03.02.37 · differential-geometry / manifolds

Homogeneous Einstein metrics on G/H

shipped3 tiersLean: none

Anchor (Master): Besse Einstein Manifolds Ch. 7; Wang-Ziller 1986 Invent. Math. 84:177; Wolf 1968 Acta Math. 120 (isotropy-irreducible); Jensen 1973 Duke Math. J. 42; Bohm-Wang-Ziller 2004 GAFA 14; Boehm-Lafuente 2023 (Alekseevsky conjecture)

Intuition Beginner

Imagine a space so symmetric that wherever you stand, and however you turn, the geometry around you looks identical. The round ball-surface — the sphere — is the friendliest example: spin it any way you like and nothing changes. A space with that much symmetry is called homogeneous, because every point can be slid onto every other point by a motion that preserves distances.

Now ask the Einstein question: is the curvature spread out so evenly that the space could model a universe in equilibrium? The Einstein condition says the Ricci curvature is the same in every direction, a fixed multiple of the metric. On a generic lumpy space this is a hard demand, a partial-differential equation that ties curvature to shape at every point.

But on a homogeneous space something wonderful happens. Because the geometry is the same everywhere, you only need to check the Einstein condition at a single point. The infinite list of equations collapses to a short list of numbers describing how the metric stretches the few independent directions. The hard analysis becomes a finite calculation.

Visual Beginner

Alt text: On the left, a round sphere carries a little frame of perpendicular arrows at one point; rotations of the sphere carry that frame, unchanged in shape, to many other points, illustrating that the geometry is identical everywhere. On the right, a squashed (stretched) sphere carries a frame whose arrows have different lengths in different directions, showing a second homogeneous shape obtained by choosing different stretching numbers for the independent directions. Both shapes are candidate Einstein metrics, and the picture conveys that on a homogeneous space the whole metric is encoded by just a handful of stretching numbers at one point rather than by a field of values across the space.

Worked example Beginner

Take the ordinary round sphere. Its symmetry group is the set of all rotations of space, and the points of the sphere are exactly the positions you can rotate a fixed starting point into. The directions you can move in from that point are all equal, because a further rotation around the point shuffles them among themselves.

So any homogeneous metric on the round sphere must treat every direction the same way: there is only one stretching number to pick, an overall size. Once you fix the size, the curvature is forced, and a short computation shows the Ricci curvature is the same in every direction. The round sphere is automatically an Einstein space, and the only freedom is how big to make it.

What this tells us: when all the directions at a point are equivalent under the symmetry, the Einstein equation has nothing to solve — there is a single size to choose and the equation balances on its own. The interesting cases are spaces where the directions split into a few independent families, so there are several stretching numbers and a genuine balancing problem among finitely many numbers.

Check your understanding Beginner

Question 1 (multiple-choice). Why does the Einstein condition become a finite problem on a homogeneous space?

Because homogeneous spaces are always flat.
Because the geometry looks the same at every point, so checking one point checks them all.
Because the Ricci curvature is always zero there.
Because such spaces have no curvature to compute.

Hint

The symmetry moves any point onto any other while preserving the metric. What does that say about the curvature at different points?

Answer

B. A motion that preserves the metric carries the curvature at one point to the curvature at any other, so the Einstein equation reads the same everywhere. Verifying it at a single point verifies it everywhere, and the metric itself is described by a few stretching numbers, so the problem is finite. (A and D are wrong: homogeneous spaces are generally curved. C is wrong: the Ricci curvature is usually nonzero.)

Question 3 (multiple-choice). A "squashed sphere" stretches one family of directions differently from the rest. How many size numbers does such a metric carry, and what does this make possible?

One number; nothing new can happen.
Two numbers; a second balancing point can appear, giving a different Einstein metric.
Infinitely many numbers; the problem stays infinite.
Zero numbers; the metric is fixed with no choices.

Hint

Two families of directions means two independent sizes. Balancing the curvature between them is one equation in those numbers.

Answer

B. Two families of directions give two size numbers. The Einstein balance between them can be solved in more than one way, so besides the round metric a second, genuinely squashed Einstein metric can appear. This is the famous extra Einstein metric on certain odd spheres.

Formal definition Intermediate+

Let $G$ be a compact (or, more generally, reductive) Lie group and $H \subset G$ a closed subgroup, so that $M = G / H$ is a homogeneous space on which $G$ acts transitively by diffeomorphisms. Write $g = Lie (G)$ and $h = Lie (H)$ . A reductive decomposition is an $Ad (H)$ -invariant complement $$ \mathfrak g = \mathfrak h \oplus \mathfrak m, \qquad \mathrm{Ad}(h),\mathfrak m \subseteq \mathfrak m ;; \forall h \in H . $$ When $G$ is compact such a complement always exists (take the orthogonal complement of $h$ with respect to a bi-invariant metric); see 07.04.07 for the symmetric-space case, where additionally $[m, m] \subseteq h$ . The subspace $m$ is canonically identified with the tangent space $T_{eH} (G / H)$ , and the action of $H$ on $m$ by $Ad$ is the isotropy representation.

Definition (invariant metric). A $G$ -invariant Riemannian metric on $G / H$ is determined by its value at the base point $eH$ , which is an $Ad (H)$ -invariant inner product $⟨ \cdot, \cdot ⟩$ on $m$ . Thus $$ {,G\text{-invariant metrics on } G/H,} ;\longleftrightarrow; {,\mathrm{Ad}(H)\text{-invariant inner products on } \mathfrak m,}. $$ Decompose the isotropy representation into irreducible summands. If $$ \mathfrak m = \mathfrak m_1 \oplus \cdots \oplus \mathfrak m_r $$ with the $m_{i}$ pairwise inequivalent irreducible $Ad (H)$ -modules, then by Schur's lemma every invariant inner product is $$ \langle,\cdot,,\cdot,\rangle = x_1, B|{\mathfrak m_1} ,\oplus, \cdots ,\oplus, x_r, B|{\mathfrak m_r}, \qquad x_i > 0, $$ where $B$ is a fixed background bi-invariant form (a multiple of the negative Killing form when $G$ is compact semisimple). The invariant metrics form an open cone of dimension $r$ — the multiplicity-free case. When summands repeat or are equivalent, the parameter space is larger but still finite-dimensional.

Definition (homogeneous Einstein metric). An invariant metric $g$ on $G / H$ is Einstein if its Ricci tensor satisfies $Ric_{g} = λ g$ for a constant $λ \in R$ (the Einstein constant), in the sense of 03.02.05. By homogeneity the Ricci tensor is itself $G$ -invariant, so it is an $Ad (H)$ -invariant symmetric form on $m$ , and the Einstein condition is a single algebraic relation at the base point.

The scalar curvature as an algebraic function. Fix a $B$ -orthonormal basis ${X_{i}}$ of $m$ adapted to the decomposition, and write $[X_i, X_j] = \sum_k c_{ij}^k X_k + (\text{$ \mathfrak h $-part})$ for the structure constants, with $m$ -components $c_{ij}^{k}$ . For the diagonal metric $g = diag (x_{1}, \dots, x_{r})$ the scalar curvature is the rational-algebraic expression $$ S(g) ;=; \tfrac12 \sum_i \frac{b_i}{x_i} ;-; \tfrac14 \sum_{i,j,k} [ijk],\frac{x_k}{x_i x_j}, $$ where $b_{i} \geq 0$ packages the Killing-form data on $m_{i}$ and $[ij k] = \sum ∣ ⟨[X_{a}, X_{b}], X_{c} ⟩ ∣^{2}$ (summed over basis vectors in $m_{i}, m_{j}, m_{k}$ ) is the symmetric structure-constant array, depending only on $G / H$ and not on the $x_{i}$ . The Einstein equation $Ric = λ g$ becomes the finite polynomial system $Ric_{i} (x) = λ x_{i}$ , $i = 1, \dots, r$ , in the $r$ unknowns $x_{i}$ (and $λ$ ).

Counterexamples to common slips

Homogeneous does not imply Einstein. A generic invariant metric on a homogeneous space need not be Einstein; the round metric on $S^{3} = S U (2)$ is, but a left-invariant Berger metric that shrinks the Hopf fibre is not. Being homogeneous only makes the Einstein equation finite, not automatic.
Invariant Einstein metrics need not be unique up to scale. Even on a fixed $G / H$ there can be several non-isometric invariant Einstein metrics. The squashed sphere supplies a second one alongside the round metric, so "the" homogeneous Einstein metric is a misnomer.
Isotropy reducible versus irreducible. When the isotropy representation is irreducible ( $r = 1$ ), there is only one invariant metric up to scale and it is forced to be Einstein (Wolf). The genuinely interesting Einstein problem lives in the reducible case $r \geq 2$ , where balancing the $x_{i}$ is a real system to solve.

Key theorem with proof Intermediate+

Theorem (Wang–Ziller variational principle). Let $G / H$ be a compact homogeneous space with a fixed background volume. On the space $M_{1}$ of $G$ -invariant Riemannian metrics of total volume $1$ , the scalar curvature $g \mapsto S (g)$ is a smooth function, and a metric $g \in M_{1}$ is Einstein if and only if it is a critical point of $S$ restricted to $M_{1}$ . Equivalently, invariant Einstein metrics are exactly the critical points of the total scalar curvature functional of 03.02.36 restricted to the finite-dimensional manifold of unit-volume invariant metrics.

Proof. The total scalar curvature (Einstein–Hilbert) functional of 03.02.36 is $$ \mathbf S(g) = \int_M S(g), dV_g, $$ and its general first variation, restricted to volume- $1$ deformations, has gradient the trace-free Ricci tensor: a unit-volume metric is a critical point of $S$ if and only if $Ric_{g} = λ g$ for the constant $λ = S (g) / n$ , $n = dim M$ . This is the variational characterisation of Einstein metrics in 03.02.36.

Specialise to $G$ -invariant metrics. Because every invariant metric is homogeneous, its scalar curvature $S (g)$ is constant over $M$ , so $$ \mathbf S(g) = S(g) \cdot \mathrm{Vol}(M, g) = S(g) $$ on the unit-volume slice $M_{1}$ . Thus on $M_{1}$ the integral functional $S$ coincides with the pointwise scalar curvature $S$ . By the principle of symmetric criticality (Palais): since $G$ acts on the space of all metrics preserving $S$ and the invariant metrics are the fixed-point set, a $G$ -invariant metric is a critical point of $S$ on all unit-volume metrics if and only if it is a critical point of the restriction $S ∣_{M_{1}} = S ∣_{M_{1}}$ . Combining the two displayed facts: $g$ is Einstein $⟺$ $g$ is $S$ -critical at unit volume $⟺$ $g$ is a critical point of $S ∣_{M_{1}}$ . Because $M_{1}$ is finite-dimensional (dimension $r - 1$ in the multiplicity-free case), this is a critical-point problem for an explicit algebraic function of finitely many variables, the scalar-curvature expression $S (x_{1}, \dots, x_{r})$ subject to the volume constraint $\prod x_{i}^{d i m m_{i}} = 1$ . $■$

The Lagrange-multiplier form is concrete: with $V (x) = \prod_{i} x_{i}^{d_{i}}$ ( $d_{i} = dim m_{i}$ ) the constraint, the Einstein metrics are the solutions of $\nabla S = μ \nabla lo g V$ , i.e. $x_{i} \partial_{x_{i}} S = μ d_{i}$ for a common multiplier $μ$ — exactly the system $Ric_{i} = λ x_{i}$ after rescaling.

Bridge. This builds toward the global existence theory: the variational reduction is the foundational reason that the analytically forbidding Einstein PDE becomes a finite critical-point problem, and it is exactly the mechanism by which the total scalar curvature functional of 03.02.36 descends from an infinite-dimensional space of metrics to the $(r - 1)$ -dimensional cone of invariant metrics. The Palais symmetric-criticality step is dual to the reduction of any symmetric variational problem to its fixed-point set, and the central insight — that on $M_{1}$ the integral functional collapses to the pointwise scalar curvature — recurs in 07.04.07 for symmetric spaces and appears again in the graph-theoretic search for new Einstein metrics via mountain-pass and minimax arguments on $M_{1}$ . Putting these together, homogeneous Einstein geometry becomes a chapter of finite-dimensional Morse theory on the cone of invariant inner products. This generalises the single-size computation on the round sphere to the full multi-parameter balancing problem.

Exercises Intermediate+

Exercise (easy, short-answer). Show that if the isotropy representation of $G/H$ is irreducible, then $G/H$ carries a unique $G$-invariant metric up to scale, and that metric is Einstein.

Hint

Use Schur's lemma: an $Ad (H)$ -invariant inner product on an irreducible real module is unique up to a positive scalar. The Ricci form is another invariant symmetric form.

Answer

Rubric. By Schur's lemma applied to the irreducible isotropy module $m$ , any two $Ad (H)$ -invariant inner products are proportional, so the invariant metric is unique up to a positive scale. The Ricci tensor of an invariant metric is itself an $Ad (H)$ -invariant symmetric bilinear form on $m$ ; by the same Schur argument it must be a scalar multiple of the metric, $Ric = λ g$ . Hence the metric is Einstein. Full credit invokes Schur on both the metric and the Ricci form. This is Wolf's isotropy-irreducible criterion.

Exercise (medium, symbolic). On $G/H$ with two isotropy summands ($r = 2$, dimensions $d_1, d_2$), the scalar curvature has the schematic form $S(x_1,x_2) = \tfrac{a_1}{x_1} + \tfrac{a_2}{x_2} - b\,\tfrac{x_2}{x_1^2}$ for constants $a_1,a_2,b > 0$. Set up the Einstein system as a single equation after imposing unit volume.

Hint

Use the Lagrange condition $x_{i} \partial_{x_{i}} S = μ d_{i}$ and eliminate $μ$ by taking the ratio of the two equations.

Answer

Rubric. Compute $x_{1} \partial_{x_{1}} S = - \frac{a _{1}}{x _{1}} + \frac{2 b x _{2}}{x _{1}^{2}}$ and $x_{2} \partial_{x_{2}} S = - \frac{a _{2}}{x _{2}} - \frac{b x _{2}}{x _{1}^{2}}$ . The Einstein condition is $x_{1} \partial_{x_{1}} S / d_{1} = x_{2} \partial_{x_{2}} S / d_{2}$ . Cross-multiplying and clearing denominators gives one polynomial relation between $x_{1}$ and $x_{2}$ ; combined with $x_{1}^{d_{1}} x_{2}^{d_{2}} = c$ it leaves a single algebraic equation in the ratio $t = x_{2} / x_{1}$ . Full credit reaches a polynomial in $t$ whose positive roots are the Einstein metrics. The number of positive roots is the number of invariant Einstein metrics up to scale.

Exercise (medium, short-answer). Explain why a normal homogeneous metric — the one induced from a bi-invariant metric on $G$ via the reductive splitting — has nonnegative sectional curvature, and why this need not make it Einstein.

Hint

For a normal metric the O'Neill formula for the Riemannian submersion $G \to G / H$ gives the curvature as the bi-invariant curvature plus a nonnegative correction. Einstein is a separate balancing condition on the Ricci eigenvalues.

Answer

Rubric. A normal metric makes $G \to G / H$ a Riemannian submersion from the bi-invariant $(G, B)$ , which has nonnegative sectional curvature; O'Neill's formula adds a nonnegative term $\frac{3}{4} ∣ [X, Y]_{m} ∣^{2}$ for horizontal $X, Y$ , so the base has nonnegative sectional curvature. But the Ricci tensor of the normal metric is generally not a multiple of the metric: its eigenvalues differ across inequivalent isotropy summands unless the structure constants happen to balance. Full credit separates the curvature-sign statement (O'Neill) from the Einstein balancing statement (equality of Ricci eigenvalues).

Exercise (hard, short-answer). State and justify the Wang–Ziller non-existence phenomenon: exhibit the structural reason that some homogeneous spaces admit no $G$-invariant Einstein metric at all, even though the variational problem is finite-dimensional.

Hint

A finite-dimensional critical-point problem can fail to have a critical point if the function has no interior maximum or critical value — the scalar curvature can tend to its supremum only on the boundary of the cone (some $x_{i} \to 0$ or $\infty$ ).

Answer

Rubric. On the open cone $M_{1}$ the scalar curvature $S$ is smooth but not proper; a critical point exists only if $S$ attains an interior maximum (or other critical value) rather than escaping to the boundary where some block collapses ( $x_{i} \to 0$ ) or expands. Wang and Ziller construct homogeneous spaces — certain $G / H$ with several isotropy summands whose structure-constant array is unbalanced — for which $sup_{M_{1}} S$ is approached only along degenerating sequences, so no invariant Einstein metric exists. Full credit identifies the loss of an interior critical point as a failure of properness / the maximum escaping to the boundary of the cone, and cites Wang–Ziller as the source.

Exercise (hard, symbolic). For the round sphere $S^n = SO(n+1)/SO(n)$, verify that the isotropy representation is irreducible and conclude that the round metric is Einstein with $\mathrm{Ric} = (n-1)g$ (unit radius).

Hint

The isotropy group $S O (n)$ acts on $m ≅ R^{n}$ by its standard representation, which is irreducible for $n \geq 2$ . Then use the previous result and compute the constant directly from the constant sectional curvature $1$ .

Answer

$Ric = (n - 1) g$ . The isotropy action of $S O (n)$ on $m ≅ T_{eH} S^{n} ≅ R^{n}$ is the standard representation, irreducible for $n \geq 2$ , so by Wolf's criterion the invariant metric is unique up to scale and Einstein. The unit sphere has constant sectional curvature $1$ ; tracing, $Ric (v, v) = \sum_{w ⊥ v} sec (v, w) = (n - 1) ∣ v ∣^{2}$ , so $Ric = (n - 1) g$ with Einstein constant $λ = n - 1$ . Full credit identifies the standard representation, invokes irreducibility, and computes the constant by tracing the sectional curvature.

Lean formalization Intermediate+

Mathlib has Lie groups, Lie algebras, the adjoint representation, and a developing Riemannian-geometry layer with the Ricci tensor on a general manifold, but it has no homogeneous-space differential geometry: no reductive decomposition, no identification of invariant metrics with $Ad (H)$ -invariant inner products, and no algebraic curvature formula for an invariant metric. The central results here are therefore not yet formalisable end-to-end. The following is the statement-level shape one would target, written in Lean-compatible pseudo-Lean. It does not compile against current Mathlib (lean_status: none).

-- Statement target (NOT compiling against current Mathlib):
-- Reductive decomposition g = h ⊕ m with Ad(H)-invariance.
variable {G : Type*} [LieGroup G] (H : ClosedSubgroup G)
variable (m : Submodule ℝ (LieAlgebra G))    -- the isotropy complement

def IsReductive : Prop :=
  IsCompl (lieAlgebra H) m ∧ ∀ h ∈ H, AdInvariant h m

-- An invariant metric ↔ an Ad(H)-invariant inner product on m:
def InvariantMetric := { q : InnerProduct ℝ m // ∀ h ∈ H, AdInvariant h q }

-- Einstein condition for an invariant metric (algebraic at the base point):
def IsHomogeneousEinstein (q : InvariantMetric H m) (lam : ℝ) : Prop :=
  ricciOfInvariantMetric q = lam • q.val

-- Wang–Ziller (target): Einstein ↔ critical point of scalar curvature
-- on the unit-volume invariant metrics.
-- theorem wang_ziller (q : InvariantMetric H m) :
--   (∃ lam, IsHomogeneousEinstein q lam)
--     ↔ IsCriticalPoint (scalarCurvature ∘ ·) (unitVolumeSlice H m) q

the Mathlib gap analysis above enumerates the missing primitives: the reductive decomposition, the invariant-metric correspondence, the algebraic Ricci and scalar-curvature formulas in the structure constants, the variational characterisation, and the structural theorems (Wolf, Wang–Ziller, Alekseevskii), none of which have an analogue in current Mathlib.

Advanced results Master

Isotropy-irreducible spaces are Einstein (Wolf). When the isotropy representation $Ad : H \to G L (m)$ is irreducible, the invariant metric is unique up to scale and automatically Einstein, by the Schur argument applied to both the metric and the Ricci form. Wolf's 1968 classification ^{[Wolf 1968]} enumerates the strongly isotropy-irreducible homogeneous spaces of compact simple groups, a list that supplies a large supply of Einstein manifolds with no continuous deformation freedom. These are the rigid endpoints of the theory: the scalar-curvature function on $M_{1}$ is constant (a single point), and the Einstein equation has nothing to balance.

Naturally reductive and normal examples. A reductive homogeneous space is naturally reductive for a metric if the geodesics through the base point are orbits of one-parameter subgroups, equivalently $⟨[X, Y]_{m}, Z ⟩ + ⟨ Y, [X, Z]_{m} ⟩ = 0$ on $m$ . Normal homogeneous metrics — induced from a bi-invariant metric on $G$ — are the leading examples; they have nonnegative sectional curvature by O'Neill and include the standard metrics on spheres, projective spaces, and isotropy-irreducible spaces. Naturally reductive Einstein metrics form the most computable family, since their Ricci tensor has a closed Killing-form expression, and Besse Ch. 7 ^{[Besse Ch. 7]} tabulates the normal-homogeneous Einstein metrics on the classical flag and Stiefel-type spaces.

Multiple Einstein metrics: the Jensen squashed sphere. The odd sphere $S^{4 n + 3} = S p (n + 1) / S p (n)$ carries, besides the round metric, a second $S p (n + 1) \cdot S p (1)$ -invariant Einstein metric obtained by shrinking the three-dimensional $S p (1)$ Hopf fibre relative to the base — the Jensen or squashed metric ^{[Jensen 1973]}. Here the isotropy representation has two inequivalent summands ( $r = 2$ : the $4 n$ -dimensional base direction and the $3$ -dimensional fibre), the scalar curvature is a function of one ratio after fixing volume, and the Einstein equation is a quadratic whose two positive roots are the round and squashed metrics. This is the canonical demonstration that homogeneous Einstein metrics are non-unique.

Flag manifolds. Generalised flag manifolds $G / H$ , where $H$ is the centraliser of a torus, are a rich proving ground: their isotropy representations split into many inequivalent summands indexed by the roots, the structure-constant array is read from the root system, and the Einstein system becomes an explicit polynomial system whose solutions have been classified for many low-rank cases. Flag manifolds also carry invariant complex structures, linking the Einstein problem to the Kähler–Einstein theory and to the representation-theoretic data of 07.04.07.

The Wang–Ziller theorems and the Alekseevskii conjecture. Wang and Ziller ^{[Wang-Ziller 1986]} proved both existence results (via the variational principle and minimax/mountain-pass arguments on $M_{1}$ ) and the first non-existence examples — homogeneous spaces with no invariant Einstein metric — settling that the finite critical-point problem can be empty. For the negative-scalar-curvature (noncompact) side, the Alekseevskii conjecture asserted that a connected homogeneous Einstein manifold with negative Einstein constant is diffeomorphic to $R^{n}$ (so it is a solvmanifold); long open, it was resolved affirmatively by Böhm and Lafuente ^{[Bohm-Lafuente 2023]}, closing the structural classification on the noncompact side.

Synthesis. The homogeneous Einstein problem is the foundational reason that an intractable nonlinear PDE collapses to finite-dimensional critical-point theory: this is exactly the variational reduction of 03.02.36, where the total scalar curvature functional, restricted by the principle of symmetric criticality to the cone of invariant inner products, becomes the explicit algebraic scalar-curvature function whose critical points are the Einstein metrics. Putting these together, the curvature machinery of 03.02.05 supplies the Ricci eigenvalues that must be balanced, the symmetric-space reductive decomposition of 07.04.07 supplies the isotropy summands that index the balancing variables, and the representation theory of $H$ — through Schur's lemma — both counts the parameters and forces the isotropy-irreducible spaces to be Einstein. The central insight that recurs across Wolf's rigidity, Jensen's multiplicity, the Wang–Ziller non-existence examples, and the resolved Alekseevskii conjecture is that the shape of the scalar-curvature function on the cone — its critical points, its behaviour on the boundary, its properness — is the single object governing existence, uniqueness, and multiplicity of invariant Einstein metrics. This generalises the one-parameter round-sphere computation to a global geometry-of-the-cone problem, and it is dual to the way Morse theory reads topology off the critical points of a function.

Full proof set Master

Proposition (invariant metrics $\leftrightarrow$ $Ad (H)$ -invariant inner products). For a reductive homogeneous space $G / H$ with decomposition $g = h \oplus m$ , restriction to the base point is a bijection between $G$ -invariant Riemannian metrics on $G / H$ and $Ad (H)$ -invariant inner products on $m ≅ T_{eH} (G / H)$ .

Proof. A $G$ -invariant metric $g$ is determined by its value $g_{eH}$ at the base point, since $g_{a H} = (L_{a}^{- 1})^{*} g_{eH}$ for any $a \in G$ by invariance, and these prescriptions are consistent precisely when $g_{eH}$ is unchanged under the stabiliser $H$ , i.e. invariant under the isotropy action $Ad (H)$ on $T_{eH} (G / H) ≅ m$ . Conversely, an $Ad (H)$ -invariant inner product on $m$ defines $g_{eH}$ , and translating by $G$ gives a well-defined $G$ -invariant metric: well-definedness across the two representatives $a$ and $ah$ of a coset is exactly the $Ad (h)$ -invariance. The two constructions are mutually inverse. $■$

Proposition (Schur parametrisation of invariant metrics). If $m = m_{1} \oplus \dots \oplus m_{r}$ with pairwise inequivalent irreducible $Ad (H)$ -modules, the cone of invariant inner products is $r$ -dimensional, given by $⨁_{i} x_{i} B ∣_{m_{i}}$ with $x_{i} > 0$ .

Proof. Let $q$ be an $Ad (H)$ -invariant inner product and $B$ the fixed background form. The relative endomorphism $A$ defined by $q (\cdot, \cdot) = B (A \cdot, \cdot)$ is $B$ -self-adjoint and commutes with $Ad (H)$ . By Schur's lemma for real representations, an intertwiner of inequivalent irreducibles vanishes off the diagonal blocks and is a scalar on each irreducible block of real type; the inequivalence and the self-adjointness force $A = ⨁ x_{i} Id_{m_{i}}$ with $x_{i} > 0$ (positivity from $q$ being an inner product). Hence $q = ⨁ x_{i} B ∣_{m_{i}}$ , and the $x_{i}$ are free positive parameters. $■$

Proposition (scalar curvature is constant on an invariant metric). For any $G$ -invariant metric on $G / H$ , the scalar curvature function $S : G / H \to R$ is constant.

Proof. The scalar curvature is a metric invariant: any isometry $ϕ$ satisfies $S \circ ϕ = S$ . The group $G$ acts by isometries of an invariant metric and transitively on $G / H$ , so for any two points $p = a H$ and $q = b H$ there is $ϕ = L_{b a^{- 1}} \in G$ with $ϕ (p) = q$ , giving $S (q) = S (ϕ (p)) = S (p)$ . Hence $S$ is constant. The same argument applies to any scalar Riemannian invariant. $■$

Proposition (variational reduction). On the unit-volume slice $M_{1}$ of invariant metrics, the total scalar curvature functional $S (g) = \int_{M} S (g) d V_{g}$ equals the pointwise scalar curvature $S (g)$ , and its critical points are the invariant Einstein metrics.

Proof. By the previous proposition $S (g)$ is constant on $M$ , so $S (g) = S (g) Vol (M, g) = S (g)$ on $M_{1}$ . The general first-variation formula for $S$ on unit-volume metrics has gradient the trace-free Ricci tensor 03.02.36, whose vanishing is the Einstein condition. By Palais's principle of symmetric criticality, a $G$ -invariant unit-volume metric is critical for $S$ among all unit-volume metrics if and only if it is critical for the restriction $S ∣_{M_{1}} = S ∣_{M_{1}}$ . Therefore the invariant Einstein metrics are exactly the critical points of $S$ on the finite-dimensional manifold $M_{1}$ . $■$

The classification theorems of Wolf, the multiplicity of the Jensen metric, and the Wang–Ziller existence/non-existence results are stated above without full proof — see Wolf ^{[Wolf 1968]}, Jensen ^{[Jensen 1973]}, and Wang–Ziller ^{[Wang-Ziller 1986]}; the Alekseevskii conjecture's resolution is in Böhm–Lafuente ^{[Bohm-Lafuente 2023]}.

Connections Master

The total scalar curvature functional and its critical points 03.02.36. This unit is the homogeneous restriction of that one: the Einstein–Hilbert functional, whose critical points among all unit-volume metrics are the Einstein metrics, descends by the principle of symmetric criticality to the finite-dimensional cone of invariant metrics, where it becomes the explicit algebraic scalar-curvature function. The variational principle proved here is precisely the finite-dimensional shadow of the infinite-dimensional variational characterisation established there.
Riemannian symmetric and homogeneous spaces 07.04.07. The reductive decomposition $g = h \oplus m$ and the isotropy representation that index the entire problem are the structures built there; symmetric spaces are the special case $[m, m] \subseteq h$ , all of which are Einstein when irreducible. The flag manifolds whose Einstein systems are read from the root system carry the invariant complex structures classified in that representation-theoretic setting.
Sectional, Ricci, and scalar curvature 03.02.05. The Ricci tensor whose eigenvalues must be balanced, and the scalar curvature that is varied, are the invariants constructed there; the algebraic Ricci formula for an invariant metric expresses those eigenvalues through the structure constants, turning a curvature computation into linear algebra on $m$ .
Lie-group representation theory (isotropy representation). Schur's lemma applied to the isotropy representation of $H$ both counts the parameters of the invariant-metric cone and forces isotropy-irreducible spaces to be Einstein; the decomposition of $Ad ∣_{H}$ into irreducibles is the representation-theoretic input that makes the Einstein equation a finite polynomial system, linking the geometry to the structure theory of compact groups 07.04.07.

Historical & philosophical context Master

The systematic study of homogeneous Einstein metrics was crystallised by Arthur Besse's encyclopaedic Einstein Manifolds (Springer, 1987), whose Chapter 7 organised the variational viewpoint, the algebraic scalar-curvature formula, and the classification of naturally reductive examples into the canonical reference for the subject ^{[Besse Ch. 7]}. The variational principle itself — that invariant Einstein metrics are the critical points of the scalar curvature on the unit-volume invariant metrics — and the first existence and non-existence theorems were established by McKenzie Wang and Wolfgang Ziller in their 1986 Inventiones paper, which turned the problem into finite-dimensional critical-point theory and exhibited homogeneous spaces admitting no invariant Einstein metric ^{[Wang-Ziller 1986]}.

Two earlier results anchor the structural picture. Joseph Wolf's 1968 Acta Mathematica classification of isotropy-irreducible homogeneous spaces showed that this large family is automatically Einstein, supplying rigid examples with no deformation freedom ^{[Wolf 1968]}; and Gary Jensen's 1973 construction of a second Einstein metric on the odd spheres — the squashed metric — demonstrated definitively that homogeneous Einstein metrics need not be unique ^{[Jensen 1973]}. The conjecture of Alekseevskii, that homogeneous Einstein manifolds of negative scalar curvature are diffeomorphic to Euclidean space, framed the noncompact theory for decades and was resolved affirmatively by Christoph Böhm and Ramiro Lafuente, completing the structural classification ^{[Bohm-Lafuente 2023]}. Philosophically the subject realises a recurring dream of geometry — that imposing enough symmetry turns an analytic existence problem into a finite, algebraic, and often completely solvable one — while the non-existence and multiplicity results show that even under maximal symmetry the answer can be intricate.

Bibliography Master

@book{Besse1987Einstein,
  author    = {Besse, Arthur L.},
  title     = {Einstein Manifolds},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {10},
  publisher = {Springer-Verlag},
  year      = {1987},
  note      = {Chapter 7: homogeneous Einstein metrics}
}

@article{WangZiller1986,
  author  = {Wang, McKenzie Y. and Ziller, Wolfgang},
  title   = {Existence and non-existence of homogeneous {E}instein metrics},
  journal = {Inventiones Mathematicae},
  volume  = {84},
  pages   = {177--194},
  year    = {1986}
}

@article{Wolf1968Isotropy,
  author  = {Wolf, Joseph A.},
  title   = {The geometry and structure of isotropy irreducible homogeneous spaces},
  journal = {Acta Mathematica},
  volume  = {120},
  pages   = {59--148},
  year    = {1968}
}

@article{Jensen1973,
  author  = {Jensen, Gary R.},
  title   = {Einstein metrics on principal fibre bundles},
  journal = {Journal of Differential Geometry},
  volume  = {8},
  pages   = {599--614},
  year    = {1973}
}

@article{BohmLafuente2023,
  author  = {B{\"o}hm, Christoph and Lafuente, Ramiro},
  title   = {Homogeneous {E}instein metrics and the {A}lekseevskii conjecture},
  journal = {Acta Mathematica},
  year    = {2023}
}

@article{BohmWangZiller2004,
  author  = {B{\"o}hm, Christoph and Wang, McKenzie and Ziller, Wolfgang},
  title   = {A variational approach for compact homogeneous {E}instein manifolds},
  journal = {Geometric and Functional Analysis (GAFA)},
  volume  = {14},
  pages   = {681--733},
  year    = {2004}
}

Prerequisites

07.04.07
03.02.05
03.02.36

Tier anchors

beginner: Besse Einstein Manifolds Ch. 7 informal; the picture of a metric that looks the same at every point
intermediate: Besse Einstein Manifolds Ch. 7 §§7.13--7.50; Wang-Ziller 1986 Invent. Math. 84; Cheeger-Ebin Ch. 3 (homogeneous spaces)
master: Besse Einstein Manifolds Ch. 7; Wang-Ziller 1986 Invent. Math. 84:177; Wolf 1968 Acta Math. 120 (isotropy-irreducible); Jensen 1973 Duke Math. J. 42; Bohm-Wang-Ziller 2004 GAFA 14; Boehm-Lafuente 2023 (Alekseevsky conjecture)

References

Besse — Einstein Manifolds (1987) · Ch. 7, §§7.13--7.118 (homogeneous Einstein metrics, the scalar-curvature functional, naturally reductive examples)
Wang, McKenzie Y. and Ziller, Wolfgang — Existence and non-existence of homogeneous Einstein metrics (1986) · Inventiones Mathematicae 84, pp. 177--194
Wolf, Joseph A. — The geometry and structure of isotropy irreducible homogeneous spaces (1968) · Acta Mathematica 120, pp. 59--148
Jensen, Gary R. — Einstein metrics on principal fibre bundles (1973) · Journal of Differential Geometry 8, pp. 599--614
Bohm, Christoph and Lafuente, Ramiro — Homogeneous Einstein metrics and the Alekseevskii conjecture (2023) · Acta Mathematica (Alekseevskii conjecture resolution)

Estimated time

beginner: 15m
intermediate: 40m
master: 78m