Left-invariant metrics on Lie groups and

submanifold geometry

Hiroshi Tamaru /田丸博士

(Hiroshima University /広島大学)

The 8th Kagoshima Algebra-Analysis-Geometry SeminarKagoshima University

20 February 2013

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0.1 Abstract▼ We are studying:

◦ geometry of left-invariant metrics on Lie groups,

◦ from the view point of submanifold geometry.

▼ This talk is organized as follows:

◦ §1: Introduction

◦ §2: Preliminaries on submanifold geometry

◦ §3: Three-dimensional case

◦ §4: Higher-dimensional examples

◦ §5: A pseudo-Riemannian version

◦ §6: Summary and Problems

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1 Introduction1.1 Left-invariant metrics (1/2) - basic notations

▼ In this talk:

◦ G : a simply-connected Lie group with dimG = n,

◦ g : the Lie algebra of G.

▼ Left-invariant metrics on G provide examples of:

◦ Einstein :⇔ Ric = c · id (∃c ∈ R),

◦ algebraic Ricci soliton

:⇔ Ric = c · id + D (∃c ∈ R, ∃D ∈ Der(g)),

◦ Ricci soliton :⇔ ric = cg + LXg (∃c ∈ R, ∃X ∈ X(G)).

▼ Fact:

◦ Einstein ⇒ algebraic Ricci soliton ⇒ Ricci soliton.

◦ g : completely solvable, Ricci soliton⇒ algebraic Ricci soliton.

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1.2 Left-invariant metrics (2/2) - theme

▼ Problem:

◦ For a given Lie group G, examine

whether G admit a “distinguished” left-invariant metric.

(e.g., Einstein, (algebraic) Ricci soliton)

▼ This is difficult in general, because:

◦ there are so many left-invariant metrics...

◦ M := {left-invariant metrics on G}� {inner products on g}� GL n(R)/O(n).

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1.3 Approach from submanifold geometry (1/2)

▼ Recall:

◦ M := {left-invariant metrics on G} � GL n(R)/O(n).

▼ Observation:

◦ All curvature information are preserved by R×Aut(g).

(g.〈·, ·〉 := 〈g−1·, g−1·〉)

▼ Def:

◦ The orbit space PM := R×Aut(g)\M is called the moduli space.

▼ Remark:

◦ For 3-dim. unimodular Lie groups,

Milnor (1976) essentially studied Aut(g)\M.

(so-called “Milnor frame”)

◦ Our framework is its generalization.

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1.4 Approach from submanifold geometry (2/2)

▼ Recall:

◦ PM := R×Aut(g)\M : the moduli space.

▼ An advantage (1):

◦ One can studyM more efficiently.

(in general, dimM >> dim PM)

▼ An advantage (2):

◦ This gives a new view point:

left-invariant metrics ↔ submanifold geometry.

◦ Also gives a new problem:

〈, 〉 is distinguished (as Riemannian metrics),

if and only if R×Aut(g).〈, 〉 is distinguished (as submanifolds)?

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1.5 An easy application (1/2)

▼ Def.:

◦ gRHn := span{e1, . . . , en} with [ e1, ej ] = ej ( j ≥ 2)

is called theLie algebra of real hyperbolic space.

(its simply-connected group acts simply-transitively onRHn)

▼ Thm. (Milnor 1976):

◦ ∀〈, 〉 on gRHn, it has constant curvaturec < 0.

▼ Comment:

◦ His proof is very direct.

◦ We can simplify the proof by usingPM as follows...

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1.6 An easy application (2/2)

▼ Recall (Milnor 1976):

◦ ∀〈, 〉 on gRHn, it has constant curvaturec < 0.

▼ Prop. (Kodama-Takahara-T. 2011):

◦ R×Aut(gRHn) =

∗ 0 · · · 0

∗ ∗ · · · ∗...

.... . .

...

∗ ∗ · · · ∗

.

◦ Hence, PM = {pt}.

▼ Comment:

◦ In order to show Milnor’s theorem,

it is enough to check only one left-invariant metric has constant

curvature c < 0.

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2 Preliminaries on submanifold geometry2.1 Cohomogeneity one actions (1/3)

▼ Note:

◦ R×Aut(g) y M.

◦ M = GL n(R)/O(n) : a noncompact symmetric space.

◦ Well-studied for “cohomogeneity one” case.

▼ Def.:

◦ H y (M, g) : of cohomogeneity one

:⇔ maximal dimensional orbits have codimension one

(⇔ the orbit spaceH\M has dimension one).

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2.2 Cohomogeneity one actions (2/3) - onRH2

▼ Fact (due to Cartan):

◦ H y RH2 : cohomogeneity one action (withH connected)

⇒ this is orbit equivalent to the actions of

K = SO(2), A =

a 0

0 a−1

| a > 0

, N =

1 b

0 1

▼ Picture:

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2.3 Cohomogeneity one actions (3/3) - geometry of orbits

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▼ Thm. (Berndt-Br uck 2002, Berndt-T. 2003):

◦ M : noncompact symmetric space, irreducible.

◦ H y M : cohomogeneity one

⇒ (type (K)) ∃1 singular orbit,

(type (A)) @ singular orbit, ∃1 minimal orbit, or

(type (N)) @ singular orbit, all orbits are congruent.

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3 Three-dimensional case3.1 Three-dimensional (1/3) - table

▼ Fact:

◦ g = span{e1, e2, e3} : 3-dim. solvable (non abelian)

is isomorphic to one of the following:

name comment brackets

h3 Heisenberg [e1, e2] = e3

r3,1 gRH3 [e1, e2] = e2, [e1, e3] = e3

r3 [e1, e2] = e2 + e3, [e1, e3] = e3

r3,a −1 ≤ a < 1 [e1, e2] = e2, [e1, e3] = ae3

r′3,a a ≥ 0 [e1, e2] = ae2 − e3, [e1, e3] = e2 + ae3

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3.2 Three-dimensional (2/3)

▼ Recall:

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▼ Prop (Hashinaga-T.):

name action metric

h3 transitive ∃ algebraic Ricci soliton

r3,1 transitive ∃ const. curvature

r3 type (N) @ Ricci soliton

r3,a type (A) ∃ algebraic Ricci soliton

r′3,a type (K) ∃ Einstein

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3.3 Three-dimensional (3/3)

▼ Recall:

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▼ Thm. (Hashinaga-T.):

◦ g : three-dimensional solvable Lie algebra.

◦ 〈, 〉 : algebraic Ricci soliton ong

⇔ R×Aut(g).〈, 〉 : minimal in M = GL n(R)/O(n).

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4 Higher dimensional examples4.1 Higher-dimensional (1/3)

▼ Note:

◦ g : three-dimensional solvable

⇒ • R×Aut(g) y M : cohomogeneity at most one,

• 〈, 〉 : algebraic Ricci soliton ⇔ R×Aut(g).〈, 〉 : minimal.

▼ Natural question:

◦ How are the higher dimensional cases?

◦ Still R×Aut(g) y M can be of cohomogeneity one?

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4.2 Higher-dimensional (2/3)

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▼ Thm. (Hashinaga-T.-Terada):

◦ g := gRH2 ⊕ Rn−2 or gRHn−1 ⊕ R, n ≥ 4

⇒ • R×Aut(g) y M : cohomogeneity one, type (K),

• 〈, 〉 : Ricci soliton ⇔ R×Aut(g).〈, 〉 : singular.

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4.3 Higher-dimensional (3/3)

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▼ Thm. (Taketomi-T.):

◦ g := span{e1, . . . , en}with [ ej , en] = ej ( j ≤ n − 2), [en−1, en] = e1 + en−1

⇒ • R×Aut(g) y M : cohomogeneity one, type (N),

• @〈, 〉 : Ricci soliton.

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5 A pseudo-Riemannian version5.1 A pseudo version (1/3) - the moduli space

▼ Note:

◦ Our framework can also be applied to

left-invariant pseudo-Riemannian metrics.

▼ Def:

◦ g : ( p + q)-dim. Lie algebra

◦ M(p,q) := {〈, 〉 : an inner product on g with signature (p, q)}� GL p+q(R)/O(p, q).

◦ PM(p,q) := R×Aut(g)\Mp+q : the Moduli space.

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5.2 A pseudo version (2/3) - a known result

▼ Recall (Milnor 1976):

◦ g := gRHn = span{e1, . . . , en} with [ e1, ej ] = ej ( j ≥ 2)

⇒ ∀〈, 〉 : Riemannian, it has constant curvaturec < 0.

◦ Alternative proof: using PM = {pt}.

▼ Thm. (Nomizu, 1979):

◦ g := gRHn

⇒ ∀〈, 〉 : Lorentz, it has constant curvature c.

(c can take any signature,c > 0, c = 0, c < 0)

▼ Comment:

◦ His proof is very direct.

◦ UsingPM(p,q), we can generalize this, and simplify the proof.

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5.3 A pseudo version (3/3) - a moduli space

▼ Prop. (Kubo-Onda-Taketomi-T.):

◦ g := gRH p+q

⇒ #PM(p,q) = 3, given by U = {I n + λE1,n | λ = 0, 1, 2}.

▼ Thm. (Kubo-Onda-Taketomi-T.):

◦ g := gRHn

⇒ ∀〈, 〉 : pseudo-Riemannian, it has a constant curvaturec.

(c can take any signature,c > 0, c = 0, c < 0)

▼ Comment:

◦ To get more examples, we need to know

isometric actions on pseudo-Riemannian symmetric spaces...

(the orbit spaces, cohomogeneity one, (hyper-)polar actions, ...)

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6 Summary and Problems6.1 Summary

▼ Story:

◦ We are interested in geometry of left-invariant metrics

(both Riemannian and pseudo-Riemannian)

◦ An approach from submanifold geometry

• one can study all metrics efficiently,

• left-invariant metrics ↔ properties of submanifolds?

▼ Results:

◦ In several cases,

〈, 〉 : (algebraic) Ricci soliton ⇔ R×Aut(g).〈, 〉 : distinguished.

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6.2 Problems▼ Problem 1:

◦ A continuation of this study, i.e.,

• check the correspondence for other cases,

((algebraic) Ricci soliton↔ distinguished orbit?)

• study the reason why there is such a correspondence.

▼ Problem 2:

◦ Apply our method to other geometric structures, e.g.,

• left-invariant complex structures,

• left-invariant symplectic structures, ...

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