ⊆ K ⊆ Gσ.

If M is a symmetric space, (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair with

σ : Iso(M)◦ → Iso(M)◦ s 7→ sp ◦ s ◦ s−1p .

Given a Riemannian symmetric pair (G ,K ), G/K is asymmetric space with respect to any G -invariant Riemannianmetric.

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Definition 5.4

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Definition 5.4

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Definition 5.4

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Definition 5.4

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Definition 5.4

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Correspondences

Given a Riemannian symmetric pair (G ,K ), let g be the Liealgebra of G and set s = σ∗,e . Then (g, s) is a orthogonalsymmetric Lie algebra.

Definition 5.5

A Riemannian symmetric pair (G ,K ) is effective if Z (G ) ∩ Kis a discrete subgroup of G .

This is equivalent to (g, s) being effective. We know everysymmetric space gives rise to a Riemannian symmetric pair;this pair is always effective.

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Correspondences

Definition 5.5

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Correspondences

Definition 5.5

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Correspondences

Definition 5.5

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Correspondences

Definition 5.5

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Correspondences

Definition 5.5

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Types Revisited

Definition 5.6

1 An effective Riemannian symmetric pair (G ,K ) is of flat,compact, or noncompact type if the correspondingeffective orthogonal symmetric Lie algebra (g, s) is offlat, compact, or noncompact type.

2 A symmetric space M is of flat, compact, or noncompacttype if the corresponding effective orthogonal symmetricLie algebra is of flat, compact, or noncompact type.

It is the second of these two conventions that we will makeuse of.

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Types Revisited

Definition 5.6

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Types Revisited

Definition 5.6

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Symmetric Space Decomposition

Observe that products of symmetric spaces are symmetric.With this observation, Cartan was able to prove the followingtheorem:

Theorem 6.1

A simply connected symmetric space M admits adecomposition

M ∼= M0 ×M+ ×M−

into symmetric spaces where M0 is of flat type, M+ is ofcompact type, and M− is of noncompact type.

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Theorem 6.1

M ∼= M0 ×M+ ×M−

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Theorem 6.1

M ∼= M0 ×M+ ×M−

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Decomposition of Symmetric Spaces

Proof sketch. By Theorem 2.2 M ∼= Iso(M)◦/Iso(M)p byfixing a basepoint p ∈ M. Recall (Iso(M)◦, Iso(M)p) is aRiemannian symmetric pair. Let (g, s) be the correspondingeffective orthogonal symmetric Lie algebra. By Theorem 5.1

g = g0 ⊕ g+ ⊕ g−.

Let G̃0, G̃+, and G̃− be the Lie groups which are the coveringspaces of the Lie groups associated to the Lie algebras above.Let K0, K+, and K− be the Lie groups corresponding to theLie subalgebras u0, u+, and u−. Then check

M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),

where G̃0/K0 is of flat type, G̃+/K+ is of compact type, andG̃−/K− is of noncompact type.

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g = g0 ⊕ g+ ⊕ g−.

M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),

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Symmetricspaces and

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SymmetricSpaces

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g = g0 ⊕ g+ ⊕ g−.

M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),

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g = g0 ⊕ g+ ⊕ g−.

M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),

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g = g0 ⊕ g+ ⊕ g−.

M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),

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Symmetricspaces and

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g = g0 ⊕ g+ ⊕ g−.

M ∼= (G̃0/K0)× (G̃+/K+)× (G̃−/K−),

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Comments

The assumption M is simply connected can be made withoutloss of generality. If we assume M is irreducible, i.e., not aproduct of symmetric spaces, then Theorem 6.1 says M iseither of compact, noncompact, or Euclidean type. So, itsuffices to classify symmetric spaces of these types. In orderto do so we will need an invariant: the rank of a symmetricspace.

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Outline

1 History

2 Symmetric Spaces

8 References

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Flats and Rank

Definition 7.1

Suppose M is an irreducible symmetric space. A totallygeodesic immersion of Rn into M is called a flat. A flat ismaximal if it is not contained in any larger flat.

Basic Lie theory shows that all maximal flats of M are of thesame dimension.

Definition 7.2

The rank of and irreducible symmetric space M is thedimension of any maximal flat.

The rank of a symmetric space plays a very important role inCartan’s classification

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Flats and Rank

Definition 7.1

Definition 7.2

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Flats and Rank

Definition 7.1

Definition 7.2

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Flats and Rank

Definition 7.1

Definition 7.2

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Flats and Rank

Definition 7.1

Definition 7.2

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Flats and Rank

Definition 7.1

Definition 7.2

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Rank and Sectional Curvature

The rank of is at least one with equality if the sectionalcurvature is positive or negative. If the sectional curvature ispositive the space is of compact type, and if the sectionalcurvature is negative the space is of noncompact type.The rank of a Euclidean type space is equal to its dimension.This implies Euclidean type spaces are isometric to Euclideanspace of that dimension.Therefore we are reduces to classifying symmetric spaces ofcompact and noncompact type.In both cases, we have two classes of symmetric spacesdescribed in terms of Riemannian symmetric pairs (G ,K ).

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Classification

Compact type:

• G = H × H where H is a simply connected compact Liegroup and K is the diagonal subgroup.

• G is the complexification of a simply connectednoncompact simple Lie group and K is the maximalcompact subgroup.

Noncompact type:

• G is a simply connected complex simple Lie group and Kis the maximal compact subgroup.

• G is a simply connected noncompact simple Lie groupand K is the maximal compact subgroup.

By the classification of Lie groups all such symmetric spacesare also classified, and this finishes Cartan’s classification.

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Classification

Compact type:

Noncompact type:

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Classification

Compact type:

Noncompact type:

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Henry Twiss

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Outline

1 History

2 Symmetric Spaces

8 References

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Thanks!

• J. Eschenburg: Lecture Notes on Symmetric Spaces,University of Augsburg.

• X. Gao: Symmetric Spaces, University of Illinois, 2014.

• P. Holmelin: Symmetric Spaces, Master’s Thesis, LundInstitute of Technology, 2005.

• P. Petersen: Riemannian Geometry Third Edition,Springer, AG, Switzerland, 2016.

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Symmetric spaces and Cartan's classification...the Killing Form Decomposition of Symmetric Spaces...

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