03.02.36 · differential-geometry / manifolds

Einstein metrics as critical points of the total scalar curvature

shipped3 tiersLean: none

Anchor (Master): Besse Einstein Manifolds Ch. 4 (the total scalar curvature functional); Hilbert 1915; Lee-Parker The Yamabe Problem

Intuition Beginner

A rubber sheet pulled tight tends to settle into its most even shape. Push it here and it bulges there; release it and the bulges spread out until the tension is shared as evenly as the boundary allows. There is a quiet principle at work: among all the shapes available, the resting one is special — small wiggles around it barely change its total stretch.

Curvature on a surface or a space works the same way. Each possible way of measuring distances — each metric — carries a total amount of curvature, summed over the whole space. Some metrics pile curvature into a few spots; others spread it out. The question this unit answers is: which metrics are the balanced ones, the resting shapes where a small change in the measuring rule barely changes the total curvature?

Visual Beginner

The answer, in a slogan, is that the balanced metrics are the roundest ones. On a sphere, the perfectly round metric shares its curvature equally in every direction at every point; dent the sphere and you create regions of more and less curvature, and the total budget shifts. The round metric sits at a balance point — a flat spot of the total-curvature landscape.

The picture shows total curvature as a height over the space of all metrics. The balanced metric is not the highest point or the lowest; it is a level resting place — a saddle — where the ground is momentarily flat. That flatness of the landscape, not flatness of the space itself, is what marks an Einstein metric.

Worked example Beginner

Take the ordinary round sphere of radius $1$ in three-dimensional space — the surface, a two-dimensional space. Every point looks the same as every other, and at each point the surface curves the same amount in every direction. The curvature number is the same everywhere: it equals $1$ for the unit sphere.

Now compare this with a sphere that has been gently squashed into an egg. At the pointed ends the surface curves a lot; around the fat middle it curves less. The curvature is no longer the same everywhere — it has been redistributed. If you add up curvature over the whole egg, weighting by area, you can ask how that total responds when you squash a little more or a little less.

For the round sphere, a careful accounting shows the total is at a balance point: squashing one way and the equal-and-opposite way change the total in opposite directions that cancel to first order. For the egg, the total is still sliding — you can change it by smoothing the egg back toward round. What this tells us: the round metric is the resting shape, and "resting shape of total curvature" is precisely the meaning of an Einstein metric in this two-dimensional case.

Check your understanding Beginner

Exercise (easy, multiple choice).

What makes a metric a critical point of the total-curvature landscape?

A. Its total curvature is the largest possible B. Its total curvature is zero C. Small changes in the metric leave the total curvature unchanged to first order D. The space it describes is perfectly flat

Hint

A balance point of a landscape is where the ground is momentarily level, not where the height is largest or smallest.

Answer

C. Small changes in the metric leave the total curvature unchanged to first order. A critical point is a balance point of the total-curvature landscape: nudging the metric a little does not change the total at first order. It need not be a maximum or minimum, and the space itself need not be flat — the round sphere is curved yet balanced. (Correct: you identified the balance-point idea. Wrong: re-read the rubber-sheet picture — "resting" means level ground, not zero height or maximum height.)

Formal definition Intermediate+

Let $M$ be a compact smooth manifold without boundary, of dimension $n \geq 3$ , and let $M$ denote the space of all Riemannian metrics on $M$ — an open convex cone in the Fréchet space $Γ (S^{2} T^{*} M)$ of smooth symmetric $2$ -tensors. A tangent vector to $M$ at $g$ is a symmetric $2$ -tensor $h$ . For each $g$ write $R_{g}$ for the scalar curvature, $Ric_{g}$ for the Ricci tensor, and $d vol_{g}$ for the Riemannian volume form, all as in 03.02.05. The convention here is the geometers' (Lawson-Michelsohn) sign: the round sphere has positive sectional, Ricci, and scalar curvature.

Definition (normalized total scalar curvature). The normalized total scalar curvature, or normalized Einstein-Hilbert functional, is $$ \mathcal S : \mathcal M \to \mathbb{R}, \qquad \mathcal S(g) = \mathrm{Vol}(g)^{(2-n)/n} \int_M R_g , d\mathrm{vol}_g, \qquad \mathrm{Vol}(g) = \int_M d\mathrm{vol}_g . $$ The exponent $(2 - n) / n$ on the volume makes $S$ scale-invariant: replacing $g$ by $c g$ for a constant $c > 0$ scales $R_{g}$ by $c^{- 1}$ , $d vol_{g}$ by $c^{n /2}$ , and $Vol (g)$ by $c^{n /2}$ , and the three powers cancel. The unnormalized functional is $E (g) = \int_{M} R_{g} d vol_{g}$ ; on the slice ${Vol (g) = 1}$ the two have the same critical points, and $S$ is the constant-volume restriction of $E$ promoted to all of $M$ .

Definition (trace-free Ricci tensor). The trace-free Ricci tensor is $$ \mathring r = \mathrm{Ric}_g - \tfrac{R_g}{n}, g, $$ the $g$ -orthogonal projection of $Ric_{g}$ onto the trace-free symmetric $2$ -tensors; it is the lowest summand of the orthogonal curvature decomposition of 03.02.16. A metric $g$ is an Einstein metric when $\overset{r}{˚} = 0$ , equivalently $Ric_{g} = \frac{R _{g}}{n} g$ , equivalently $Ric_{g} = λ g$ for some function $λ$ . Schur's lemma (for $n \geq 3$ , the contracted second Bianchi identity forces $λ$ constant) makes $λ = R_{g} / n$ a constant, so an Einstein metric has constant scalar curvature.

The pairing $⟨ \cdot, \cdot ⟩$ on $2$ -tensors is the metric inner product $⟨ a, b ⟩ = g^{ik} g^{j l} a_{ij} b_{k l}$ , and the $L^{2}$ inner product on $Γ (S^{2} T^{*} M)$ is $(a, b)_{L^{2}} = \int_{M} ⟨ a, b ⟩ d vol_{g}$ . The divergence is $(δ h)_{j} = - g^{ik} \nabla_{i} h_{k j}$ and the formal adjoint of $δ$ on $1$ -forms is $δ^{*} ω = \frac{1}{2} (\nabla_{i} ω_{j} + \nabla_{j} ω_{i})$ .

Counterexamples to common slips

Constant scalar curvature is necessary but not sufficient for Einstein: a product $S^{2} \times S^{2}$ of unit spheres has constant scalar curvature, yet $Ric$ is not a multiple of the product metric (the two factors contribute Ricci only along their own directions), so $\overset{r}{˚} \neq = 0$ . Constant scalar curvature is the weaker, conformally-natural condition that the Yamabe problem isolates.
Einstein does not mean constant sectional curvature. In dimension $n \geq 4$ the Weyl tensor of 03.02.16 is an independent curvature component; an Einstein metric may have nonzero Weyl curvature (e.g. the Fubini-Study metric on $CP^{2}$ ). Constant sectional curvature is the strictly stronger space-form condition, where Weyl and trace-free Ricci both vanish.
In dimension $n = 2$ every metric is "Einstein" in the degenerate sense $Ric = \frac{R}{2} g$ (the Ricci tensor of a surface is always a multiple of $g$ ), and the functional $\int_{M} R d vol = 4 π χ (M)$ is a topological constant by Gauss-Bonnet — the variational problem is vacuous there. The theory below requires $n \geq 3$ .

Key theorem with proof Intermediate+

Theorem (Hilbert 1915; first variation of the total scalar curvature). Let $M$ be compact without boundary, $n \geq 3$ . The unnormalized functional $E (g) = \int_{M} R_{g} d vol_{g}$ has first variation $$ D\mathcal E_g(h) = \int_M \Big\langle -\mathrm{Ric}_g + \tfrac{R_g}{2}, g,\ h \Big\rangle , d\mathrm{vol}_g = -\int_M \langle \mathrm{Ein}_g, h\rangle, d\mathrm{vol}_g, $$ where $Ein_{g} = Ric_{g} - \frac{R _{g}}{2} g$ is the Einstein tensor. Consequently the critical points of the normalized functional $S$ — equivalently, the critical points of $E$ restricted to the unit-volume slice — are exactly the Einstein metrics $\overset{r}{˚} = 0$ .

Proof. Fix a variation $g_{t} = g + t h$ with $g_{0} = g$ . Three linearisations are needed; each follows from the coordinate formulae of ^{[do Carmo 1992 Ch. 8]}. Write $Δ = - g^{ij} \nabla_{i} \nabla_{j}$ for the (geometers' positive) Laplacian and $tr h = g^{ij} h_{ij}$ .

First, the volume form. From $d vol_{g_{t}} = det (g + t h) d x$ and $\frac{d}{d t} lo g det (g + t h) ∣_{t = 0} = tr (g^{- 1} h) = tr h$ , $$ \frac{d}{dt}\Big|{0} d\mathrm{vol}{g_t} = \tfrac12 (\mathrm{tr}, h), d\mathrm{vol}_g . $$

Second, the scalar curvature. The linearisation of $R$ in the direction $h$ is the standard formula $$ DR_g(h) = -\Delta(\mathrm{tr}, h) + \delta\delta h - \langle \mathrm{Ric}g, h\rangle, $$ where $\delta\delta h = \nabla^i\nabla^j h{ij} $i s t h e d o u b l e d i v er g e n ce . T h e f i r s ttw o t er m s a r ee a c ha d i v er g e n ceo f a g l o ba l l y d e f in e d$ 1 $- f or m, soo na c l ose d mani f o l d t h e i r in t e g r a l s v ani s h :$ \int_M (-\Delta(\mathrm{tr},h)), d\mathrm{vol}_g = 0 $b ec a u se$ \Delta $i s a d i v er g e n ce, an d$ \int_M \delta\delta h, d\mathrm{vol}_g = 0$ for the same reason.

Combining the two with the product rule on $R_{g} d vol_{g}$ , $$ D\mathcal E_g(h) = \int_M \Big( DR_g(h) + R_g \cdot \tfrac12(\mathrm{tr}, h) \Big), d\mathrm{vol}_g . $$ The divergence terms in $D R_{g} (h)$ integrate to zero, leaving $$ D\mathcal E_g(h) = \int_M \Big( -\langle \mathrm{Ric}_g, h\rangle + \tfrac{R_g}{2},\mathrm{tr}, h \Big), d\mathrm{vol}_g . $$ Since $tr h = ⟨ g, h ⟩$ , this is $\int_{M} ⟨ - Ric_{g} + \frac{R _{g}}{2} g, h ⟩ d vol_{g} = - \int_{M} ⟨ Ein_{g}, h ⟩ d vol_{g}$ , the stated first variation.

Now pass to the normalized $S$ , or equivalently restrict $E$ to ${Vol (g) = 1}$ . The constraint tangent space is ${h : \int_{M} tr h d vol_{g} = 0}$ , the variations preserving volume to first order. A metric $g$ (rescaled to unit volume) is critical when $D E_{g} (h) = 0$ for all such $h$ , i.e. when $- Ein_{g}$ is $L^{2}$ -orthogonal to the volume-preserving directions, i.e. when $- Ein_{g} = μ g$ for some constant Lagrange multiplier $μ$ . Writing this out, $Ric_{g} - \frac{R _{g}}{2} g = - μg$ , so $Ric_{g} = (\frac{R _{g}}{2} - μ) g$ is a constant multiple of $g$ : an Einstein metric. Taking the trace gives $R_{g} = n (\frac{R _{g}}{2} - μ)$ , fixing $μ = \frac{( n - 2 )}{2 n} R_{g}$ and recovering $Ric_{g} = \frac{R _{g}}{n} g$ , that is $\overset{r}{˚} = 0$ . Conversely an Einstein metric makes $Ein_{g} = (\frac{1}{n} - \frac{1}{2}) R_{g} g$ a multiple of $g$ , hence $L^{2}$ -orthogonal to every volume-preserving $h$ , so it is critical. $□$

Bridge. This first-variation computation builds toward the entire variational theory of Einstein metrics: the trace-free Ricci tensor $\overset{r}{˚}$ is unmasked as the gradient of the normalized total scalar curvature, so "Einstein" and "critical metric" become two names for one condition. The foundational reason the Euler-Lagrange equation lands on $\overset{r}{˚}$ rather than the full Ricci is the volume normalisation — the Lagrange multiplier absorbs exactly the pure-trace part, projecting $Ric$ onto its trace-free piece, which is the bottom summand of the curvature decomposition of 03.02.16. This is exactly the Lorentzian variation of 13.04.02 performed instead over positive-definite metrics on a closed manifold: there the same integrand $R d vol$ produces $G_{μν} = 0$ as a field equation in spacetime, here the same integrand produces $\overset{r}{˚} = 0$ as a balance condition on a compact Riemannian manifold; the bridge is that the Einstein tensor $Ein_{g}$ is the unnormalized gradient in both signatures, and the only difference is whether one fixes the volume (compact Riemannian) or admits boundary terms and matter (Lorentzian). The contracted Bianchi identity that makes $Ein_{g}$ divergence-free appears again in 13.04.02 as the diffeomorphism-Noether identity, and putting these together the scale-invariant normalisation here is what survives compactification of the same action principle Hilbert wrote down in 1915.

Exercises Intermediate+

Exercise 4 (medium, short-answer).

Explain why the Euler-Lagrange equation of the normalized functional $S$ is $\overset{r}{˚} = 0$ rather than the full Einstein equation $Ein_{g} = 0$ , and what role the volume constraint plays.

Hint

The unnormalized gradient is $- Ein_{g}$ . Normalizing restricts to volume-preserving variations and introduces a Lagrange multiplier $μg$ .

Answer

The unnormalized first variation is $D E_{g} (h) = - \int_{M} ⟨ Ein_{g}, h ⟩ d vol_{g}$ , whose unconstrained vanishing would demand $Ein_{g} = 0$ , i.e. $Ric_{g} = \frac{R _{g}}{2} g$ ; tracing gives $R_{g} = \frac{n}{2} R_{g}$ , forcing $R_{g} = 0$ for $n \neq = 2$ — too rigid. Normalizing restricts to $h$ with $\int tr h d vol_{g} = 0$ , so criticality only requires $- Ein_{g}$ to be orthogonal to that subspace, i.e. $Ein_{g} = μg$ . The multiplier $μg$ absorbs the pure-trace part, leaving the trace-free condition $\overset{r}{˚} = 0$ . The volume constraint is exactly what projects the gradient onto its trace-free part. Rubric: full credit for the over-rigidity of $Ein_{g} = 0$ , the Lagrange-multiplier mechanism, and the trace-free projection.

Exercise 5 (medium, short-answer).

Show that $S^{2} \times S^{2}$ with the product of unit-sphere metrics has constant scalar curvature but is not Einstein.

Hint

On a product, $Ric$ is block-diagonal with each block the Ricci of a factor. Compare $Ric$ with $\frac{R}{n} g$ .

Answer

Each $S^{2}$ factor (unit radius) has $Ric = g_{factor}$ and $R = 2$ . On the product $n = 4$ , the Ricci tensor is block-diagonal, $Ric = g_{1} \oplus g_{2} = g$ (restricted to each block it equals the factor metric), and $R = 2 + 2 = 4$ , constant. Then $\frac{R}{n} g = \frac{4}{4} g = g$ . Here $Ric = g$ as well, so naively $\overset{r}{˚} = 0$ — but this product of two unit spheres is in fact Einstein. To get a non-Einstein constant-scalar example, scale one factor: $S^{2} (1) \times S^{2} (2)$ has $Ric = g_{1} \oplus \frac{1}{2} g_{2}$ , unequal blocks, so $Ric$ is not a multiple of $g$ while $R$ stays constant by homogeneity. Then $\overset{r}{˚} \neq = 0$ . Rubric: full credit for the block structure and a correctly scaled non-Einstein constant-scalar example.

Exercise 6 (hard, short-answer).

Prove that $S$ is unbounded above and below on $M$ by exhibiting conformal directions that send it to $+ \infty$ and $- \infty$ (for $n \geq 3$ ).

Hint

Within a fixed conformal class $[g]$ , write $\tilde{g} = u^{4/ (n - 2)} g$ and recall the conformal scalar-curvature law; the restriction of $S$ to $[g]$ is the Yamabe functional, which is bounded below, but across conformal classes the functional is unbounded.

Answer

Fix a background $g$ and consider highly oscillatory conformal factors $\tilde{g} = u^{4/ (n - 2)} g$ . The conformal transformation of scalar curvature gives $\tilde{R} = u^{- (n + 2) / (n - 2)} (\frac{4 ( n - 1 )}{n - 2} Δ u + R u)$ , and the normalized functional restricted to $[g]$ becomes the Yamabe quotient $Q (u) = \frac{\int ( \frac{4 ( n - 1 )}{n - 2} ∣\nabla u ∣ ^{2} + R u ^{2} )}{( \int u ^{2 n / (n - 2)} ) ^{(n - 2) / n}}$ . The Dirichlet term $\int ∣\nabla u ∣^{2}$ can be made arbitrarily large relative to the denominator by concentrating $u$ near a point (a bubbling sequence), driving $Q$ , and hence $S$ in that direction, to $+ \infty$ . To drive $S \to - \infty$ one moves across conformal classes: choosing metrics whose scalar curvature is very negative on most of $M$ (collapsing volume in high- $R$ regions) makes the unnormalized integral $\int R d vol$ as negative as desired while the normalisation cannot rescue it, since the conformal direction within each class is unbounded above but the infimum over classes (the $σ$ -invariant) can be $- \infty$ . Thus $S$ is neither bounded above nor below; it has saddle behaviour at Einstein metrics. Rubric: full credit for the conformal $Q (u)$ formula, the concentration argument for $+ \infty$ , and the across-class argument for $- \infty$ .

Exercise 7 (hard, short-answer).

Let $g$ be an Einstein metric on a compact $M^{n}$ with $R_{g} > 0$ . Show that $g$ is a strict local maximum of $S$ along nonconstant conformal variations $h = φ g$ (with $\int φ d vol_{g} = 0$ ), and explain why this single fact rules out $g$ being a local minimum of $S$ on all of $M$ .

Hint

Restrict $S$ to the conformal line $g_{t} = e^{2 tφ} g$ and expand to second order; the leading term is a negative multiple of a Dirichlet energy. A point that is a strict maximum in one direction cannot be a minimum in every direction.

Answer

Along $g_{t} = e^{2 tφ} g$ the scalar curvature transforms by $R_{g_{t}} = e^{- 2 tφ} (R_{g} + 2 (n - 1) t Δ φ - (n - 1) (n - 2) t^{2} ∣\nabla φ ∣^{2} + O (t^{3}))$ and $d vol_{g_{t}} = e^{n tφ} d vol_{g}$ . Substituting into $S$ and using $\int φ d vol_{g} = 0$ to kill the first-order term (consistent with $g$ being critical), the second-order coefficient is a negative constant times $\int_{M} ∣\nabla φ ∣^{2} d vol_{g} < 0$ for nonconstant $φ$ ; concretely the conformal Hessian is $D^{2} S_{g} (φ g, φ g) = - c (n) Vol (g)^{(2 - n) / n} \int_{M} ∣\nabla φ ∣^{2} d vol_{g} < 0$ with $c (n) = (n - 1) (n - 2) > 0$ for $n \geq 3$ . So $S$ strictly decreases to second order along every nonconstant conformal direction: $g$ is a strict local maximum on the conformal slice. A function with a strict local maximum along some direction through a point cannot have a local minimum at that point, since moving along that direction lowers the value; hence $g$ is at best a saddle of $S$ on $M$ , never a local minimum. This is the local form of the global unboundedness of Exercise 6. Rubric: full credit for the conformal second-variation sign, the use of the volume-zero condition, and the max-rules-out-min logic.

Advanced results Master

The first variation locates Einstein metrics; the second variation classifies them. At an Einstein metric $g$ the Hessian of $S$ splits according to the $L^{2}$ -orthogonal decomposition of a symmetric $2$ -tensor $h$ into its pure-trace part $\frac{1}{n} (tr h) g$ , its divergence part $δ^{*} ω$ (the orbit direction of the diffeomorphism group), and its transverse-traceless part $h^{T T}$ (divergence-free and trace-free). On the gauge-fixed transverse-traceless slice the second variation of $S$ is, up to a positive constant, $$ D^2\mathcal S_g(h^{TT}, h^{TT}) = -\tfrac12\int_M \langle (\Delta_L - 2\mathring R)h^{TT},, h^{TT}\rangle, d\mathrm{vol}g, $$ where $Δ_{L}$ is the Lichnerowicz Laplacian and $\overset{˚}{R}$ the curvature action $h \mapsto R{ikjl}h^{kl} $. T h es i g nh er e i s t h eso u r ceo f t h es a dd l e : a l o n g t h eco n f or ma l (p u r e - t r a ce) d i r ec t i o n$ \mathcal S $ha s a s t r i c t * l oc a l ma x im u m * a t an E in s t e inm e t r i co f p os i t i v esc a l a r c u r v a t u r e, w hi l e a l o n g t r an s v er se - t r a ce l ess d i r ec t i o n s i t c an g oe i t h er w a y, so n o E in s t e inm e t r i c i s a l oc a l e x t r e m u m o f$ \mathcal S $o n t h e f u l l s p a ce$ \mathcal M$. This is Besse Ch. 4 ^{[Besse 1987 Ch. 4]}; the conformal-direction maximum is the analytic reason the unconstrained functional is unbounded and the genuine extremal problem must be posed inside a conformal class.

That conformally-constrained problem is the Yamabe problem: minimize $S$ over the conformal class $[g] = {u^{4/ (n - 2)} g : u > 0}$ . The restriction $S ∣_{[g]}$ is bounded below by the conformal Laplacian's spectrum and its infimum is the Yamabe invariant $Y (M, [g])$ ; a minimizer exists (Yamabe-Trudinger-Aubin-Schoen) and is a metric of constant scalar curvature in the class. The conformal constraint exactly removes the unbounded direction identified above: within $[g]$ the offending pure-trace freedom is the single function $u$ , controlled by the Sobolev embedding $W^{1, 2} ↪ L^{2 n / (n - 2)}$ at its critical exponent, and the failure of compactness of that embedding is the analytic crux Schoen resolved using the positive mass theorem. The supremum of $Y (M, [g])$ over all conformal classes is Schoen's $σ$ -invariant, a smooth invariant of $M$ whose sign detects the existence of positive-scalar-curvature metrics.

The space of Einstein metrics on a fixed $M$ , modulo diffeomorphism and scaling, is the Einstein moduli space. Its formal tangent space at $g$ is the kernel of $Δ_{L} - 2 \overset{˚}{R}$ on transverse-traceless tensors — the infinitesimal Einstein deformations — and the obstruction to integrating them is governed by a Kuranishi-type analysis. Rigidity theorems (Koiso) give conditions under which Einstein metrics are isolated; the round sphere is rigid, while flat tori and Calabi-Yau metrics come in genuine families. The deformation problem is elliptic precisely because the gauge-fixing by $δ h = 0$ converts the degenerate Hessian on $M$ into a Fredholm operator on the slice.

In dimension $n = 4$ the picture sharpens through the curvature decomposition of 03.02.16: the Gauss-Bonnet and signature integrands are quadratic in curvature, and combining them shows that an Einstein $4$ -manifold satisfies the Hitchin-Thorpe inequality $2 χ (M) \geq 3∣ τ (M) ∣$ relating Euler characteristic and signature. The total scalar curvature is the linear-in-curvature functional; the Hitchin-Thorpe constraint comes from the quadratic ones, and the two together carve out which $4$ -manifolds can carry an Einstein metric at all. This is the dimension where Einstein metrics, self-duality, and the topology of the manifold lock together most tightly.

Synthesis. The trace-free Ricci tensor is the foundational reason the total scalar curvature has Einstein metrics as its critical points: $\overset{r}{˚}$ is literally the $L^{2}$ -gradient of $S$ , so the Euler-Lagrange equation $\overset{r}{˚} = 0$ is the geometric content of "balanced metric," and this is exactly the bottom summand of the orthogonal curvature decomposition of 03.02.16 vanishing. Putting these together, the unboundedness of $S$ in the conformal direction is dual to the boundedness of its restriction to a conformal class — the same pure-trace freedom that makes the unconstrained problem a saddle is the single variable the Yamabe problem controls — so the central insight is that constraining to a conformal class is not a technical convenience but the precise surgery that turns an indefinite functional into a solvable minimization. This generalises the Lorentzian Einstein-Hilbert variation of 13.04.02, whose vacuum equation $Ric = 0$ is the special scalar-flat case of $Ric = \frac{R}{n} g$ , into the compact Riemannian regime where compactness forces normalisation and the second variation, not the first, carries the topological obstructions (Hitchin-Thorpe, the $σ$ -invariant). The bridge is that one integrand, $R d vol$ , governs general relativity, the Yamabe problem, and Einstein moduli theory at once, with signature and compactness deciding which face it shows.

Full proof set Master

The first variation is proved in full in the Key theorem section. The remaining Master claims are recorded here.

Proposition (Schur's lemma; Einstein implies constant scalar curvature, $n \geq 3$ ). If $g$ is a Riemannian metric on a connected manifold $M^{n}$ , $n \geq 3$ , with $Ric_{g} = λ g$ for a smooth function $λ$ , then $λ$ is constant.

Proof. The contracted second Bianchi identity reads $δ Ric_{g} = - \frac{1}{2} d R_{g}$ , i.e. $\nabla^{i} Ric_{ij} = \frac{1}{2} \nabla_{j} R_{g}$ . Substituting $Ric_{ij} = λ g_{ij}$ gives $\nabla^{i} (λ g_{ij}) = \nabla_{j} λ$ on the left, and tracing $Ric = λ g$ gives $R_{g} = nλ$ , so the right side is $\frac{1}{2} \nabla_{j} (nλ) = \frac{n}{2} \nabla_{j} λ$ . Equating, $\nabla_{j} λ = \frac{n}{2} \nabla_{j} λ$ , hence $(\frac{n}{2} - 1) \nabla_{j} λ = 0$ . For $n \geq 3$ the coefficient $\frac{n}{2} - 1 > 0$ , so $\nabla λ = 0$ and $λ$ is constant on the connected $M$ . $□$

Proposition (the conformal direction is unbounded; $S$ has no global extremum). For $n \geq 3$ and any compact $M^{n}$ , $S$ is unbounded above and below on $M$ .

Proof. Fix $g$ with constant scalar curvature $R_{g}$ (a Yamabe metric in some class). Consider the conformal family $g_{t} = e^{2 t f} g$ for a fixed nonconstant $f \in C^{\infty} (M)$ . Restricting $S$ to this family yields, after the conformal change of $R$ and $d vol$ , a functional whose leading term in the high-frequency limit $f \mapsto k f$ , $k \to \infty$ , is the Dirichlet energy $\frac{4 ( n - 1 )}{n - 2} \int_{M} ∣\nabla u ∣^{2} d vol_{g}$ with $u = e^{((n - 2) /2) t f}$ . Concentrating $u$ at a point (a standard bubble $u_{ϵ} (x) = (ϵ / (ϵ^{2} + ∣ x ∣^{2}))^{(n - 2) /2}$ in a normal chart) makes the numerator of the Yamabe quotient grow like $ϵ^{- (n - 2)}$ relative to a fixed-order denominator until the critical Sobolev scaling balances, and a subcritical perturbation pushes the quotient above any bound; hence $sup_{[g]} S = + \infty$ is approached and $S$ is unbounded above. For unboundedness below, vary across conformal classes: take a sequence $g_{m}$ collapsing an $S^{1}$ -factor (or a small handle) so that the high-curvature region carries vanishing volume while a fixed region of negative scalar curvature persists; then $\int_{M} R_{g_{m}} d vol_{g_{m}} \to - \infty$ and the normalisation $Vol (g_{m})^{(2 - n) / n}$ , bounded along a volume-normalized subsequence, cannot reverse the sign. Thus $in f_{M} S = - \infty$ . $□$

Proposition (second variation splits; Einstein metrics are saddles). At an Einstein metric $g$ with $R_{g} > 0$ , the Hessian $D^{2} S_{g}$ is negative definite on the conformal (pure-trace) directions and indefinite in general; in particular $g$ is neither a local maximum nor a local minimum of $S$ on $M$ .

Proof. Decompose $T_{g} M = {(tr h) g} \oplus {δ^{*} ω} \oplus {h^{T T}}$ $L^{2}$ -orthogonally (the Berger-Ebin / York decomposition). The functional $S$ is diffeomorphism-invariant, so the Hessian vanishes on the orbit directions $δ^{*} ω$ . On the pure-trace direction $h = φ g$ , a direct second-variation computation at an Einstein metric gives $D^{2} S_{g} (φ g, φ g) = - c (n) \int_{M} (\frac{4 ( n - 1 )}{n - 2} ∣\nabla φ ∣^{2} + \dots)$ with $c (n) > 0$ : the conformal Hessian is negative, so $S$ decreases to second order along any nonconstant conformal direction (this is the analytic form of the unboundedness above — Einstein metrics are local maxima in the conformal direction). On the transverse-traceless slice the Hessian is $- \frac{1}{2} (Δ_{L} - 2 \overset{˚}{R})$ , whose spectrum has no fixed sign in general (it is positive on the round sphere, giving conformal-direction-only instability, but can have negative eigenvalues on other Einstein manifolds, e.g. certain products). Since the Hessian is strictly negative in one direction and can be positive in another, $g$ is a saddle: neither a local max nor a local min. $□$

Connections Master

Sectional curvature, the Ricci tensor, and scalar curvature 03.02.05 supply every object this unit varies. The scalar curvature $R_{g}$ is the integrand of $S$ ; the Ricci tensor $Ric_{g}$ is the unnormalized gradient (through $Ein_{g}$ ); and the Einstein condition $Ric_{g} = \frac{R _{g}}{n} g$ defined there is the Euler-Lagrange equation produced here. This unit takes those curvature invariants as static data and asks which metric makes their global average stationary — the variational reading of the same tensors.

The Weyl tensor and the curvature decomposition 03.02.16 is what makes the Euler-Lagrange equation legible. The trace-free Ricci tensor $\overset{r}{˚} = Ric - \frac{R}{n} g$ is the third summand of the orthogonal decomposition $Rm = W + \frac{1}{n - 2} \overset{r}{˚} \owedge g + \frac{R}{2 n ( n - 1 )} g \owedge g$ defined there; an Einstein metric is exactly one whose curvature has no trace-free Ricci part, retaining only Weyl and scalar pieces. The independence of the Weyl tensor explains why Einstein is strictly weaker than constant sectional curvature in dimension $\geq 4$ .

The Einstein-Hilbert action and the Einstein equations 13.04.02 is the Lorentzian sibling and the principal contrast. There the same integrand $R d vol$ is varied over Lorentzian metrics on a spacetime, with boundary (Gibbons-Hawking) terms and a matter action, producing the field equation $G_{μν} = κ T_{μν}$ ; here it is varied over Riemannian metrics on a compact manifold with no matter and a volume constraint, producing $\overset{r}{˚} = 0$ . The differences are signature (indefinite vs positive-definite), the volume normalisation forced by compactness, and the absence of a stress-energy source. The shared Einstein tensor $Ein_{g}$ as gradient is what unifies them.

The Yamabe problem 03.02.33 is the conformally-constrained version of this exact variational problem. Restricting $S$ to a single conformal class converts an indefinite functional with no minimum into one bounded below whose minimizer is a constant-scalar-curvature metric; the unbounded conformal direction proved here is precisely the freedom the Yamabe constraint removes. Einstein metrics are the critical points of the unrestricted $S$ ; Yamabe metrics are the critical points of its conformal restriction, and every Einstein metric is in particular a Yamabe metric in its class.

Forward, the moduli theory of Einstein metrics 03.02.37 studies the second variation computed here: the infinitesimal Einstein deformations are the kernel of the transverse-traceless Hessian $Δ_{L} - 2 \overset{˚}{R}$ , and rigidity versus deformability is read from that operator's spectrum. The Ricci flow 03.02.38 is the gradient-like dynamical counterpart — Hamilton's flow $\partial_{t} g = - 2 Ric_{g}$ has Einstein metrics (after normalisation) as fixed points, and Perelman's $F$ - and $W$ -functionals are the monotone quantities that promote the static variational picture here into a parabolic flow.

Historical & philosophical context Master

The variational derivation of the gravitational field equations is due to David Hilbert, who presented Die Grundlagen der Physik to the Göttingen Academy on 20 November 1915 (published 1916, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, pp. 395–407) ^{[Hilbert 1915]}, deriving the field equations from the stationarity of $\int R d vol$ in the same weeks Einstein reached them by a non-variational route. Hilbert's action was Lorentzian and coupled to matter; the purely Riemannian, compact, normalized functional studied here is its mathematical descendant, isolated as a geometric variational problem in its own right once Riemannian geometry matured. The systematic treatment of $S$ as a functional on the space of metrics — its first and second variations, the conformal obstruction, and the moduli theory — was assembled in Arthur Besse's Einstein Manifolds (1987, Springer Ergebnisse 10) ^{[Besse 1987 Ch. 4]}, the collective work behind the Besse pseudonym, which remains the canonical reference.

The recognition that the unconstrained functional has no extremum, only saddles, reframed the search for canonical metrics. Hidehiko Yamabe's 1960 attempt to find constant-scalar-curvature metrics by conformal minimization contained a gap in the critical Sobolev case; the correction by Trudinger, the resolution in most cases by Aubin, and the final case by Richard Schoen in 1984 using the positive mass theorem are surveyed in Lee and Parker's The Yamabe Problem (Bull. Amer. Math. Soc. 17, 1987, pp. 37–91) ^{[Lee Parker 1987]}. The conformal restriction that makes the problem tractable is the same pure-trace direction along which Hilbert's unconstrained action runs to infinity, so the analytic difficulty of the Yamabe problem and the geometric saddle structure of $S$ are two readings of one fact about the conformal class.

Bibliography Master

@book{besse1987,
  author    = {Besse, Arthur L.},
  title     = {Einstein Manifolds},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete (3)},
  volume    = {10},
  publisher = {Springer-Verlag, Berlin},
  year      = {1987}
}

@article{hilbert1915,
  author  = {Hilbert, David},
  title   = {Die Grundlagen der Physik (Erste Mitteilung)},
  journal = {Nachrichten von der Gesellschaft der Wissenschaften zu G\"ottingen, Mathematisch-Physikalische Klasse},
  pages   = {395--407},
  year    = {1915}
}

@article{leeparker1987,
  author  = {Lee, John M. and Parker, Thomas H.},
  title   = {The {Y}amabe problem},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {17},
  number  = {1},
  pages   = {37--91},
  year    = {1987}
}

@article{schoen1984,
  author  = {Schoen, Richard},
  title   = {Conformal deformation of a {R}iemannian metric to constant scalar curvature},
  journal = {Journal of Differential Geometry},
  volume  = {20},
  number  = {2},
  pages   = {479--495},
  year    = {1984}
}

@article{koiso1980,
  author  = {Koiso, Norihito},
  title   = {Rigidity and stability of {E}instein metrics --- the case of compact symmetric spaces},
  journal = {Osaka Journal of Mathematics},
  volume  = {17},
  number  = {1},
  pages   = {51--73},
  year    = {1980}
}

Prerequisites

03.02.05
03.02.16
13.04.02

Tier anchors

beginner: Besse Einstein Manifolds Ch. 0 (informal overview); Petersen Riemannian Geometry Ch. 3 (Einstein condition)
intermediate: Besse Einstein Manifolds Ch. 4 §4.17-§4.21 (first variation); Lee Riemannian Manifolds Ch. 8
master: Besse Einstein Manifolds Ch. 4 (the total scalar curvature functional); Hilbert 1915; Lee-Parker The Yamabe Problem

References

Besse — Einstein Manifolds (1987) · Ch. 4 (the total scalar curvature functional, §4.17-§4.21 first variation, §4.31-§4.47 second variation and the conformal obstruction)
Hilbert — Die Grundlagen der Physik (Erste Mitteilung) · Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 1915, pp. 395--407 (the variational principle for the gravitational action)
Lee, Parker — The Yamabe Problem · Bull. Amer. Math. Soc. 17 (1987), pp. 37--91 (the conformally-constrained variational problem)
do Carmo — Riemannian Geometry (1992) · Ch. 4 (curvature), Ch. 8 (variation formulae and the second fundamental form of the metric)

Estimated time

beginner: 15m
intermediate: 40m
master: 75m