Some geometric and topological features of symmetric R...
Transcript of Some geometric and topological features of symmetric R...
Some geometric and topological features
of symmetric R-spaces
Geodesics and moment maps
in collaboration with Jurgen Berndt and Anna Fino
Index numbers: Riem. invariants for flag manifolds
Index number=∑
(Z2-Betti numbers)
(Takeuchi, Sanchez)
• Aim:
- interpretation within symplectic topology and Morsetheory
- sketch of an alternative simple proof
- interplay with submanifold geometry
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A complex flag manifold can be realized as a coadjointorbit of a connected, compact, semisimple real Liegroup
=⇒ carries a natural symplectic structure
Moreover a real flag manifold =
connected component of the fixed point set of an anti-symplectic involution on “its complexification” (whichis a complex flag manifold).
• Main tools:
- moment mappings of Hamiltonian torus actions
- convexity theorems :
by Atiyah, Guillemin-Sternberg and Kostant
(for Hamiltonian torus actions on cpt symplectic mflds)
by Duistermaat
(for fixed point sets of antisymplectic involutions)
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COMPLEX FLAG MANIFOLD =
orbit of the adjoint representation of a compact Liegroup K
Example:
K = SU(3)
k = su(3) = {A ∈ gl(3, C) | A = −tA, trA = 0}
Cartan subalgebra t ={diag (λ1, λ2, λ3) | λ1+λ2+λ3 = 0}
• H=diag(λ1, λ2, λ3), λ1 6= λ2 6= λ3
=⇒ Ad(SU(3))·H=“full flag manifold”
Manifold F1,2,3 of full flags V1 ⊂ V2 ⊂ V3 in C3
• H=diag(λ1, λ2, λ3), λ1 = λ2 6= λ3
=⇒ Ad(SU(3))·H=“partial flag manifold”
Manifold of partial flags V1 = V2 ⊂ V3 in C3 ∼= CP 2
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Geometry of complex flag manifoldsas submanifolds of Euclidean space
M = Ad(K)X ∼= K/KX ↪→ (k,−B( , ))
standard immersion of a flag manifold
TXM = imad(X) kX = ker ad(X)
reductive decomposition (⊥): k = kX ⊕ TXM
−→ kX = ker ad(X) = νXM
X regular ⇐⇒ kX as minimal dim
principal orbit (full flag manifold)
kX =: t Cartan subalgebra
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Example: K = SU(3)
t = {diag (λ1, λ2, λ3) | λ1 + λ2 + λ3 = 0}
General fact (Ad(SU(3))X) ∩ t =
Weyl group orbit
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F1,2,3 CP2
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Geometric property:
t meets all orbits orthogonally
polar action
“There is a linear subspace which meets all orbits orthogonally”
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One can choose X ∈ t
• the orbits are parallel and equidistant
• If M = Ad(K)H is a principal orbit,
ξ ∈ t = νXM induces a ∇⊥-parallel normal field −→νM is flat
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Spit k into real root spaces (w.r. to t):
[T, Xα] = α(T )Yα , [T, Yα] = −α(T )Xα , T ∈ t
complex (Kahler) structure on complex flag manifoldJ : Xα 7→ Yα , Yα 7→ −Xα
On a principal orbit M = Ad(K)H
THM = span {Xα, Yα}
AξXα = − α(ξ)α(H)Xα AξYα = − α(ξ)
α(H)Yα
−→ M isoparametric submanifold
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M isoparametric submanifold of Rn: (Terng)
• νM flat
• the eigenvalues of A relative to parallel normal fields
are constant.
−→ {Aξ}ξ∈νM simult. diagonalisable
common eigenvalues: λi(ξ) =: 〈ni, ξ〉, i = 1, ..., g
ni: curvature normals
common eigendistributions: Ei, i = 1, ..., g
are autoparallel
−→ integrable with totally geodesic leaves
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Si(p) leaf through p ∈M= sphere of dimension dimEi
−→ M is foliated by spheres
A full flag manifold is isoparametric
Thorbergsson: codimM ≥ 3,
M isoparametric full and irreducible
=⇒ full flag manifold
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Endpoint map:
tξ : M → Rn
x 7→ x + ξ(x) = exp(ξ(x))
focal point in direction ξ = critical value of tξ .
x + ξ(x) focal point in dir. of ξ ⇐⇒ ker(id−Aξ(x)) isnon trivial
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In general: if ξ ∈ νM parallel normal isoparametricsection (Aξ has constant eigenvalues)
⇒ the image of tξ , Mξ = {x + ξ(x) | x ∈M}
is a submanifold of Rn
dimMξ = dimM − dim(ker(id−Aξ(x)))
parallel focal manifold
TpMξ = sum of the eigendistributions Ej “non focalised”( 6∈ ker(id−Aξ))
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→ one has submersions
π : M →Mξ : x 7→ x + ξ(x),
π−1(p) isoparametric in νpMξ
(consequence of Olmos’ normal holonomy theorem)
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Aξ =
〈n1, ξ〉 . . . 0
. . .
..
.0 . . . 〈ng, ξ〉
.
the focal points belong to
`i(p) := {1− 〈ni(p), ξ(p)〉 = 0} focal hyperplanes
Coxeter group (of a complete isoparam. subm.)
Terng= finite group generated by reflections σi w.r. to `i
(= Weyl group for flag manifold)
Fact: σi(p)= antipodal point of p in Si(p)
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Height functions and topology of cx flag manifolds
M = Ad(K) ·H (with H ∈ t)
X ∈ t regular element −→
Critical set = (Ad(K) ·H) ∩ t
all critical points of fX are non degenerate ≡ fX Morsefunction
Index of a critical point = 2 × number of hyperplanescrossed by a line joining X to the critical point
Example for K = SU(3)
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H regular (F1,2,3) H singular (CP 2)
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Morse theory=⇒
bi(M) = number of critical points of index i
=⇒
fX is a perfect Morse function
fX determines a cell decomposition ≡ Bruhat
decomposition
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k-Symmetric structure:
∀ complex flag manifold M ∃ k0 ≥ 2 :
∀k ≥ k0 ∃ k-symmetric struct. {θ(k)x | x ∈ M} on M
(Jimenez)
k-symmetry θ(k)x = isometry of M of order k
for which x is an isolated fixed point.
REAL FLAG MANIFOLD =
orbit of the isotropy representation of a symmetricspace G/K
A complex flag manifold is a real flag manifold
[take G/K = G(= G×G/∆)]
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BUT
Important fact:
we can regard a real flag manifold as a real form of acomplex flag manifold
=⇒
∃ a natural immersion of a real flag manifold M into
a complex flag manifold MC “its complexification”
M ↪→MC
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Index numbers
• 2-number of a Riem. mfld M Chen, Nagano
“#2(M)”
#2(M)=maximalf cardinality of subsets A ⊂M :∀ pair of points x, y ∈ A ∃ a closed geodesic γ in Mon which x and y are antipodal
i.e. γ : [0,2b] → M is a closed geodesic of length 2b: x = γ(0), y = γ(b).
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sγ∨
y = γ(b)
x = γ(0) = γ(2b)
Takeuchi: M symmetric R-space
(=particular real flag mfld which is also symmetric)
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=⇒
#2(M) =∑
bZ2
i (= dimH∗(M, Z2))
bZ2
i : Z2-Betti numbers M endowed with the normalmetric
Note: For a symmetric space M ,
#2(M) = maximal possible cardinality of the 2-setsA2 ⊂M :
∀x ∈ A2 the symmetry sx of M fixes every point ofA2.
symmetric space ≡ 2-symmetric space =⇒
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• k-number of a k-symmetric space M Sanchez
“#k(M)”
#k(M)=maximal cardinality of the k-sets Ak ⊂M :
∀x ∈ Ak the symmetry θ(k)x fixes every point of Ak .
Sanchez: M complex flag manifold
=⇒
#k(M) =∑
bi (= dimH∗(M, Z))
bi : Z-Betti numbers
M endowed with the normal metric
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• Index number of a flag manifold M Sanchez
“#I(M)”
#I(M) = maximal cardinality of p-sets Ap ⊂M :
∀x ∈ Ap the symmetry θ(p)x in the p-symmetric struc-
ture on MC fixes every point of Ap.
Sanchez: M real flag manifold
=⇒
#I(M) =∑
bZ2
i (= dimH∗(M, Z2))
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Symplectic structure on complex flag manifolds
Adjoint orbits ←→ Coadjoint orbits
Ad : k ∈ K 7→ Ad(k) : k→ k Ad∗ : k ∈ K 7→ Ad∗(k) : k∗ → k∗
(Ad∗(K) · η)X := η(Ad(k)X)
Ad(K) ·H ←→ Ad∗(K) · η (η = 〈H, 〉)
Symplectic structure on M = Ad∗(K) · η:
ωη(ad∗Xη, ad∗Y η) := η([X, Y ])
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T := a maximal torus of K
Important fact: The natural K -action on M restrictsto a Hamiltonian action of T on M = Ad∗(K) · η
Hamiltonian action of a group G on a symplectic man-ifold N
⇐⇒
∃ Lie alg. hom (g, [ , ])→ (C∞(M), { , } : X 7→ fX :
∀X ∈ g the Killing vector field X∗ induced by theaction =
Hamiltonian vector field XfXassociated with fX
{fX}X∈g
f : M → g∗
momentum map
If G = T and M = Ad∗(K) · η
=⇒ momentum map ≡ {the height functions }
fX : M → R : p 7→ 〈p, X〉
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Convexity Theorems
Guillermin - Sternberg, Atiyah
(N, ω) compact symplectic manifold
T torus acting on N in a Hamiltonian way
f : N → t∗ momentum mapping of the HamiltonianT -action
=⇒
• f(F ) is finite F = fixed point set of the T -action• f(N) = convex hull of f(F );• dimH∗(N, Z) = dimH∗(CX, Z) ∀X ∈ t,(CX = critical point set of fX )
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For a complex flag manifold (=coadj. orbit η ↔ H)
take T =maximal torus
f : Ad∗(K) · η ↪→ k∗proj−→ t∗ =⇒
f(F ) = (Ad∗(K) · η) ∩ t = Weyl group orbit
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Duistermaat
(N, ω) compact symplectic manifoldT torus acting on N in a Hamiltonian wayτ antisymplectic involution on N Q= fixed pointset 6= ∅ :fX ◦ τ = fX ∀X ∈ t
=⇒
• f(Q) = f(N) is a convex polytope• dimH∗(Q, Z2) = dimH∗(CX, Z2) ∀X ∈ t,CX = set of critical points of fX|Q
We can replace Q by any connected component Q0
of Q
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Proof and geom. interpretation of Sanchez Thm forcomplex flag manifolds
Basic algebraic fact: k-set Ak ⊂ Ad(K) ·H =⇒
∃ Cartan subalgebra t : Ak = t ∩ (Ad(K) ·H)
convexity=⇒
#k(Ad(K) ·H) = cardinality of A =
= cardinality of t ∩ (Ad(K) ·H) == n.of vertices of convexpolytope =
=∑
bi =
= n.of cells inBruhat decomposition == minimal number of cells needed to
have aCW − complex structure
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M := Ad (K)p symmetric irreducible R-space(G/K symmetric space of compact type)
(M endowed with normal metric =induced metric - by irred.)
−→ M is a focal mfld of a principal orbit M of theisotropy repr. of a symmetric space G/K .
M is isoparametric
π : M →M projection π(p) = p + ξ(p),ξ parallel normal field
TM = ⊕Ei
Ei eigendistributions of the shape operator of M
TM = ⊕ non focalized eigendistributions Ei
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NOTE: A non focalized eigendistribution Ei a leafSi(q) homothetically sent to M .
[π∗|Ei= (id−Aξ)|Ei
is a multiple of the identity] −→
• M is foliated by totally geodesic spheres
• σi ∈W , (reflections w.r. to `i)q ∈M 7→ antipodal point in Si(q).
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Si(q)
q
σi(q)
maximal 2-set A2 = a ∩M = νpM ∩M = W · q
MOMENTUM MAP ≡ projection ⊥ M → νpM = a
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=⇒ The half closed geodesics connecting
points in a maximal 2-set A2 = W · q which
correspond one each other by reflections σi
with respect to the walls `i of the Weyl cham-
bers are mapped by the moment map µ into
lines in a = νpM .
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K = SU(3) (CP 2)
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