Group actions on symmetric spaces related to left-invariant geometric structures

(1)

...

related to left-invariant geometric structures

Hiroshi Tamaru

Hiroshima University

RIMS Workshop

“Development of group actions and submanifold theory”

25/June/2014

(2)

. . . . . .

.

...

In view of left-invariant geometric structures, the following actions are interesting:

(I, II) isometric actions on noncpt Riem. symmetric sp.

(III) nonisometric actions on (symmetric) R-spaces.

(IV) isometric actions on pseudo-Riem. symmetric sp.

.

Aim of this talk

..

...

In this talk, we

mention our framework and results.

propose some problems on groups actions.

Hiroshi Tamaru (Hiroshima) 25/June/2014 2 / 37

(3)

.

...We are interested in left-invariant geometric structures.

.

Good Point (1)

..

...

Relatively easier to treat.

Many arguments can be reduced to the Lie algebra level.

.

Good Point (2)

..

...

Left-invariant geometric structures provide examples of Einstein / Ricci soliton metrics,

(generalized) complex / symplectic structures, ...

(4)

. . . . . .

.

Central Problem

..

...

For a given Lie group, determine whether it admits a

“good” left-invariant geometric structure or not.

.

Remark

..

...

For a given Lie group and a given geometric structure, one can directly study its property, e.g.,

calculate the curvatures of a left-invariant metric, check the integrability condition of an almost complex structure, ...

But this does not mean that the above problem is easy.

(5)

.

Central Problem (recall)

..

...

For a given Lie group, determine whether it admits a

“good” left-invariant geometric structure or not.

.

What is diﬃcult? (1)

..

...

There are so many Lie algebras...

{G : simply-connected Lie group with dimG = n}

∼= {g : Lie algebra with dimg = n}

∼= {[,] ∈ ∧²(Rⁿ)^∗ ⊗Rⁿ : satisfying Jacobi identity}.

(6)

. . . . . .

What is diﬃcult? (2)

..

...

The space of the left-invariant structures on G is large.

For examples, if dimG = n then

{left-invariant Riem. metrics on G}

∼= {inner products on g := Lie(G)}

∼= GL_n(R)/O(n).

{left-invariant metrics on G with signature (p,q)}

∼= GL_n(R)/O(p,q).

. ...

These spaces are symmetric spaces.

We study our problem in terms of groups actions!

(7)

Slogan

..

...Left-invariant metrics vs isom. actions on GL_n(R)/O(n).

.

Problem (I)

..

...

M = G/K : Riem. symmetric space of noncompact type.

Consider H ↷G/K : isometric action.

Then, what the orbit space H\M can be?

.

Note

..

...

In this section, we mention:

What are known?

How related to the study of left-invariant metrics?

(8)

. . . . . .

...

Consider K,A,N ↷ RH² = SL2(R)/SO(2), where K = SO(2),

A=

{( a 0 0 −a

)

| a > 0 }

, N =

{( 1 b 0 1

)

| b ∈ R }

.

Then the orbits look like

&%

'$

m b

type (K)

[0,+∞)

&%

'$

type (A) R

&%

'$

e

type (N) R

(9)

.

Picture for R H

²

..

...

&%

'$

m b

type (K)

[0,+∞)

&%

'$

type (A) R

&%

'$

e

type (N) R

.

Theorem (Berndt-Br¨ uck (2001))

..

...

H ↷G/K : cohomogeneity one (with H connected).

⇒ H\M ∼= R or [0,+∞).

(10)

. . . . . .

...

Fix G : Lie group with dimG = n.

We now consider left-invariant Riemannian metrics on G. .

Def.

..

...

The space of left-invariant Riemannian metrics:

fM:= {left-invariant Riem. metrics on G}

∼= {inner products on g := Lie(G)}

∼= GL_n(R)/O(n).

.

Note

..

...

GLn(R) = GL(g) ↷Mf by g.⟨·,·⟩ := ⟨g⁻¹(·),g⁻¹(·)⟩.

(11)

. ...

R^× := {c ·id∈ GL_n(R) | c ∈ R̸=0},

Aut(g) := {φ ∈ GL_n(R) | φ[·,·] = [φ(·), φ(·)]}. .

Def.

..

...

The moduli space of left-invariant Riem. metrics:

PM := R^×Aut(g)\fM (orbit space).

.

Remark

..

...

R^×Aut(g) ↷Mf gives rise to an isometry up to scaling.

(12)

. . . . . .

Prop. (Hashinaga-T. (preprint))

..

...

g := (R³,[,]) with [e₁,e₂] = e₂ (others = 0).

⟨,⟩⁰ : the canonical inner product.

⇒

R^×Aut(g) ↷ fM is of cohomogeneity one.

PM =



R^×Aut(g).(



 1 1

λ 1



.⟨,⟩⁰) | λ ∈ R



. .

Proof

..

...

Very direct matrix calculations.

A general theory gives a certification.

(13)

.

Cor. (Hashinaga-T. (preprint))

..

...

g := (R³,[,]) with [e₁,e₂] = e₂ (others = 0).

⟨,⟩ : any inner product.

∃⇒λ ∈ R, ∃k > 0, ∃{x₁,x₂,x₃} o.n.b. w.r.t. k⟨,⟩:

[x₁,x₂] = x₂ −λx₃, others = 0.

.

Note

..

...

{x₁,x₂,x₃} is a generalization of “Milnor frames”.

Hence, this is called a Milnor-type theorem.

(14)

. . . . . .

Result (Hashinaga-T.-Terada)

..

...

∀n ≥ 3,

we construct g of dimension n such that R^×Aut(g) ↷ fM : cohomogeneity one.

we determine all possible Ricci signatures on them.

.

Result (Hashinaga-T.)

..

...

∀g : 3-dim., solvable,

we construct Milnor-type theorems for g.

we (re)classify left-invariant Ricci soliton metrics.

.

...Key Point: for cohomogeneity one actions, PM is easy!

(15)

Problem (I)-1

..

...

For Riemannian symmetric spaces of noncompact type, classify (possible topological type of) orbit spaces of

cohomogeneity two actions, (hyper)polar actions, ...

.

Problem (I)-2

..

...

Classify g such that R^×Aut(g) ↷Mf are cohomogeneity one or two actions, (hyper)polar actions, ...

(∃ examples by Taketomi (2014))

(16)

. . . . . .

Slogan

..

...Left-invariant metrics vs isom. actions on GL_n(R)/O(n).

.

Problem (II)

..

...

Consider H ↷G/K : isometric action.

Then, are there “distinguished” orbits?

.

Note

..

...

What are known?

How related to the study of left-invariant metrics?

(17)

. . . . . .

Thm. (Berndt-T. (2003))

..

...

M = G/K : irr. Riem. symmetric space of noncpt type.

H ↷G/K : cohomogeneity one (with H connected).

⇒ it satisfies one of the following:

(K) ∃! singular orbit.

(A) ̸ ∃ singular orbit, ∃! minimal orbit.

(N) ̸ ∃ singular orbit, all orbits are congruent.

&%

'$

m b

type (K)

[0,+∞)

&%

'$

type (A) R

&%

'$

e

type (N) R

.

Note

.. ⇒ ∃ a unique “distinguished” orbit.

(18)

. . . . . .

.

...This picture fits very nicely to “algebraic Ricci solitons”.

.

Def.

..

...

(g,⟨,⟩) : algebraic Ricci soliton (ARS) :⇔ ∃c ∈ R, ∃D ∈ Der(g) : Ric = c ·id+D. .

...

Der(g) := {D ∈ gl(g) | D[·,·] = [D(·),·] + [·,D(·)]}. .

Note

..

...

left-invariant Einstein ⇒ algebraic Ricci soliton.

algebraic Ricci soliton ⇒ Ricci soliton (next page).

(in many cases, the converse also holds)

(19)

.

Prop. (Lauret (2011))

..

...

Let us consider

(g,⟨,⟩) : algebraic Ricci soliton (Ric = c ·id+D), (G,g) : corresponding simply-connected one.

Then one has

exp(tD) ∈ Aut(g) for ∀t ∈ R,

∃φ_t ∈ Aut(G) : (dφ_t)_e = exp(tD), X ∈ X(G) by X_p := _dt^d φ_t(p)|^t=0,

ric_g = cg −(1/2)L_Xg (i.e., Ricci soliton).

(20)

. . . . . .

.

Thm. (Hashinaga-T.)

..

...

g : 3-dimensional, solvable. Then,

⟨,⟩ on g is an algebraic Ricci soliton (ARS)

⇔ fM⊃ R^×Aut(g).⟨,⟩ : minimal.

.

Note

..

...

Mf = GL_n(R)/O(n) is a Riem. symmetric space (w.r.t. a natural GL_n(R)-invariant metric).

In these cases, R^×Aut(g) ↷Mf is of cohom. ≤1.

(21)

.

...

&%

'$

m b

type (K)

[0,+∞)

&%

'$

type (A) R

&%

'$

e

type (N) R

.

Thm. (Hashinaga-T.): more precise

..

...

Let g : 3-dimensional, solvable. Then,

(K)-type: ARS ⇔ R^×Aut(g).⟨,⟩ : singular.

(A)-type: ARS ⇔ R^×Aut(g).⟨,⟩ : minimal.

(N)-type: ̸ ∃ ARS.

(22)

. . . . . .

.

Fact

..

...

For all known g with R^×Aut(g) ↷ fM cohom. one, we have checked that “ARS ⇔ minimal”.

.

Thm. (Hashinaga (to appear))

..

...

∃g : 4-dim., solvable :

both implications of “ARS ⇔ minimal” do not hold.

.

Expectation

..

...

ARS ⇔ R^×Aut(g).⟨,⟩ is “distinguished” in some sense?

(cf. ⟨,⟩ : biinvariant ⇒ R^×Aut(g).⟨,⟩ is totally geod.)

(23)

.

Problem (II)-1

..

...

For Riemannian symmetric spaces of noncompact type, study the geometry of orbits of

cohomogeneity one actions (with H not connected), cohomogeneity two actions, (hyper)polar actions, ...

.

Problem (II)-2

..

...

Property of g can be understood by group actions?

ARS can be characterized by submanifolds?

Tasaki-Umehara invariant for 3-dim. Lie algebras?

(24)

. . . . . .

.

Def.

..

...

L : semisimple Lie group, with trivial center, L ⊃ Q : parabolic subgroup.

Then M := L/Q is called an R-space.

.

Problem (III)

..

...

Let M = L/Q : R-space, H ⊂ L.

Study the action H ↷ M = L/Q.

.

Note

..

...

In this section, we mention: An easy example.

(Motivation will be mentioned in the next section.)

(25)

.

Example

..

...

RPⁿ is an R-space:

RPⁿ⁻¹ = SL_n(R)/















∗ ∗ · · · ∗ 0 ∗ · · · ∗ ... ... ... ...

0 ∗ · · · ∗













 .

.

Side Remark

..

...

R-spaces can be realized as orbits of s-reps.

RPⁿ⁻¹ is an orbit of the s-rep. of SL_n(R)/SO(n).

(26)

. . . . . .

.

Set Up

..

...

We consider SO(p,q) ↷ RPⁿ⁻¹ (with n = p+q).

RPⁿ⁻¹ = SL_n(R)/Q, and

SL_n(R) ⊃SO(p,q) : symmetric.

(an analogy of “Hermann actions”?) .

Remark

..

...

⟨,⟩0 : canonical inner product on Rⁿ, signature (p,q).

Then, SO(p,q) ↷ Rⁿ preserves ⟨,⟩⁰.

(27)

.

Prop.

..

...

SO(p,q) ↷ RPⁿ⁻¹ has three orbits:

O⁺ := {[v] ∈ RPⁿ⁻¹ | ⟨v,v⟩⁰ > 0}, O⁰ := {[v] ∈ RPⁿ⁻¹ | ⟨v,v⟩0 = 0}, O⁻ := {[v] ∈ RPⁿ⁻¹ | ⟨v,v⟩⁰ < 0}. .

Note

..

...

O⁺, O⁻ are open.

(28)

. . . . . .

Problem (III)-1

..

...

Study the action H ↷ M = L/Q.

Construct interesting examples.

What happens if L ⊃H is symmetric?

When it has an open orbit?

.

Problem (III)-2

..

...

Study the geometry of orbits.

H.p can be inhomogeneous w.r.t. Isom(M), but have some “nice” properties?

(29)

Slogan

..

...

Left-invariant pseudo-Riemannian metrics vs isometric actions on GL_n(R)/O(p,q).

.

Problem (IV)

..

...

M = G/K : pseudo-Riemannian symmetric space.

Study isometric actions H ↷ M. .

...

An easy example.

How related to left-invariant pseudo-Riem. metrics.

How related to R-spaces.

(30)

. . . . . .

.

Prop.

..

...

The following action has exactly three orbits:

Q :=















∗ ∗ · · · ∗ 0... ∗ 0













 ↷ GL_p+q(R)/O(p,q).

.

Proof

..

...

The orbit space coincides with the orbit space of O(p,q) ↷GL_p+q(R)/Q = RP^p+q⁻¹.

(31)

.

...Fix G : Lie group with dimG = n = p +q.

.

Def.

..

...

The space of left-inv. metrics with signature (p,q):

Mf_p,q := {left-invariant metrics on G with (p,q)}

∼= {inner products on g:= Lie(G) with (p,q)}

∼= GLn(R)/O(p,q).

The moduli space of left-inv. metrics with (p,q):

PM_p,q := R^×Aut(g)\fMp,q (orbit space).

(32)

. . . . . .

.

Lem.

..

...

Let g be one of the following:

H³ : Heisenberg group.

G_RHⁿ : the group acting simply-transitively on RHⁿ. Then R^×Aut(g) is parabolic (= the previous Q).

.

Thm. (Kubo-Onda-Taketomi-T.)

..

...

On the above Lie groups,

∀(p,q) ∈ N² with n = p+q,

∃ exactly three left-invariant metrics with (p,q).

(33)

Thm. (Kubo-Onda-Taketomi-T.): recall

..

...

On the above Lie groups,

∀(p,q) ∈ N² with n = p+q,

∃ exactly three left-invariant metrics with (p,q).

.

Comments

..

...

H³ : known by Rahmani (1992), method is diﬀerent.

G_R_Hⁿ : known by Nomizu (1979) for Lorentzian case. The case of generic signature is new.

.

Side Remark

..

...For G_R_Hⁿ, any left-invariant metric has const. curvature.

(34)

. . . . . .

.

Problem (IV)-1

..

...

Study isometric actions H ↷ M.

First of all, study the case when H is parabolic.

(↔ symmetric actions on R-spaces.) .

Problem (IV)-2

..

...

Construct “nice” actions on M = G/K. Let M^′ = G/K^′ : Riemannian symmetric.

If H ↷M^′ is nice, then so is H ↷M?

(35)

.

Comment

..

...

Our framework is

left-inv. metrics vs actions on symmetric spaces.

theory of symmetric spaces is quite useful.

This would also be useful to study left-invariant complex structures:

{almost complex strs} ∼= GL_2n(R)/GL_n(C).

left-invariant symplectic structures:

{nondegenerate 2-forms} ∼= GL_2n(R)/Sp_2n(R).

and so on...

(36)

. . . . . .

.

Comment

..

...

Our framework motivates to study

isometric actions on noncpt Riem. symmetric sp.

nonisometric actions on (symmetric) R-spaces.

isometric actions on pseudo-Riem. symmetric sp.

.

...

This would provide

new examples of actions (from Aut(g)).

particular examples of actions with applications.

new ideas or new notions?

(37)

.

...

.

1.. Berndt, J., Br¨uck, M.: Crelle (2001).

.

2.. Berndt, J., D´ıazRamos, J. C., Tamaru, H.: JDG (2010).

.

3.. Berndt, J., Tamaru, H.: JDG (2003).

.

4.. Hashinaga, T.: Hiroshima Math. J., to appear.

.

5.. Hashinaga, T., Tamaru, H.: preprint (2013).

.

6.. Hashinaga, T., Tamaru, H., Terada, K.: preprint (2012).

.

7.. Kodama, H., Takahara, A., Tamaru, H.: Manuscripta Math. (2011).

.

8.. Kubo, A., Onda, K., Taketomi, Y., Tamaru, H.: in preparation.

.

9.. Lauret, J.: Crelle (2011).

.

10.. Nomizu, K.: Osaka J. Math. (1979).

.

11.. Rahmani, S.: J. Geom. Phys. (1992).

.

12.. Taketomi, Y.: Topology Appl. (Special Issues), to appear.

.

13.. Taketomi, Y., Tamaru, H.: in preparation.

.

14.. Tasaki, H., Umehara, M.: Proc. Amer. Math. Soc. (1992).

.

... Thank you!