5 Milnor’s formulas
Also closed, and also PDE-free. Milnor’s 1976 paper computes the curvature of left-invariant metrics in terms of structure constants; nothing from it had been formalized. Everything in this chapter is proved.
Conventions. Ours are \(R(x,y) = \nabla _x\nabla _y - \nabla _y\nabla _x - \nabla _{[x,y]}\) and \(\operatorname {Ric}(x,y) = \operatorname {tr}(w \mapsto R(w,x)y)\), as in Chapter 2. Milnor flips both the curvature sign and the contraction slot, so his numerical values agree with ours — the two flips cancel. Had only one flipped, every eigenvalue below would be silently negated.
\(\langle \nabla _x y, z\rangle = \tfrac 12\big(\langle [x,y],z\rangle - \langle [y,z],x\rangle + \langle [z,x],y\rangle \big)\).
With \(\alpha _{ijk} = \langle [e_i,e_j], e_k\rangle \) for an orthonormal basis, \(\operatorname {Ric}\) expands as a double sum in the \(\alpha _{ijk}\) — in any dimension, for any orthonormal basis. This is the computational backbone of the paper.
In dimension three, a basis with \([e_2,e_3] = \lambda _1 e_1\), \([e_3,e_1] = \lambda _2 e_2\), \([e_1,e_2] = \lambda _3 e_3\).
Existence is proved — see Theorem 34.
In a Milnor frame \(\operatorname {Ric}\) is diagonal, with
equivalently \(r_i = \tfrac 12\big(\lambda _i^2 - (\lambda _j-\lambda _k)^2\big)\).
Every \(3\)-dimensional unimodular metric Lie algebra admits a Milnor frame.
In dimension three the alternating bracket factors as \([x,y] = L(x \times y)\) through the isomorphism \(\Lambda ^2\mathfrak {g} \cong \mathfrak {g}\); unimodularity says exactly that \(L\) is self-adjoint; and the spectral theorem supplies an orthonormal eigenbasis, whose eigenvalues are the \(\lambda _i\). Dimension three is essential — it is where \(\Lambda ^2\mathfrak {g} \cong \mathfrak {g}\) comes from.
Every \(3\)-dimensional unimodular metric Lie algebra admits a basis in which \(\operatorname {Ric}\) is diagonal with \(r_i = 2\mu _j\mu _k\) — with no frame assumed.
The Heisenberg algebra (\(\lambda _1 = \lambda _2 = 0 \neq \lambda _0\)) has principal Ricci curvatures \((\lambda _0^2/2,\, -\lambda _0^2/2,\, -\lambda _0^2/2)\): strictly mixed signs. An instance of Milnor’s Theorem 2.4.
Every \(3\)-dimensional unimodular metric Lie algebra admits an orthonormal Milnor frame in which
with no frame assumed. At Lie-algebra level raising the index is a single composition with \(g^{-1}\), because the metric is carried as a map into the dual.
Sanity check: the round \(SU(2)\) has \(\lambda _i = 2\), hence \(\mu _i = 1\) and \(\operatorname {scal}= 6\) — the unit \(3\)-sphere.
At a diagonal metric \(\operatorname {diag}(A,B,C)\) in a Milnor frame the field \(g\mapsto -2\operatorname {Ric}(g)\) is again diagonal — the ansatz is invariant — and
cyclically. This is the classical Isenberg–Jackson system: Ricci flow on the three-dimensional unimodular Lie groups, written out.
Proving the \(3\)-dimensional case for a diagonal rather than orthonormal metric is what removes the square roots and makes this derivable.
Stated at the level of the field’s value at diagonal metrics. That solutions remain diagonal needs a second Picard–Lindelöf pass through the positive octant, and is not proved.