03.02.38 · differential-geometry / manifolds

Infinitesimal Einstein deformations, the Lichnerowicz Laplacian on 2-tensors, and Koiso rigidity

shipped3 tiersLean: none

Anchor (Master): Besse Einstein Manifolds Ch. 12; Koiso 1980 Osaka J. Math. 17 and 1982 Osaka J. Math. 19 and 1983 Invent. Math. 73; Berger-Ebin 1969 J. Diff. Geom. 3

Intuition Beginner

An Einstein metric is a way of curving a space so the curvature is the same in an averaged sense at every point and in every direction — the geometric version of a perfectly balanced shape. The round sphere is one. So is the flat torus, and so are some far stranger high-dimensional spaces. The question this unit answers is: once you have one such balanced shape, how many genuinely different balanced shapes sit right next to it?

Sometimes the answer is none. You can push and pull on the round sphere all you like, but the only nearby balanced shapes are the same sphere viewed at a different size or from a different angle. Nothing new appears. We call the sphere rigid.

Other times a whole family appears. Certain six-dimensional spaces from string theory, and a famous four-dimensional surface called K3, come in continuous families of balanced shapes, like a dial you can turn. Turning the dial gives you a different Einstein metric every time, none of them just a relabelling of another.

The tool that decides between these two outcomes is a single bending-energy operator. You feed it a tiny proposed change to the metric, and it reports back whether that change keeps the balance to first order. The changes it accepts are the directions the dial could turn.

Visual Beginner

Alt text: Two panels contrast rigid and deformable Einstein shapes. On the left a round sphere is shown with small inward arrows that all bounce back to the same sphere, illustrating that every nearby balanced shape is just a rotation or a change of size, so the sphere admits no new balanced neighbours. On the right a K3 surface is drawn as a complicated handled shape with a control dial; as the dial turns, the shape passes through a continuous family of genuinely different balanced shapes, illustrating a positive-dimensional moduli space. The picture encodes the central dichotomy: some Einstein metrics are isolated points, others belong to whole families.

Worked example Beginner

Take the round unit sphere in three dimensions and ask: which tiny changes to its metric keep it balanced to first order?

First throw out the changes that do not count. Rescaling the whole sphere — making it slightly bigger everywhere — keeps it balanced but is the same shape at a new size, so it does not count as new. Sliding the sphere around by a smooth motion of the points also changes the metric on paper while leaving the actual shape untouched, so those do not count either. What is left after removing rescalings and motions is the set of honestly new directions.

For the round sphere, that leftover set is empty. Every remaining candidate change, when fed to the balance-checking operator, is rejected: it would break the even curvature. So the only way to stay balanced near the round sphere is to rescale or to move, and both of those give you the sphere back.

What this tells us: the round sphere has no new nearby balanced shapes. It is an isolated point in the catalogue of balanced shapes. The same verdict holds for the standard metric on complex projective space. By contrast, when you run the identical check on K3, the leftover set is not empty — there are honest new directions, and the catalogue near K3 is a whole region rather than a lone point.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $(M, g)$ is a closed (compact, no boundary) Riemannian manifold of dimension $n$ , with Levi-Civita connection $\nabla$ , Riemann curvature tensor $R$ , and Ricci tensor $Ric$ . The sign convention is that of unit 03.02.05 and Besse: $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ , so that the round sphere has positive sectional curvature; the rough Laplacian is $\nabla^{*} \nabla = - tr_{g} \nabla^{2}$ , a non-negative operator, matching unit 03.02.15. The metric $g$ is Einstein if $Ric = λ g$ for a constant $λ$ , the Einstein constant. We work on the bundle $S^{2} T^{*} M$ of symmetric $2$ -tensors, with the pointwise inner product induced by $g$ and the $L^{2}$ inner product $⟨ h, k ⟩_{L^{2}} = \int_{M} ⟨ h, k ⟩ d V_{g}$ .

Definition (curvature action on symmetric $2$ -tensors). The Riemann tensor acts on $S^{2} T^{*} M$ by the symmetric endomorphism $\overset{˚}{R}$ , given in a local orthonormal frame $(e_{i})$ by $$ (\mathring R, h){ij} = \sum{k,l} R_{ikjl}, h_{kl}. $$ This is the natural extension of the curvature operator to the second symmetric power; it is self-adjoint with respect to the pointwise inner product.

Definition (Lichnerowicz Laplacian). The Lichnerowicz Laplacian on $S^{2} T^{*} M$ is the second-order operator $$ \Delta_L h = \nabla^\nabla h - 2,\mathring R, h + \mathrm{Ric}\cdot h + h\cdot\mathrm{Ric}, $$ where $(Ric \cdot h)_{ij} = \sum_{k} Ric_{ik} h_{k j}$ and $(h \cdot Ric)_{ij} = \sum_{k} h_{ik} Ric_{k j}$ . On an Einstein manifold with $Ric = λ g$ the last two terms collapse to $2 λh$ , so $$ \Delta_L h = \nabla^\nabla h - 2,\mathring R, h + 2\lambda h. $$ This operator is self-adjoint and elliptic, with the same principal symbol $∣ ξ ∣^{2} id$ as the rough Laplacian. It is not the Hodge Laplacian: although $Δ_{L}$ agrees with $Δ_{Hodge}$ on functions and on $1$ -forms (where it is the Weitzenböck-corrected Hodge operator), on symmetric $2$ -tensors the two differ by curvature terms that have no Hodge counterpart, because $S^{2} T^{*} M$ is not a summand of the exterior algebra. It is also distinct from the spinor Lichnerowicz operator $\neq \partial^{2} = \nabla^{*} \nabla + Scal /4$ of unit 03.09.08: that operator acts on sections of the spinor bundle and carries only the scalar curvature, whereas $Δ_{L}$ acts on symmetric $2$ -tensors and carries the full Riemann tensor through $\overset{˚}{R}$ . Both are named for Lichnerowicz; they are different operators on different bundles ^{[Besse Ch. 12]}.

Definition (Berger-Ebin splitting). Let $δ : S^{2} T^{*} M \to Ω^{1} (M)$ be the divergence, $(δ h)_{i} = - \sum_{j} \nabla_{j} h_{ij}$ , with formal adjoint the symmetrised covariant derivative $δ^{*} : Ω^{1} (M) \to S^{2} T^{*} M$ , $(δ^{*} ω)_{ij} = \frac{1}{2} (\nabla_{i} ω_{j} + \nabla_{j} ω_{i})$ . On a closed manifold there is an $L^{2}$ -orthogonal splitting ^{[Berger-Ebin 1969]} $$ S^2 T^M = \big(C^\infty(M)\cdot g\big)\ \oplus\ \delta^\big(\Omega^1(M)\big)\ \oplus\ \mathrm{TT}, $$ where the first summand is the pure-trace (conformal) part, the second is the image of the Lie-derivative directions $δ^{*} ω = \frac{1}{2} L_{ω^{♯}} g$ generated by infinitesimal diffeomorphisms, and $TT = ker δ \cap ker tr_{g}$ is the space of transverse-traceless tensors. The first two summands are the gauge directions: a trace change rescales, and a Lie derivative is the infinitesimal action of the diffeomorphism group, so neither alters the underlying geometry.

Definition (space of infinitesimal Einstein deformations). For an Einstein metric $g$ with constant $λ$ , the space of infinitesimal Einstein deformations is $$ \varepsilon(g) = \big{, h \in \mathrm{TT} \ :\ \Delta_L h = 2\lambda h ,\big}. $$ A metric is infinitesimally rigid (or, in Koiso's terminology, has no infinitesimal deformations) if $ε (g) = 0$ , and infinitesimally deformable otherwise. Because $Δ_{L}$ is elliptic on a closed manifold, $ε (g)$ is finite-dimensional. The elements of $ε (g)$ are the candidate tangent directions to the moduli space of Einstein metrics; the eigenvalue equation $Δ_{L} h = 2 λh$ is exactly the linearisation of the Einstein condition restricted to the gauge-fixed slice TT.

Counterexamples to common slips

$Δ_{L}$ is not the Hodge Laplacian on $2$ -tensors. The two agree on functions and $1$ -forms but split on $S^{2} T^{*} M$ ; conflating them drops the $\overset{˚}{R}$ curvature term and gives the wrong spectrum. The defining equation $Δ_{L} h = 2 λh$ becomes meaningless if $Δ_{L}$ is replaced by $Δ_{Hodge}$ .
$Δ_{L}$ is not the spinor Lichnerowicz operator. Unit 03.09.08's operator $\nabla^{*} \nabla + Scal /4$ acts on spinors and carries only scalar curvature; the present $Δ_{L}$ carries the full Riemann tensor. Same name, different bundle, different curvature content.
Forgetting to restrict to TT. The equation $Δ_{L} h = 2 λh$ has many solutions coming from trace and Lie-derivative directions; only the transverse-traceless solutions are genuine geometric deformations. Counting all eigenvectors overcounts by the gauge directions.

Key theorem with proof Intermediate+

Theorem (linearised Einstein operator on the gauge slice). Let $(M^{n}, g)$ be a closed Einstein manifold with $Ric = λ g$ . The kernel of the linearisation of the Einstein equation, restricted to the transverse-traceless slice TT, is exactly $ε (g) = {h \in TT : Δ_{L} h = 2 λh}$ . Consequently $ε (g)$ is finite-dimensional, and it is the candidate tangent space to the space of Einstein metrics at $g$ modulo rescaling and diffeomorphism.

Proof. Write $Ric (g)$ for the Ricci tensor as a map on metrics, and let $g_{t} = g + t h$ with $h \in S^{2} T^{*} M$ . The linearisation of the Ricci operator at $g$ is the classical formula ^{[Besse Ch. 1]} $$ \mathrm{Ric}'_g(h) = \tfrac12 \Delta_L h - \delta^* \delta h - \tfrac12 \nabla d(\operatorname{tr}_g h), $$ where $Δ_{L}$ is the Lichnerowicz Laplacian, $δ$ the divergence, $δ^{*}$ its adjoint, and $\nabla d$ the Hessian acting on the function $tr_{g} h$ . The Einstein condition $Ric (g_{t}) = λ (t) g_{t}$ linearises, after differentiating in $t$ at $t = 0$ and writing $λ^{'} (0) = μ$ , to $$ \mathrm{Ric}'_g(h) = \mu, g + \lambda h. $$

Now impose the gauge $h \in TT$ , so $tr_{g} h = 0$ and $δ h = 0$ . The last two terms of the linearised Ricci formula vanish: $δ^{*} δ h = 0$ because $δ h = 0$ , and $\nabla d (tr_{g} h) = 0$ because $tr_{g} h = 0$ . The linearised equation reduces to $$ \tfrac12 \Delta_L h = \mu, g + \lambda h. $$ Take the $g$ -trace of both sides. Since $h$ is traceless and $Δ_{L}$ preserves the trace decomposition (it commutes with $tr_{g}$ on an Einstein background), $tr_{g} (Δ_{L} h) = Δ (tr_{g} h) = 0$ , while $tr_{g} (μg + λh) = n μ$ . Therefore $μ = 0$ . The equation becomes $$ \Delta_L h = 2\lambda h, $$ which is the defining equation of $ε (g)$ . Conversely any $h \in TT$ solving $Δ_{L} h = 2 λh$ satisfies the linearised Einstein equation with $μ = 0$ . Because $Δ_{L}$ is a self-adjoint elliptic operator on the closed manifold $M$ and $2 λ$ is a fixed real number, the eigenspace is finite-dimensional. The gauge slice TT is, by the Berger-Ebin splitting, a complement to the rescaling and diffeomorphism directions, so $ε (g)$ is the candidate tangent to the moduli space at $g$ . $■$

Bridge. This computation builds toward the global moduli theory: the finite-dimensional eigenspace $ε (g)$ is the linear shadow whose second-order obstructions cut out the actual premoduli space, and the gauge-fixing that produced it appears again in 03.02.36, where Einstein metrics are realised as critical points of the normalised total scalar curvature and the TT slice is the natural domain of the second variation. The foundational reason the messy linearised Ricci formula collapses to a clean eigenvalue equation is exactly the Berger-Ebin orthogonal splitting: the trace and Lie-derivative directions are precisely the terms that drop, so working on TT is dual to quotienting by rescaling and diffeomorphism. Putting these together, the central insight is that $Δ_{L}$ on TT is the Hessian of the Einstein functional in disguise, and its $2 λ$ -eigenspace is the kernel through which every genuine deformation must pass. This is exactly the pattern by which the Bochner machinery of 03.02.15 converts a geometric existence question into the spectral question of whether a curvature-corrected Laplacian has a prescribed eigenvalue.

Exercises Intermediate+

Exercise (medium, short-answer). Explain in two or three sentences why a positive-dimensional space $\varepsilon(g)$ is only a candidate tangent to the moduli space, not the actual tangent, naming what can go wrong.

Hint

First-order solvability of a nonlinear equation does not guarantee second-order solvability.

Answer

Rubric. $ε (g)$ solves only the linearised Einstein equation; an actual curve of Einstein metrics tangent to $h \in ε (g)$ must also satisfy the equation to second and higher order. Koiso's obstruction is a quadratic map $ε (g) \to (TT)$ whose vanishing is necessary for $h$ to integrate; when it does not vanish, $h$ is an infinitesimal deformation that does not extend, and the premoduli space is singular or lower-dimensional. Full credit names the integrability obstruction and the second-order failure.

Exercise (hard, short-answer). On the round sphere $S^n$ ($n\ge 3$) the curvature operator $\mathring R$ on TT tensors is a positive multiple of the identity, and one finds $\Delta_L h \ge c\, h$ with $c > 2\lambda$ for every nonzero $h\in\mathrm{TT}$. Use this spectral gap to conclude that $\varepsilon(S^n) = 0$.

Hint

$ε (g)$ is the $2 λ$ -eigenspace of $Δ_{L}$ on TT. What does the strict bound $Δ_{L} \geq c > 2 λ$ say about that eigenspace?

Answer

Rubric. $ε (S^{n})$ is by definition ${h \in TT : Δ_{L} h = 2 λh}$ , i.e. the eigenspace of $Δ_{L}$ for the eigenvalue $2 λ$ . If $⟨ Δ_{L} h, h ⟩ \geq c ∥ h ∥^{2}$ with $c > 2 λ$ for all nonzero $h \in TT$ , then the smallest TT-eigenvalue of $Δ_{L}$ strictly exceeds $2 λ$ , so $2 λ$ is not an eigenvalue and the eigenspace is $0$ . Hence $ε (S^{n}) = 0$ and the round sphere is infinitesimally rigid. Full credit identifies $ε$ as the $2 λ$ -eigenspace and uses the strict gap.

Exercise (hard, short-answer). On a Calabi-Yau or hyperkähler manifold the TT deformations split into pieces detected by Hodge theory, and the dimension of $\varepsilon(g)$ matches a Dolbeault cohomology number. Explain qualitatively why K3 then has $\varepsilon(g)\ne 0$ while $S^n$ does not, in terms of which cohomology is available.

Hint

K3 carries nonzero $(1, 1)$ and $(2, 0)$ cohomology; the sphere's relevant cohomology vanishes.

Answer

Rubric. On a Ricci-flat Kähler (Calabi-Yau) manifold the infinitesimal Einstein deformations are governed by harmonic representatives of complex-structure and Kähler-class cohomology; the relevant groups $H^{1, 1}$ and $H^{0} (Λ^{2, 0})$ are nonzero on K3, producing a positive-dimensional $ε (g)$ that integrates to the $58$ -dimensional moduli of Ricci-flat K3 metrics. On $S^{n}$ the corresponding cohomology vanishes (the sphere has no such harmonic forms), so no TT eigenvector at $2 λ$ survives and $ε (S^{n}) = 0$ . Full credit links nonvanishing cohomology to a nonzero $ε (g)$ and the sphere's vanishing cohomology to rigidity.

Advanced results Master

Koiso's second-order integrability obstruction. An element $h \in ε (g)$ is the first-order term of a genuine curve $g_{t}$ of Einstein metrics only if it survives a sequence of obstructions, the first of which is quadratic. Expanding the Einstein equation along $g_{t} = g + t h + \frac{t ^{2}}{2} h_{2} + \dots$ on the TT slice, the order- $t^{2}$ term requires solving $\frac{1}{2} (Δ_{L} - 2 λ) h_{2} = Q (h, h)$ for a TT tensor $h_{2}$ , where $Q$ is an explicit quadratic expression in $h$ and its covariant derivatives built from the second variation of Ricci. Self-adjointness of $Δ_{L} - 2 λ$ makes this solvable exactly when $Q (h, h)$ is $L^{2}$ -orthogonal to $ε (g) = ker (Δ_{L} - 2 λ) ∣_{TT}$ . The map $Ψ : ε (g) \to ε (g)^{*}$ , $Ψ (h) = [k \mapsto ⟨ Q (h, h), k ⟩]$ , is the Koiso obstruction; $h$ is second-order integrable iff $Ψ (h) = 0$ ^{[Koiso 1983]}. Higher obstructions are governed by the same Fredholm alternative against $ε (g)$ , organised by the Kuranishi/implicit-function method on the gauge slice.

The premoduli space is real-analytic and possibly singular. Because the Einstein operator and the gauge-fixing are real-analytic, the Kuranishi map identifies the local moduli of Einstein structures near $g$ with the zero set of a real-analytic map $Φ : U \subset ε (g) \to ε (g)$ whose two-jet is the Koiso obstruction $Ψ$ . The premoduli space — Einstein metrics near $g$ modulo diffeomorphism and rescaling — is therefore locally the analytic variety $Φ^{- 1} (0)$ . It is smooth of dimension $dim ε (g)$ when all obstructions vanish, but it can be a singular analytic set: cusps, transverse intersections, and lower-dimensional components all occur. Koiso exhibited Einstein metrics for which $ε (g) \neq = 0$ yet the obstruction is nonzero, so that the infinitesimal deformations do not integrate and the metric is rigid despite being infinitesimally deformable ^{[Koiso 1982]}.

Rigidity of the symmetric models. For the round sphere $S^{n}$ and complex projective space $CP^{n}$ with the Fubini-Study metric, $ε (g) = 0$ : there are no nonzero TT solutions of $Δ_{L} h = 2 λh$ . The proof for $S^{n}$ uses the explicit spectrum of $Δ_{L}$ on the symmetric space, where the lowest TT eigenvalue strictly exceeds $2 λ$ ; for $CP^{n}$ Koiso's representation-theoretic analysis of $Δ_{L}$ on the irreducible compact symmetric space gives the same strict gap ^{[Koiso 1980]}. These metrics are isolated points of the moduli space — rigid in the strongest sense, with no first-order deformations to obstruct.

Deformability of Ricci-flat Kähler models. By contrast, K3 surfaces and Calabi-Yau manifolds carry nonzero $ε (g)$ , and there the obstructions vanish: the deformations integrate to genuine families. For Ricci-flat Kähler metrics the TT eigenspace at $2 λ = 0$ is computed by Hodge theory — infinitesimal Einstein deformations correspond to harmonic $(1, 1)$ -forms and to complex-structure deformations $H^{1} (M, T^{1, 0} M)$ — and Yau's solution of the Calabi conjecture supplies the actual metrics, so the moduli space is smooth of the expected dimension. For K3 this yields the $57$ -real-dimensional moduli of Ricci-flat Kähler metrics in a fixed complex structure, part of the larger Einstein moduli space; hyperkähler and Calabi-Yau $3$ -folds behave analogously ^{[Besse Ch. 12]}.

Synthesis. The Lichnerowicz Laplacian is the central insight that organises this entire dichotomy: the foundational reason some Einstein metrics are rigid and others deform is the location of $2 λ$ in the TT-spectrum of $Δ_{L}$ . This is exactly the spectral question that the Bochner machinery of 03.02.15 is built to answer, and it generalises the harmonic-form vanishing theorems there from $1$ -forms to symmetric $2$ -tensors with the full curvature correction $\overset{˚}{R}$ replacing the Ricci term. The Koiso obstruction is dual to the Kuranishi obstruction in deformation theory: putting these together, the premoduli space is the zero set of a real-analytic Kuranishi map whose linear part is $ε (g)$ and whose quadratic part is $Ψ$ , exactly as a complex-structure moduli space is cut out inside $H^{1} (M, T)$ by the Maurer-Cartan obstruction. The rigidity of $S^{n}$ and $CP^{n}$ and the deformability of K3 are then two readings of one spectral fact, and the curvature decomposition of 03.02.16 is what controls $\overset{˚}{R}$ on the TT slice and hence which side of the dichotomy a given Einstein manifold falls on. This appears again in 03.02.36, where the same TT slice is the domain of the second variation of the total scalar curvature, and the bridge is that $Δ_{L} ∣_{TT} - 2 λ$ is the Hessian of the Einstein functional.

Full proof set Master

Proposition (gauge reduction of the linearised Einstein equation). Let $(M^{n}, g)$ be closed Einstein with $Ric = λ g$ . If $h \in TT$ satisfies the linearised Einstein equation $Ric_{g}^{'} (h) = μg + λh$ for some constant $μ$ , then $μ = 0$ and $Δ_{L} h = 2 λh$ .

Proof. The linearised Ricci operator is $Ric_{g}^{'} (h) = \frac{1}{2} Δ_{L} h - δ^{*} δ h - \frac{1}{2} \nabla d (tr_{g} h)$ ^{[Besse Ch. 1]}. For $h \in TT$ both $δ h$ and $tr_{g} h$ vanish, so $Ric_{g}^{'} (h) = \frac{1}{2} Δ_{L} h$ . The hypothesis becomes $\frac{1}{2} Δ_{L} h = μg + λh$ . On an Einstein background $Δ_{L}$ preserves the orthogonal decomposition into trace and trace-free parts, and $tr_{g} (Δ_{L} h) = Δ_{g} (tr_{g} h) = 0$ since $tr_{g} h = 0$ . Taking $tr_{g}$ of $\frac{1}{2} Δ_{L} h = μg + λh$ gives $0 = n μ + λ tr_{g} h = n μ$ , so $μ = 0$ . Hence $\frac{1}{2} Δ_{L} h = λh$ , i.e. $Δ_{L} h = 2 λh$ . $■$

Proposition (finite-dimensionality of $ε (g)$ ). On a closed Einstein manifold, $ε (g)$ is finite-dimensional.

Proof. The operator $Δ_{L}$ is a second-order differential operator on sections of $S^{2} T^{*} M$ with principal symbol $σ (Δ_{L}) (ξ) = ∣ ξ ∣_{g}^{2} id$ , the same as the rough Laplacian, hence elliptic. On a closed manifold an elliptic self-adjoint operator has discrete spectrum with finite-dimensional eigenspaces. The restriction to the closed subspace TT is again Fredholm, since TT is preserved by $Δ_{L}$ on an Einstein background (the divergence and trace constraints are conserved by $Δ_{L}$ there). Therefore $ε (g)$ , the $2 λ$ -eigenspace of $Δ_{L} ∣_{TT}$ , is finite-dimensional. $■$

Proposition (the Koiso obstruction is well-defined and is the second-order integrability condition). Let $h \in ε (g)$ . There exists a TT tensor $h_{2}$ solving $\frac{1}{2} (Δ_{L} - 2 λ) h_{2} = Q (h, h)$ , where $Q (h, h)$ is the order- $t^{2}$ term of the gauge-fixed Einstein equation along $g_{t} = g + t h + \frac{t ^{2}}{2} h_{2}$ , if and only if $⟨ Q (h, h), k ⟩_{L^{2}} = 0$ for all $k \in ε (g)$ . The pairing $Ψ (h) (k) = ⟨ Q (h, h), k ⟩_{L^{2}}$ is well-defined on $ε (g)$ .

Proof. The operator $A = Δ_{L} - 2 λ$ on TT is self-adjoint elliptic on the closed manifold, so by the Fredholm alternative its image is the $L^{2}$ -orthogonal complement of its kernel, $im A ∣_{TT} = (ker A ∣_{TT})^{⊥} = ε (g)^{⊥}$ . The equation $\frac{1}{2} A h_{2} = Q (h, h)$ is solvable for $h_{2} \in TT$ precisely when $Q (h, h) \in im A = ε (g)^{⊥}$ , i.e. when $⟨ Q (h, h), k ⟩ = 0$ for every $k \in ε (g)$ . The construction of $Q$ from the second derivative of the Einstein operator in the gauge slice shows $Q (h, h)$ is TT, so the pairing against $k \in ε (g)$ is the relevant one and $Ψ (h)$ is well-defined as an element of $ε (g)^{*}$ . Hence $Ψ (h) = 0$ is exactly the second-order integrability condition. $■$

Proposition (real-analyticity of the premoduli space). Near a closed Einstein metric $g$ , the premoduli space of Einstein structures is locally homeomorphic to the zero set $Φ^{- 1} (0)$ of a real-analytic map $Φ : U \subset ε (g) \to ε (g)$ whose two-jet at $0$ is the Koiso obstruction.

Proof. Fix the diffeomorphism gauge by the slice theorem (Ebin), restricting to metrics with $δ_{g} (\cdot) = 0$ relative to $g$ , and the rescaling gauge by fixing volume. On this slice the Einstein equation $E (g^{'}) = Ric (g^{'}) - λ g^{'} = 0$ is a real-analytic map between Banach spaces of sections whose linearisation at $g$ has kernel $ε (g)$ and closed image of finite codimension equal to $dim ε (g)$ (self-adjointness). The Lyapunov-Schmidt / Kuranishi reduction solves the complementary equation by the analytic implicit function theorem, producing a real-analytic finite-dimensional reduction $Φ : U \subset ε (g) \to ε (g)$ with $Φ^{- 1} (0)$ the local premoduli space. The quadratic term of $Φ$ is the Koiso obstruction $Ψ$ by the previous proposition. Real-analyticity is inherited from the polynomial-in-curvature, real-analytic structure of $Ric$ and the analytic implicit function theorem ^{[Koiso 1983]}. $■$

The rigidity statements for $S^{n}$ and $CP^{n}$ — that $ε (g) = 0$ — are stated above; their proofs are the explicit eigenvalue computations of $Δ_{L} ∣_{TT}$ on the relevant rank-one compact symmetric spaces and are given in full in Koiso ^{[Koiso 1980]}. The Ricci-flat Kähler computation identifying $ε (g)$ with $H^{1, 1} \oplus H^{1} (M, T^{1, 0} M)$ is stated above without proof — see Besse ^{[Besse Ch. 12]}.

Connections Master

Bochner technique and curvature vanishing theorems 03.02.15. The Lichnerowicz Laplacian is the $S^{2} T^{*} M$ analogue of the Weitzenböck-corrected operators of that unit: $Δ_{L} = \nabla^{*} \nabla - 2 \overset{˚}{R} + (Ric terms)$ is a rough Laplacian plus a curvature endomorphism, and the rigidity proofs are Bochner arguments showing the curvature term forces the $2 λ$ -eigenspace to vanish. The vanishing theorems there for harmonic $1$ -forms are the codimension-one prototype of the vanishing $ε (g) = 0$ here.
Weyl tensor and the curvature decomposition 03.02.16. The endomorphism $\overset{˚}{R}$ acting on TT tensors is controlled by the irreducible pieces of the Riemann tensor decomposed there; on an Einstein manifold the Ricci part is fixed by $λ$ , so the deformation behaviour is governed by the Weyl and scalar parts. Whether $\overset{˚}{R}$ pushes the TT-spectrum above or down to $2 λ$ — and hence rigidity versus deformability — is read off that decomposition.
Einstein metrics as critical points 03.02.36. This unit is the deformation theory of the critical points produced there: the TT slice is the natural domain of the second variation of the normalised total scalar curvature, and $Δ_{L} ∣_{TT} - 2 λ$ is exactly that Hessian. The variational characterisation supplies the functional whose degenerate critical directions are $ε (g)$ .
The spinor Lichnerowicz operator and the Dirac operator 03.09.08. That unit's operator $\neq \partial^{2} = \nabla^{*} \nabla + Scal /4$ shares the Weitzenböck shape and the name Lichnerowicz, but acts on the spinor bundle and carries only scalar curvature, where the present $Δ_{L}$ acts on symmetric $2$ -tensors and carries the full Riemann tensor. The contrast is structural: both are curvature-corrected Laplacians, but on different bundles, and conflating them is the most common error in the literature.
Sectional, Ricci, and scalar curvature 03.02.05. The Einstein condition $Ric = λ g$ , the curvature tensor $R$ feeding $\overset{˚}{R}$ , and the sign convention all come from that unit; the entire deformation problem is a question about a metric on the bundle built from $R$ .

Historical & philosophical context Master

The decomposition of the space of symmetric $2$ -tensors into trace, Lie-derivative, and transverse-traceless parts is due to Marcel Berger and David Ebin, "Some decompositions of the space of symmetric tensors on a Riemannian manifold" (Journal of Differential Geometry 3, 1969, 379–392), which isolated the gauge directions and made the linearised Einstein equation tractable ^{[Berger-Ebin 1969]}. The Lichnerowicz Laplacian on $2$ -tensors descends from André Lichnerowicz's work on propagation and curvature operators; it is a different operator from the spinor Lichnerowicz formula of his harmonic-spinor theorem, though both bear his name.

The systematic deformation theory is the achievement of Norihito Koiso in a sequence of Osaka Journal of Mathematics and Inventiones papers. "Rigidity and stability of Einstein metrics, the case of compact symmetric spaces" (Osaka J. Math. 17, 1980, 51–73) established the rigidity of the rank-one symmetric models including $S^{n}$ and $CP^{n}$ ; "Rigidity and infinitesimal deformability of Einstein metrics" (Osaka J. Math. 19, 1982, 643–668) constructed examples with nonintegrable infinitesimal deformations; and "Einstein metrics and complex structures" (Inventiones Mathematicae 73, 1983, 71–106) supplied the second-order obstruction and the real-analytic premoduli structure ^{[Koiso 1983]}. Arthur Besse's Einstein Manifolds (Springer, 1987), the collective Bourbaki-style monograph, gathered this material into Chapter 12 and made it the standard reference ^{[Besse Ch. 12]}. The deformability of Ricci-flat Kähler manifolds rests on Shing-Tung Yau's 1978 proof of the Calabi conjecture, which converted the Hodge-theoretic count of infinitesimal deformations into actual families of Einstein metrics.

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      = {Ch. 12 (moduli of Einstein structures); Ch. 1 (curvature conventions)}
}

@article{Koiso1980Rigidity,
  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}
}

@article{Koiso1982Deformability,
  author  = {Koiso, Norihito},
  title   = {Rigidity and infinitesimal deformability of {E}instein metrics},
  journal = {Osaka Journal of Mathematics},
  volume  = {19},
  number  = {3},
  pages   = {643--668},
  year    = {1982}
}

@article{Koiso1983Complex,
  author  = {Koiso, Norihito},
  title   = {{E}instein metrics and complex structures},
  journal = {Inventiones Mathematicae},
  volume  = {73},
  number  = {1},
  pages   = {71--106},
  year    = {1983}
}

@article{BergerEbin1969,
  author  = {Berger, Marcel and Ebin, David},
  title   = {Some decompositions of the space of symmetric tensors on a {R}iemannian manifold},
  journal = {Journal of Differential Geometry},
  volume  = {3},
  pages   = {379--392},
  year    = {1969}
}

@article{Yau1978Calabi,
  author  = {Yau, Shing-Tung},
  title   = {On the {R}icci curvature of a compact {K}{\"a}hler manifold and the complex {M}onge-{A}mp{\`e}re equation, {I}},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {31},
  number  = {3},
  pages   = {339--411},
  year    = {1978}
}

Prerequisites

03.02.15
03.02.16
03.02.36

Tier anchors

beginner: Besse Einstein Manifolds Ch. 0 (the moduli question, informal); needham-style picture of a family of shapes
intermediate: Besse Einstein Manifolds Ch. 12 §§12.9--12.30; Koiso 1983 Osaka J. Math.; Lee Introduction to Riemannian Manifolds Ch. 8 (curvature operators)
master: Besse Einstein Manifolds Ch. 12; Koiso 1980 Osaka J. Math. 17 and 1982 Osaka J. Math. 19 and 1983 Invent. Math. 73; Berger-Ebin 1969 J. Diff. Geom. 3

References

Besse — Einstein Manifolds (1987) · Ch. 12 (moduli of Einstein structures; the Lichnerowicz Laplacian; infinitesimal deformations); Ch. 1 (curvature operator conventions)
Koiso — Rigidity and stability of Einstein metrics, the case of compact symmetric spaces (1980) · Osaka J. Math. 17, pp. 51--73
Koiso — Rigidity and infinitesimal deformability of Einstein metrics (1982) · Osaka J. Math. 19, pp. 643--668
Koiso — Einstein metrics and complex structures (1983) · Invent. Math. 73, pp. 71--106 (second-order integrability obstruction)
Berger-Ebin — Some decompositions of the space of symmetric tensors on a Riemannian manifold (1969) · J. Differential Geom. 3, pp. 379--392 (the splitting of $S^2 T^*M$)

Estimated time

beginner: 16m
intermediate: 44m
master: 85m