Theory of metric Lie groups andH-surfaces in homogeneous3 ... · Example (The metric Lie group R) The real numbers R with its usual metric and group operation + is a Lie group with

Theory of metric Lie groups and H-surfaces inhomogeneous 3-manifolds.

William H. Meeks IIIUniversity of Massachusetts at Amherst

Based on joint work with Mira, Pérez, Ros and Tinaglia.

Definition

A 2-dimensional submanifold with constant mean curvature H ≥ 0 in aRiemannian 3-manifold is called an H-surface.

Definition

If the isometry group of a Riemannian manifold Y acts transitively, thenY is called homogeneous.

Definition

A Lie group with left invariant metric is called a metric Lie group.

Notation and Language

Y = simply connected homogeneous 3-manifold.

X = simply connected 3-dimensional Lie group with left invariantmetric (X is a metric Lie group).

H(Y) = Inf{max |HM| : M = immersed closed surface in Y}, wheremax |HM| denotes max of absolute mean curvature function HM.

The number H(Y) is called the critical mean curvature of Y.

Ch(Y) = InfK⊂Y compactArea(∂K)

Volume(K)= Cheeger constant of Y.

Goals of Lecture 1

Classify simply connected homogeneous 3-manifolds Y.

Present the basic theory of simply connected, 3-dimensional metric Liegroups X.

Express metric and bases for left and right invariant vector fields inexplicit coordinates for X = R2 oA R = a semidirect product.Give formulae for the Levi-Civita connection ∇, the principal Riccicurvature vectors and the principal Ricci curvature values for X.

Goals of Lecture 2

Classification and quasi-isometric classification of the possible Y.

Right cosets of 2-dimensional subgroups H ⊂ X and the existence ofalgebraic open book decompositions.

Uniqueness and embeddedness of minimal spheres in X ≈ SU(2).Isoperimetric domains and the isoperimetric profile of Y.

Explain the formula: Ch(Y) = 2H(Y) for non-compact Y.

Discuss CMC foliations, Isoperimetric Inequality Conjectures, StabilityConjecture, Product CMC Foliation Conjectures.

Goals of Lecture 3

Classification of algebraic open book decompositions.

The left invariant Lie group Gauss map and the embeddedness problemfor H-spheres.

The Meeks-Tinaglia curvature and radius estimates for H-disks.

Existence and uniqueness of H-spheres in X (Meeks-Mira-Pérez-Ros).

Discuss various conjectures on embedded H-surfaces in Y.

Goals of Lecture 4

The H-potential for X and a Weierstrass representation for H-surfaces inX (Meeks-Mira-Pérez-Ros).

Outline of the proof of the existence and uniqueness of H-spheres in X(Meeks-Mira-Pérez-Ros).

The Plateau problem in X.

Other theoretical techniques and open problems.

Theorem (Simply connected homogeneous 3-dimensional Y)

If Y is a simply connected homogenous 3-manifold, then:

Y is isometric to a metric Lie group(Lie group with left invariant metric); these examples form a 3-parameterfamily of non-isometric homogeneous 3-manifolds.

or

Y is isometric to S2(κ)× R for some κ > 0.

Theorem (Simply connected 3-dimensional Lie groups X)

Every X is isomorphic to SU(2) = {group of unit length quaterions} or to theuniversal covering group of a 3-dimensional subgroup of the 6-dim affine groupF = {f(x) = Ax + b : R2 → R2 | b ∈ R2, A ∈ GL(2,R)}, which is the naturalsemidirect product of R2 with GL(2,R) = Aut(R2).

The 3-dimensional subgroups of F are one of the following types:

SL(2,R) = {A ∈ GL(2,R) | det(A) = 1}.The semidirect product of the normal subgroup R2 ⊂ F of translationswith any particular 1-parameter subgroup Γ of GL(2,R).

Remark

These two theorems will be proved at the beginning of Lecture 2.

Definition

A Lie group G is a smooth manifold whose group operation ∗satisfies the following:

∗ : G× G→ G (a, b) 7−→ a ∗ b

I : G→ G a 7→ a−1

are both smooth mappings. We will frequently use themultiplicative notation ab to denote a ∗ b.

The respective left and right multiplications by a ∈ G are given by:

la : G→ G x 7−→ ax

ra : G→ G x 7−→ xa.

Given a, p ∈ G, let (la)∗ : TpG→ TapG (resp (ra)∗) denote thedifferential of la (resp ra).

For vp ∈ TpG, let avp (resp. vpa) denote the vector(la)∗(vp) ∈ TapG (resp. vpa = (ra)∗(vp) ∈ TpaG).

Definition

A vector field X is called left invariant if ∀a ∈ G, X = aX , orequivalently, for each p ∈ G, Xap = aXp.

A vector field Y is called right invariant if ∀a ∈ G, Y = Ya.

The vector space L(G) of left invariant vector fields on G can beidentified with the tangent space TeG at the identity element e ∈ G.

L(G) is a Lie algebra under the Lie bracket of vector fields, i.e., forX ,Y ∈ L(G), then [X ,Y ] ∈ L(G).

L(G) is called the Lie algebra of G.

Remark (1-parameter subgroups)

For each X ∈ L(G), the integral curve G(X ) passing through theidentity is the image of a 1-parameter subgroup, i.e., there is a Liegroup homomorphism expXe : R→ G(X ) ⊂ G .

In this case Xe is the velocity vector exp′Xe

(0) of expXe (t) at e.

Let M(n,R) = Rn2 be the set of real n × n matrices, which is a Liealgebra under the operation [A,B] = AB − BA.

When G is a subgroup of GL(n,R) ⊂M(n,R) = Rn2 , thenL(G) = TeG can be identified with the subspace ofM(n,R) = TeGL(2,R) of tangent vectors to G at the identitymatrix e = In.

When G ⊂ GL(2,R) and Xe = A ∈M(n,R) = TeG,

expXe (t) = exp(tA) =∞∑n=0

tnAn

n!∈ G(X ) ⊂ G.

This explains the notation for the group homomorphismexpXe : R→ G, for X ∈ L(G).

Theorem (Ado’s Theorem)

Given a n-dimensional real Lie algebra L, then for some k ∈ N, L isisomorphic to a subalgebra L′ of the Lie algebra of k × k matricesM(k ,R) under the bracket of matrices.

L is isomorphic to the Lie algebra of the connected Lie subgroup Gof GL(k,R) generated by the elements

exp(tA) =∞∑n=0

tnAn

n!∈ GL(k ,R) for t ∈ R, A ∈ L′.

Therefore, every simply connected Lie group is the universalcovering group of some subgroup G ⊂ GL(k ,R) for some k ∈ N.

Definition

Given an X ∈ L(G) with related 1-parameter subgroup G(X ) ⊂ G,then X is the velocity vector field associated to the 1-parametergroup of diffeomorphisms obtained by letting G(X ) act on G fromthe right.

The velocity field of the 1-parameter group of diffeomorphismsobtained on G by the left action of G(X ) on G is the right invariantvector field KX , where KXe = Xe .

Let R(G) denote the n-dimensional vector space of right invariantvector fields, where n is the dimension of G.

R(G) is also a Lie algebra under the Lie bracket of vector fields.

The vector field KX is the Killing field associated to the 1-parametergroup of isometries G(X ) for ANY left invariant metric on G.

K(X) = space of Killing fields in X is a Lie algebra isomorphic tothe Lie algebra of the isometry group Iso(X) of X.

Definition (And Facts)

A Riemannian metric on Lie group G is called left invariant if forall a ∈ G, la : G→ G is an isometry of G.

Each left invariant metric on G is obtained by taking a metric 〈, 〉eon TeG and defining for a ∈ G and v,w ∈ TaG,〈v,w〉 = 〈a−1v, a−1w〉e .

We call a Lie group Y with a left invariant metric, a metric Liegroup.

The related metric Lie group is complete.

The universal cover Ỹ with the pulled back metric is a metric Liegroup and Y = Ỹ/π1(Y), where we identify π1(Y) with a discretenormal abelian subgroup of Y acting on Y by left translation.

Example (The metric Lie group R)

The real numbers R with its usual metric and group operation + isa Lie group with a left invariant metric.

In this case L(R) = R(R) are just the constant vector fieldsX tp = (p, t), where we consider the tangent bundle TR of R to beR× R.

By taking X 10 = (0, 1) ∈ T0R, expX 10 : R→ R is a groupisomorphism, which is essentially the identity map.

Theorem (Simply connected homogeneous 1 and 2-manifolds)

R with its usual metric is the unique simply connectedhomogeneous 1-manifold.

Every simply connected homogeneous 2-manifold has constantsectional curvature τ ∈ R and is isometric to R2 for τ = 0, to S2(τ)for τ > 0, or to the ”hyperbolic plane” H(τ) for τ < 0.

Except for S2(τ), the simply connected homogeneous manifolds ofdimension ≤ 2 are isometric to metric to Lie groups.

Example (The metric Lie group H = RoA=(1) R)

Here we consider A = (1) to be the 1× 1 identity matrix.

The notation RoA=(1) R means we have a semidirect product of Rwith R via the homomorphism σ : R→ Aut(R) = R− {0}, whereσ(t) = eAt = et . Let σt denote the automorphism σ(t).

Then the non-commutative group operation ? on H is(a, b) ? (c , d) = (a + σb(c), b + d) = (a + e

bc , b + d).

Choose the metric at (Te=(0,0)H) = R× R to be the usual one, i.e.,|∂x |(0, 0) = |∂t |(0, 0) = 1 and extend it to a left invariant metric.

The metric Lie group H has constant Gaussian curvature −1.

Orbits of the 1-parameter group of left translations by elements in{0}oR have constant geodesic curvature less than 1 in H with theorbit ({0}oR) ? (0, 0) = {0}oR being a geodesic.

Orbits of the 1-parameter group of left translations by elements inR× {0} are parallel horocycles in H orthogonal to {0} × R.(a,b) 7→ ebx + a is an isomorphism with the subgroup K of affinetransformations of R: K = {f(x) = ax + b | a > 0, b ∈ R}.

Orbits in H of actions of 1-parameter subgroups of PSL(2,R).

Here H = Ro(1) R is represented by the unit disk model.

By left translation, we consider the Lie group H to be a subgroup ofPSL(2,R) = the isometry group of H = R×(1) R.

Left: Elliptic subgroup in PSL(2,R) of rotations fixing e = (0, 0).

Center: Hyperbolic translations along the geodesic {0}oR,corresponding to left translations by elements in {0} × R.

Right: Parabolic rotations/translations around the point the topboundary point at infinity in H, corresponding to left translations byelements in R× {0}.

Definition

S̃L(2,R) = P̃SL(2,R) is the universal covering of the projectivelinear group PSL(2,R) = SL(2,R)/{±I2}.

The Lie algebra of any of the groups SL(2,R), S̃L(2,R), PSL(2,R)is L = sl(2,R) = {B ∈M2(R) | Trace(B) = 0}.

PSL(2,R) is a simple group, i.e., it does not contain normalsubgroups other than the trivial ones.

S̃L(2,R) is a not a simple group. Its center is Z = π1(PSL(2,R)).

We can view PSL(2,R) as the group of orientation preservingisometries of the hyperbolic plane.

Using the upper half-plane model for H, these are transformationsof the type

z ∈ H 7→ az + bcz + d

(a, b, c , d ∈ R, ad − bc = 1).

We can also view PSL(2,R) as the unit tangent bundle T1H of thehyperbolic plane.

This is because an orientation preserving isometry of H is uniquelydetermined by the image of a point and the image under itsdifferential of a given unitary vector tangent at that point.

This view of PSL(2,R) = T1H as an S1-bundle over H (and henceof P̃SL(2,R) as a R-bundle over H) defines naturally the so calledstandard metric on PSL(2,R) (resp. on S̃L(2,R) = P̃SL(2,R)).

The isometry group of SL(2,R) with its standard metric hasdimension 4.

A basis for sl(2,R) is E1 =(

1 00 −1

), E2 =

(0 11 0

),

E3 =

(0 −11 0

)with 1-parameter subgroups Γ1 =

(et 00 e−t

),

Γ2 =

(cosh t sinh tsinh t cosh t

), Γ3 =

(cos t − sin tsin t cos t

), respectively.

Here Γ1 and Γ2 are hyperbolic 1-parameter subgroups of SL(2,R)and Γ3 is an elliptic 1-parameter subgroup.

The left invariant metric of S̃L(2,R) for the ”orthonormal” basisE1,E2,E3 of sl(2,R) is the E(κ, τ)-metric for the unit tangentbundle of H2(−4).

The Lie bracket operation is

[E1,E2] = −2E3, [E2,E3] = 2E1, [E3,E1] = 2E2.

So E1,E2,E3 is a unimodular basis (defined later on) for S̃L(2,R)with structure constants c1 = 2, c2 = 2, c3 = −2.

Left invariant metrics ga,b,c with E′1 = aE1, E

′2 = bE2, E3 = cE3,

a, b, c > 0.

S̃L(2,R) = T̃1H with usual metric and proj. Π: S̃L(2,R)→ H.Here Π: S̃L(2,R)→ H is the related Riemannian submersion.E1,E2,E3 ⊂ Te S̃L(2,R) = orth. basis, E3 tangent to Π−1(Π(e)).Invar. metrics ga,b,c with E

′1 = aE1,E

′2 = bE2,E3 = cE3, a, b, c > 0.

Two-dimensional subgroups of PSL(2,R).

H type subgroups. For each θ ∈ S1 = ∂H, Hθ denotes thesubgroup of isometries that fix θ.

Each element of Hθ is a parabolic rotation around θ or a translationalong a geodesic with one of its endpoints being θ.

Each Hθ is isomorphic to the nontrivial semidirect product RoA R,where A = I1 = (1), with curvature −1.

RoA {0} ⊂ RoA R is a normal subgroup corresponding to theparabolic subgroup fixing θ.{0}oA R ⊂ RoA R is a subgroup corresponding to translationalong a geodesic.

Two representations of a subgroup Hθ of P̃SL(2,R), θ ∈ ∂∞H.

Left: As the semidirect product Ro(1) R.

Right: The set of orientation preserving isometries of H which fix θ.

Each of the blue curves in the right picture corresponds to the orbitof one of the hyperbolic 1-parameter subgroups of Ro(1) R.

Each of the red curves in the right picture corresponds to an orbit ofthe normal subgroup Ro(1) {0}.

Example (SU(2) and the Special Orthogonal Group SO(3))

The special orthogonal group SO(3) of rotations about the origin inR3.

π1(SO(3)) = Z2 and the universal covering group of SO(3) isSU(2) of unit length quaternions S3 ⊂ R4.

SU(2) is isomorphic to the group of special unitary matrices.

Since left multiplication by a unit length quaternion is an isometryof R4, the usual metric on S3 with constant sectional curvature 1 isa left invariant metric.

In this case SO(3) = S3/{±1} is seen to have a quotient leftinvariant metric of constant sectional curvature 1.

The 1-parameter subgroups of SO(3) are the circle subgroups givenby all rotations around some fixed axis in R3.

Example (SU(2) and the Special Orthogonal Group)

Let T1(S2) = {(x , y) ∈ R3 × R3 | ‖x‖ = ‖y‖ = 1, x ⊥ y} be theunit tangent bundle of S2, which can be viewed as a Riemanniansubmanifold of TR3 = R3 × R3.

The natural constant curvature metric on SO(3) is also the oneinduced from the diffeomorphism F : SO(3)→ T1(S2) given by

F (c1, c2, c3 = c1 × c2) = (c1, c2) ∈ T1(S2) ⊂ R3 × R3,

where c1, c2, c3 are the columns of the corresponding matrix inSO(3).

Semidirect Products: Lie Groups of the form R2 oA R

Given a matrix A ∈M2(R), t ∈ R and p ∈ R2, consider thehomomorphism σ : R→ GL(2,R) = Aut(R2) given by

σ(t)(p) = etAp =

( ∞∑n=0

tnAn

n!

)p.

For simplicity we denote σ(t) : R2 → R2 by σt : R2 → R2.

The group operation ? of the semidirect product R2 oA R, where +is the group operation of R2 and R, is given by

(p1, z1) ? (p2, z2) = (p1 +σz1(p2), z1 + z2) = (p1 + (ez1Ap2), z1 + z2).

We denote the corresponding group by R2 oA R, which is thesemidirect product of R2 and the universal cover Γ = R ofσ(R) ⊂ GL(2,R).

Metric Lie Groups of the form R2 oA R

Given a matrix A, the group R2 oA R has an associated leftinvariant metric by taking the metric at the tangent space to theidentity element (0, 0, 0) to be the product metric of R2 × R.

Given A =

(0 00 0

)∈M2(R), then ezA =

(1 00 1

). This

produces the usual direct product of groups, which in our case isR3 = R2 × R with its usual metric.

Taking A =

(1 00 1

), then ezA =

(ez 00 ez

)and one recovers

the hyperbolic three-space H3 with its usual metric.

Taking A =

(1 00 0

), then ezA =

(ez 00 1

). This gives

R2 oA R = H2 × R with its usual product metric.

The examples Ẽ(2),Sol3,Nil3

Here are some other particular cases depending on A:

If A =

(0 −11 0

), then ezA =

(cos z − sin zsin z cos z

)and

R2 oA R = Ẽ(2), the universal cover of the group of rigid motionsof the plane with the flat metric.

If A =

(−1 00 1

), then ezA =

(e−z 0

0 ez

)and

R2 oA R = Sol3 with its usual metric.

If A =

(0 10 0

), then ezA =

(1 z0 1

), and

R2 oA R = Nil3, the Heisenberg group of upper triangular matriceswith its usual metric: 1 a c0 1 b

0 0 1

.

R2 oA R: Metric and bases for the Lie algebras L(X) and R(X)

Consider A =

(a bc d

).

For R2 oA R, choose coordinates (x , y) ∈ R2, z ∈ R.Then ∂x , ∂y , ∂z is a parallelization of R2 oA R.Taking derivatives at t = 0 of the left and right multiplication by(p1, z1) = (t, 0, 0) ∈ X (resp. by (0, t, 0), (0, 0, t)), we obtain thefollowing basis {F1,F2,F3} of the right invariant and basis {E1,E2,E3} ofleft invariant vector fields on X:

F1 = ∂x , F2 = ∂y , F3(x , y , z) = (ax + by)∂x + (cx + dy)∂y + ∂z . (1)

E1(x , y , z) = a11(z)∂x + a21(z)∂y , E2(x , y , z) = a12(z)∂x + a22(z)∂y , E3 = ∂z , (2)

where we have denoted

ezA =

(a11(z) a12(z)a21(z) a22(z)

). (3)

The expression of 〈, 〉 w.r.t. the basis {∂x , ∂y , ∂z} is given by:

〈, 〉 ={(

a21(−z)2 + a22(−z)2)dx2 +

(a11(−z)2 + a12(−z)2

)dy2

}+ dz2

+ (a11(−z)a21(−z) + a12(−z)a22(−z)) (dx ⊗ dy + dy ⊗ dx) . (4)

Levi-Civita connection ∇ for the canonical left invariant metric ofX = R2 oA R

For A =

(a bc d

):

∇E1E1 = aE3 ∇E1E2 = b+c2 E3 ∇E1E3 = −aE1 −b+c2 E2

∇E2E1 = b+c2 E3 ∇E2E2 = dE3 ∇E2E3 = −b+c2 E1 − dE2

∇E3E1 = c−b2 E2 ∇E3E2 =b−c2 E1 ∇E3E3 = 0.

z 7→ (x0, y0, z) is a geodesic in X for every (x0, y0) ∈ R2, which is thefixed point set of the isometry (x , y , z) 7→ (2x0 − x , 2y0 − y , z).So the minimal vertical planes in the coordinates R3 of R2 oA R areinvariant under rotation by π around each vertical line in them.

If b = c = 0, then reflection in planes ”parallel” to the vertical coordinateplanes are isometries.

The mean curvature of each leaf of foliation F = {R2 oA {z} | z ∈ R}with respect to the unit normal field E3 is the constant H = Trace(A)/2.

Scaling A by λ > 0 to obtain λA, changes H into λH = same effect asscaling the ambient metric by 1/λ.

Definition

A Lie group G is unimodular if and only if for every X ∈ L(G) = the Liealgebra of G, the automorphism

adX : L(G)→ L(G), adX (Y ) = [X ,Y ]has trace 0. Otherwise, G is called non-unimodular.

Theorem (Milnor)

A simply connected 3-dimensional Lie group X is non-unimodular if andonly if it is a semidirect product R2 oA R, where the matrix A hasnon-zero trace.

After scaling the metric Lie group X so that Trace(A) = 2, then Acan be expressed uniquely as:

A =

(1 + a −(1− a)b

(1 + a)b 1− a

),

where a, b ≥ 0.The Milnor D-invariant D(X) = det(A) characterizes the groupstructure of X if A 6= I2, where I2 is the identity matrix.

Theorem (Milnor)

Let X be a 3-dimensional non-unimodular metric Lie group.

The set {X ∈ L(X) | Trace(adX ) = 0} is a normal Lie subalgebracalled the unimodular kernel K(X).

K(X) is isomorphic to the commutative Lie algebra of R2.

For any orthonormal basis {E1,E2,E3} of L(X) such that {E1,E2} isan orthonormal basis for K(X), then the metric Lie group isisomorphic to the semidirect product R2 oA R, for

A =

(a bc d

),

where A is the matrix for the induced linear map

adE3 : K(X)→ K(X).In other words, [E3,E1] = aE1 + cE2 and [E3,E2] = bE1 + dE2.

Theorem (Classification of non-unimodular metric Lie groups, Milnor )

Let X be a metric non-unimodular Lie group which is expressed as asemidirect product R2 oA R. After choosing a good orthonormal basisE1,E2 for the unimodular kernel K(X), then:

After scaling the metric Lie group X so that Trace(A) = 2, then Ais expressed uniquely as:

A =

(1 + a −(1− a)b

(1 + a)b 1− a

),

where a, b ≥ 0.

D(X) = det(A) characterizes the group structure of X if A 6= I2.12Trace(A) = 1 is the mean curvature of the planes R

2 o {z} forz ∈ R.

Basis {E1,E2,E3 = E1 × E2} of L(X) are the principal Riccicurvature directions with eigenvalues:

Ric(E1) = −2(1 + a(1 + b2)

)Ric(E2) = −2

(1− a(1 + b2)

)Ric(E3) = −2

(1 + a2(1 + b2)

).

Theorem (Classification of unimodular metric semidirect products)

Let X be a metric unimodular Lie group which is expressed as a nontrivialsemidirect product R2 oA R; in other words, A 6= 0. After choosing agood orthonormal basis E1,E2,E3 for L(X) so that [E1,E2] = 0, then:

Trace(A) = 0 which equals the mean curvature of the planesR2 o {z} for z ∈ R.

After scaling the metric Lie group X, A can be expressed uniquelyas:

A =

(0 ± 1aa 0

)for a > 0 or A =

(0 10 0

).

If det(A) = −1, then the group is Sol3 and these A give all the leftinvariant metrics (up to scaling).

If det(A) = 1, then the group is Ẽ(2) and these A give all the leftinvariant metrics (up to scaling).

If det(A) = 0, then the group is Nil3 and A gives the unique leftinvariant metric (up to scaling).

Unimodular groups

Suppose that X is unimodular.

Then there exists an orthonormal basis {E1,E2,E3} of L(X) suchthat

[E2,E3] = c1E1, [E3,E1] = c2E2, [E1,E2] = c3E3. (5)

for certain c1, c2, c3 ∈ R.

These ci depend on the chosen left invariant metric, but their signsdetermine the underlying unimodular Lie algebra.

This follows from the following fact: if we change the left invariantmetric by changing the lengths of E1,E2,E3, say we let declarebcE1, acE2, abE3 to be orthonormal for a, b, c > 0, then the newconstants in (5) are a2c1, b2c2, c2c3.

The six cases for the Lie algebra of an unimodular three-dimensionalLie group are listed in the next table.

Three-dimensional unimodular Lie groups

Signs of c1, c2, c3 Lie group+, +, + SU(2)

+, +, – S̃L(2,R)+, +, 0 Ẽ(2)+, –, 0 Sol3+, 0, 0 Nil30, 0, 0 R3

Here c1, c2, c3 are the structure constants related to an orthonormalunimodular basis E1,E2,E3 of the metric Lie algebra, and whichdiagonalizes the Ricci curvature form.

The last 4 groups in the table can be expressed as semidirectproducts.

Three-dimensional, simply connected unimodular metric Lie groups

Signs c1, c2, c3 dim Iso(X) = 3 dim Iso(X) = 4 dim Iso(X) = 6

+, +, + SU(2) S3Berger = E(κ > 0, τ) S3(κ)+, +, – S̃L(2,R) E(κ < 0, τ) Ø+, +, 0 Ẽ(2) Ø (Ẽ(2), flat)

+, –, 0 Sol3 Ø Ø

+, 0, 0 Ø Nil3 = E(0, τ) Ø0, 0, 0 Ø Ø R3

Each horizontal line corresponds to a unique Lie group structure;when all the structure constants are different, the isometry group ofX is 3-dimensional.

If two or more constants agree, the isometry group of X hasdimension 4 or 6.

We have used the standard notation E(κ, τ) for the total space ofthe Riemannian submersion with bundle curvature τ over acomplete simply connected surface of constant curvature κ.

Levi-Civita connection ∇ for a unimodular X

It is convenient to introduce new constants µ1, µ2, µ3 ∈ R by

µ1 =1

2(−c1+c2+c3), µ2 =

1

2(c1−c2+c3), µ3 =

1

2(c1+c2−c3).

The Levi-Civita connection ∇ for the metric associated to theseconstants µi is given by

∇E1E1 = 0 ∇E1E2 = µ1E3 ∇E1E3 = −µ1E2∇E2E1 = −µ2E3 ∇E2E2 = 0 ∇E2E3 = µ2E1∇E3E1 = µ3E2 ∇E3E2 = −µ3E1 ∇E3E3 = 0.

The symmetric Ricci tensor associated to the metric diagonalizes inthe basis {E1,E2,E3} with eigenvalues:

Ric(E1) = 2µ2µ3, Ric(E2) = 2µ1µ3, Ric(E3) = 2µ1µ2.

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Theory of metric Lie groups andH-surfaces in homogeneous3 ... · Example (The metric Lie group R) The real numbers R with its usual metric and group operation + is a Lie group with