Theory of Metric Lie Groups and H-Surfaces in Homogeneous 3

Home , Isometry group

Theory of metric Lie groups andH-surfaces in homogeneous3-manifolds. William H. Meeks III University of Massachusetts at Amherst Based on joint work with Mira, P´erez,Ros and Tinaglia.

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

Deﬁnition If the isometry group of a Riemannian manifold Y acts transitively, then Y 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 invariant metric (X is a metric Lie group).

H(Y) = Inf{max |HM| : M = immersed closed surface in Y}, where max |HM| denotes max of absolute mean curvature function HM. The number H(Y) is called the critical mean curvature of Y. Area(∂K) Ch(Y) = Inf = Cheeger constant of Y. K⊂Y compact Volume(K) Goals of Lecture 1 Classify simply connected homogeneous 3-manifolds Y. Present the basic theory of simply connected, 3-dimensional metric Lie groups X. Express metric and bases for left and right invariant vector ﬁelds in 2 explicit coordinates for X = R oA R = a semidirect product. Give formulae for the Levi-Civita connection ∇, the principal Ricci curvature 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 of algebraic 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, Stability Conjecture, Product CMC Foliation Conjectures. Goals of Lecture 3 Classification of algebraic open book decompositions. The left invariant Lie group Gauss map and the embeddedness problem for 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 in X (Meeks-Mira-P´erez-Ros). Outline of the proof of the existence and uniqueness of H-spheres in X (Meeks-Mira-P´erez-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-parameter family of non-isometric homogeneous 3-manifolds. or 2 Y is isometric to S (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 the universal covering group of a 3-dimensional subgroup of the 6-dim aﬃne group 2 2 2 F = {f(x) = Ax + b: R → R | b ∈ R , A ∈ GL(2, R)}, which is the natural 2 2 semidirect product of R with GL(2, R) = Aut(R ). The 3-dimensional subgroups of F are one of the following types: SL(2, R) = {A ∈ GL(2, R) | det(A) = 1}. 2 The semidirect product of the normal subgroup R ⊂ F of translations with 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 groupG 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 the multiplicative 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 the diﬀerential 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 , or equivalently, 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 be identified with the tangent space Te G at the identity element e ∈ G. L(G) is a Lie algebra under the Lie bracket of vector fields, i.e., for X , 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 the identity is the image of a 1-parameter subgroup, i.e., there is a Lie

group homomorphism expXe : R → G(X ) ⊂ G . In this case X is the velocity vector exp0 (0) of exp (t) at e. e Xe Xe

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

2 When G is a subgroup of GL(n, R) ⊂ M(n, R) = Rn , then L(G) = Te G can be identiﬁed with the subspace of M(n, R) = Te GL(2, R) of tangent vectors to G at the identity matrix e = In.

When G ⊂ GL(2, R) and Xe = A ∈ M(n, R) = Te G, ∞ X tnAn exp (t) = exp(tA) = ∈ G(X ) ⊂ G. Xe n! n=0 This explains the notation for the group homomorphism

expXe : R → G, for X ∈ L(G). Theorem (Ado’s Theorem)

Given a n-dimensional real Lie algebra L, then for some k ∈ N, L is isomorphic to a subalgebra L0 of the Lie algebra of k × k matrices M(k, R) under the bracket of matrices. L is isomorphic to the Lie algebra of the connected Lie subgroup G of GL(k, R) generated by the elements

∞ X tnAn exp(tA) = ∈ GL(k, ) for t ∈ , A ∈ L0. n! R R n=0

Therefore, every simply connected Lie group is the universal covering 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-parameter group of diffeomorphisms obtained by letting G(X ) act on G from the right. The velocity field of the 1-parameter group of diffeomorphisms obtained on G by the left action of G(X ) on G is the right invariant X X vector field K , where Ke = Xe . Let R(G) denote the n-dimensional vector space of right invariant vector 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 K X is the Killing field associated to the 1-parameter group of isometries G(X ) for ANY left invariant metric on G. K(X) = space of Killing fields in X is a Lie algebra isomorphic to the 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 for all a ∈ G, la : G → G is an isometry of G.

Each left invariant metric on G is obtained by taking a metric h, ie on Te G and deﬁning for a ∈ G and v, w ∈ TaG, −1 −1 hv, wi = ha v, a wie . We call a Lie group Y with a left invariant metric, a metric Lie group. The related metric Lie group is complete. The universal cover Ye with the pulled back metric is a metric Lie group and Y = Ye/π1(Y), where we identify π1(Y) with a discrete normal 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 + is a Lie group with a left invariant metric.

In this case L(R) = R(R) are just the constant vector ﬁelds t Xp = (p, t), where we consider the tangent bundle TR of R to be R × R. 1 By taking X = (0, 1) ∈ T0 , exp 1 : → is a group 0 R X0 R R isomorphism, which is essentially the identity map.

Theorem (Simply connected homogeneous 1 and 2-manifolds)

R with its usual metric is the unique simply connected homogeneous 1-manifold. Every simply connected homogeneous 2-manifold has constant sectional 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 of dimension ≤ 2 are isometric to metric to Lie groups. Example (The metric Lie group H = R oA=(1) R) Here we consider A = (1) to be the 1 × 1 identity matrix.

The notation R oA=(1) R means we have a semidirect product of R with R via the homomorphism σ : R → Aut(R) = R − {0}, where At t σ(t) = e = e . Let σt denote the automorphism σ(t). Then the non-commutative group operation ? on H is b (a, b) ? (c, d) = (a + σb(c), b + d) = (a + e c, 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} o R have constant geodesic curvature less than 1 in H with the orbit ({0} o R) ? (0, 0) = {0} o R being a geodesic. Orbits of the 1-parameter group of left translations by elements in R × {0} are parallel horocycles in H orthogonal to {0} × R. (a, b) 7→ ebx + a is an isomorphism with the subgroup K of aﬃne transformations 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 = R o(1) R is represented by the unit disk model. By left translation, we consider the Lie group H to be a subgroup of PSL(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} o R, corresponding to left translations by elements in {0} × R. Right: Parabolic rotations/translations around the point the top boundary point at infinity in H, corresponding to left translations by elements in R × {0}. Definition

SLf(2, R) = PSLg (2, R) is the universal covering of the projective linear group PSL(2, R) = SL(2, R)/{±I2}. The Lie algebra of any of the groups SL(2, R), SLf(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 normal subgroups other than the trivial ones.

SLf(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 preserving isometries of the hyperbolic plane.

Using the upper half-plane model for H, these are transformations of the type az + b z ∈ 7→ (a, b, c, d ∈ , ad − bc = 1). H cz + d R

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

This is because an orientation preserving isometry of H is uniquely determined by the image of a point and the image under its diﬀerential of a given unitary vector tangent at that point. 1 This view of PSL(2, R) = T1H as an S -bundle over H (and hence of PSLg (2, R) as a R-bundle over H) deﬁnes naturally the so called standard metric on PSL(2, R) (resp. on SLf(2, R) = PSLg (2, R)). The isometry group of SL(2, R) with its standard metric has dimension 4. 1 0 0 1 A basis for sl(2, ) is E = , E = , R 1 0 −1 2 1 0 0 −1 et 0 E = with 1-parameter subgroups Γ = , 3 1 0 1 0 e−t cosh t sinh t cos t − sin t Γ = , Γ = , respectively. 2 sinh t cosh t 3 sin t cos t

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 SLf(2, R) for the ”orthonormal” basis E1, E2, E3 of sl(2, R) is the E(κ, τ)-metric for the unit tangent bundle 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 SLf(2, R) with structure constants c1 = 2, c2 = 2, c3 = −2. 0 0 Left invariant metrics ga,b,c with E1 = aE1, E2 = bE2, E3 = cE3, a, b, c > 0. SLf(2, R) = Tg1H with usual metric and proj. Π: SLf(2, R) → H. Here Π: SLf(2, R) → H is the related Riemannian submersion. −1 E1, E2, E3 ⊂ Te SLf(2, R) = orth. basis, E3 tangent to Π (Π(e)). 0 0 Invar. metrics ga,b,c with E1 = aE1, E2 = bE2, E3 = cE3, a, b, c > 0. SLf(2, R) = Tg1H with usual metric and proj. Π: SLf(2, R) → H. Here Π: SLf(2, R) → H is the related Riemannian submersion. −1 E1, E2, E3 ⊂ Te SLf(2, R) = orth. basis, E3 tangent to Π (Π(e)). 0 0 Invar. metrics ga,b,c with E1 = aE1, E2 = bE2, E3 = cE3, a, b, c > 0. SLf(2, R) = Tg1H with usual metric and proj. Π: SLf(2, R) → H. Here Π: SLf(2, R) → H is the related Riemannian submersion. −1 E1, E2, E3 ⊂ Te SLf(2, R) = orth. basis, E3 tangent to Π (Π(e)). 0 0 Invar. metrics ga,b,c with E1 = aE1, E2 = bE2, E3 = cE3, a, b, c > 0. SLf(2, R) = Tg1H with usual metric and proj. Π: SLf(2, R) → H. Here Π: SLf(2, R) → H is the related Riemannian submersion. −1 E1, E2, E3 ⊂ Te SLf(2, R) = orth. basis, E3 tangent to Π (Π(e)). 0 0 Invar. metrics ga,b,c with E1 = aE1, E2 = bE2, E3 = cE3, a, b, c > 0. SLf(2, R) = Tg1H with usual metric and proj. Π: SLf(2, R) → H. Here Π: SLf(2, R) → H is the related Riemannian submersion. −1 E1, E2, E3 ⊂ Te SLf(2, R) = orth. basis, E3 tangent to Π (Π(e)). 0 0 Invar. metrics ga,b,c with E1 = aE1, E2 = bE2, E3 = cE3, a, b, c > 0. Two-dimensional subgroups of PSL(2,R).

1 H type subgroups. For each θ ∈ S = ∂H, Hθ denotes the subgroup of isometries that ﬁx θ.

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

Each Hθ is isomorphic to the nontrivial semidirect product R oA R, where A = I1 = (1), with curvature −1. R oA {0} ⊂ R oA R is a normal subgroup corresponding to the parabolic subgroup ﬁxing θ. {0} oA R ⊂ R oA R is a subgroup corresponding to translation along a geodesic. Two representations of a subgroup Hθ of PSLg (2, R), θ ∈ ∂∞H.

Left: As the semidirect product R o(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 orbit of one of the hyperbolic 1-parameter subgroups of R o(1) R. Each of the red curves in the right picture corresponds to an orbit of the normal subgroup R o(1) {0}. Example (SU(2) and the Special Orthogonal Group SO(3)) The special orthogonal group SO(3) of rotations about the origin in R3.

π1(SO(3)) = Z2 and the universal covering group of SO(3) is SU(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 isometry of R4, the usual metric on S3 with constant sectional curvature 1 is a left invariant metric.

In this case SO(3) = S3/{±1} is seen to have a quotient left invariant metric of constant sectional curvature 1. The 1-parameter subgroups of SO(3) are the circle subgroups given by all rotations around some ﬁxed axis in R3. Example (SU(2) and the Special Orthogonal Group)

2 3 3 Let T1(S ) = {(x, y) ∈ R × R | kxk = kyk = 1, x ⊥ y} be the unit tangent bundle of S2, which can be viewed as a Riemannian submanifold of TR3 = R3 × R3. The natural constant curvature metric on SO(3) is also the one 2 induced from the diﬀeomorphism F : SO(3) → T1(S ) given by

2 3 3 F (c1, c2, c3 = c1 × c2) = (c1, c2) ∈ T1(S ) ⊂ R × R ,

where c1, c2, c3 are the columns of the corresponding matrix in SO(3). 2 Semidirect Products: Lie Groups of the form R oA R 2 Given a matrix A ∈ M2(R), t ∈ R and p ∈ R , consider the homomorphism σ : R → GL(2, R) = Aut(R2) given by

∞ ! X tnAn σ(t)(p) = etAp = p. n! n=0

2 2 2 2 For simplicity we denote σ(t): R → R by σt : R → R . 2 The group operation ? of the semidirect product R oA R, where + is the group operation of R2 and R, is given by

z1A (p1, z1) ? (p2, z2) = (p1 + σz1 (p2), z1 + z2) = (p1 + (e p2), z1 + z2).

2 We denote the corresponding group by R oA R, which is the semidirect product of R2 and the universal cover Γ = R of σ(R) ⊂ GL(2, R). 2 Metric Lie Groups of the form R oA R 2 Given a matrix A, the group R oA R has an associated left invariant metric by taking the metric at the tangent space to the identity element (0, 0, 0) to be the product metric of R2 × R. 0 0 1 0 Given A = ∈ M ( ), then ezA = . This 0 0 2 R 0 1 produces the usual direct product of groups, which in our case is R3 = R2 × R with its usual metric. 1 0 ez 0 Taking A = , then ezA = and one recovers 0 1 0 ez

the hyperbolic three-space H3 with its usual metric. 1 0 ez 0 Taking A = , then ezA = . This gives 0 0 0 1

2 2 R oA R = H × R with its usual product metric. The examples Ee(2), Sol3, Nil3 Here are some other particular cases depending on A: 0 −1 cos z − sin z If A = , then ezA = and 1 0 sin z cos z

2 R oA R = Ee(2), the universal cover of the group of rigid motions of the plane with the ﬂat metric.

−1 0 e−z 0 If A = , then ezA = and 0 1 0 ez

2 R oA R = Sol3 with its usual metric. 0 1 1 z If A = , then ezA = , and 0 0 0 1

2 R oA R = Nil3, the Heisenberg group of upper triangular matrices with its usual metric:  1 a c   0 1 b  . 0 0 1 2 R oA R: Metric and bases for the Lie algebras L(X) and R(X) ab Consider A = . cd 2 2 For R oA R, choose coordinates (x, y) ∈ R , z ∈ R. 2 Then ∂x , ∂y , ∂z is a parallelization of R 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 the following basis {F1, F2, F3} of the right invariant and basis {E1, E2, E3} of left 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 a (z) a (z) ezA = 11 12 . (3) a21(z) a22(z)

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

2 2 2 2 2 2 2 h, i = a21(−z) + a22(−z) dx + a11(−z) + a12(−z) dy + dz

+ (a11(−z)a21(−z) + a12(−z)a22(−z)) (dx ⊗ dy + dy ⊗ dx) . (4) Levi-Civita connection ∇ for the canonical left invariant metric of 2 X = R oA R ab For A = : cd

b+c b+c ∇E1 E1 = a E3 ∇E1 E2 = 2 E3 ∇E1 E3 = −a E1 − 2 E2 b+c b+c ∇E2 E1 = 2 E3 ∇E2 E2 = d E3 ∇E2 E3 = − 2 E1 − d E2 c−b b−c ∇E3 E1 = 2 E2 ∇E3 E2 = 2 E1 ∇E3 E3 = 0.

2 z 7→ (x0, y0, z) is a geodesic in X for every (x0, y0) ∈ R , which is the fixed point set of the isometry (x, y, z) 7→ (2x0 − x, 2y0 − y, z). 3 2 So the minimal vertical planes in the coordinates R of R oA R are invariant under rotation by π around each vertical line in them. If b = c = 0, then reflection in planes ”parallel” to the vertical coordinate planes are isometries. 2 The mean curvature of each leaf of foliation F = {R 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 as scaling the ambient metric by 1/λ. Definition A Lie group G is unimodular if and only if for every X ∈ L(G) = the Lie algebra 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 and 2 only if it is a semidirect product R oA R, where the matrix A has non-zero trace. After scaling the metric Lie group X so that Trace(A) = 2, then A can be expressed uniquely as:

1 + a −(1 − a)b A = , (1 + a)b 1 − a

where a, b ≥ 0. The Milnor D-invariant D(X) = det(A) characterizes the group structure 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 subalgebra called 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} is an orthonormal basis for K(X), then the metric Lie group is 2 isomorphic to the semidirect product R oA R, for ab A = , cd 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 a 2 semidirect product R oA R. After choosing a good orthonormal basis E1, E2 for the unimodular kernel K(X), then: After scaling the metric Lie group X so that Trace(A) = 2, then A is expressed uniquely as:

1 + a −(1 − a)b A = , (1 + a)b 1 − a

where a, b ≥ 0.

D(X) = det(A) characterizes the group structure of X if A 6= I2. 1 2 2 Trace(A) = 1 is the mean curvature of the planes R o {z} for z ∈ R.

Basis {E1, E2, E3 = E1 × E2} of L(X) are the principal Ricci curvature directions with eigenvalues: 2 Ric(E1) = −2 1 + a(1 + b ) 2 Ric(E2) = −2 1 − a(1 + b ) 2 2 Ric(E3) = −2 1 + a (1 + b ) .

Theorem (Classiﬁcation of unimodular metric semidirect products) Let X be a metric unimodular Lie group which is expressed as a nontrivial 2 semidirect product R oA R; in other words, A 6= 0. After choosing a good orthonormal basis E1, E2, E3 for L(X) so that [E1, E2] = 0, then: Trace(A) = 0 which equals the mean curvature of the planes R2 o {z} for z ∈ R. After scaling the metric Lie group X, A can be expressed uniquely as: 0 ± 1 0 1 A = a for a > 0 or A = . a 0 0 0

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

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

If det(A) = 0, then the group is Nil3 and A gives the unique left invariant metric (up to scaling). Unimodular groups Suppose that X is unimodular.

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

[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 signs determine the underlying unimodular Lie algebra. This follows from the following fact: if we change the left invariant metric by changing the lengths of E1, E2, E3, say we let declare bcE1, acE2, abE3 to be orthonormal for a, b, c > 0, then the new 2 2 2 constants in (5) are a c1, b c2, c c3. The six cases for the Lie algebra of an unimodular three-dimensional Lie group are listed in the next table. Three-dimensional unimodular Lie groups

Signs of c1, c2, c3 Lie group +, +, + SU(2) +, +, – SLf(2, R) +, +, 0 Ee(2) +, –, 0 Sol3 +, 0, 0 Nil3 0, 0, 0 R3

Here c1, c2, c3 are the structure constants related to an orthonormal unimodular basis E1, E2, E3 of the metric Lie algebra, and which diagonalizes the Ricci curvature form. The last 4 groups in the table can be expressed as semidirect products. Three-dimensional, simply connected unimodular metric Lie groups

Signs c1, c2, c3 dim Iso(X) = 3 dim Iso(X) = 4 dim Iso(X) = 6 3 3 +, +, + SU(2) SBerger = E(κ > 0, τ) S (κ) +, +, – SLf(2, R) E(κ < 0, τ) Ø +, +, 0 Ee(2) Ø (Ee(2), ﬂat)

+, –, 0 Sol3 Ø Ø

+, 0, 0 Ø Nil3 = E(0, τ) Ø 0, 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 of X is 3-dimensional. If two or more constants agree, the isometry group of X has dimension 4 or 6.

We have used the standard notation E(κ, τ) for the total space of the Riemannian submersion with bundle curvature τ over a complete simply connected surface of constant curvature κ. Levi-Civita connection ∇ for a unimodular X

It is convenient to introduce new constants µ1,µ 2,µ 3 ∈ R by 1 1 1 µ = (−c +c +c ),µ = (c −c +c ),µ = (c +c −c ). 1 2 1 2 3 2 2 1 2 3 3 2 1 2 3

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

∇E1 E1 = 0 ∇E1 E2 = µ1E3 ∇E1 E3 = −µ1E2

∇E2 E1 = −µ2E3 ∇E2 E2 = 0 ∇E2 E3 = µ2E1

∇E3 E1 = µ3E2 ∇E3 E2 = −µ3E1 ∇E3 E3 = 0.

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

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