Abstract
Any 6-dimensional strict nearly Kähler manifold is Einstein with positive scalar curvature. We compute the coindex of the metric with respect to the Einstein–Hilbert functional on each of the compact homogeneous examples. Moreover, we show that the infinitesimal Einstein deformations on \(F_{1,2}={\text {SU}}(3)/T^2\) are not integrable into a curve of Einstein metrics.
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1 Introduction
The special case of dimension 6 has been a primary focus of nearly Kähler geometry since Nagy showed that every nearly Kähler manifold is locally isometric to a Riemannian product of 6-dimensional nearly Kähler manifolds, nearly Kähler homogeneous spaces and twistor spaces over positive scalar curvature quaternionic-Kähler manifolds [16]. Moreover, nearly Kähler manifolds that are non-Kähler (so-called strict nearly Kähler manifolds) of dimension 6 exhibit other notable properties, such as carrying a real Killing spinor and thus being Einstein with positive scalar curvature.
On a compact manifold M, Einstein metrics can be variationally characterized as critical points of the total scalar curvature functional S (also called Einstein–Hilbert action), defined on the set of all Riemannian metrics on M of a fixed volume. Given a compact Einstein manifold (M, g), one can ask whether g locally maximizes S (after restricting to a suitable subclass of Riemannian metrics). Such an Einstein metric g is called stable with respect to S. The linearized problem considers the Hessian \(S_g''\) of the Einstein–Hilbert action at g. Accordingly, an Einstein metric g is called linearly stable if \(S_g''\le 0\) on the space of tt-tensors (i.e.,trace- and divergence-free symmetric 2-tensors on M). A closely related notion is that of infinitesimal deformability of the Einstein metric g—it is called infinitesimally deformable if \(S_g''\) is degenerate on tt-tensors.
A compact, 6-dimensional, strict nearly Kähler manifold (M, g, J) with scalar curvature normalized to \({\text {scal}}_g=30\) (hence Einstein constant \(E=5\)) is called a Gray manifold, after Gray, who studied them in the 70s. The stability and infinitesimal deformability of Einstein metrics on Gray manifolds have already been investigated. In [18], Semmelmann et al. show linear instability if the second or third Betti number does not vanish—in fact, the coindex of g (see Sect. 2.2 for a definition) is bounded below by \(b_2+b_3\). Moroianu and Semmelmann [14] give a description of the space of infinitesimal Einstein deformations in terms of eigenspaces of the Hodge Laplacian on coclosed primitive (1, 1)-forms. The present article generalizes this result to a similar description of eigenspaces of the Lichnerowicz Laplacian on tt-tensors to arbitrary eigenvalues not exceeding a certain threshold (see Lemma 3.2).
Homogeneous Gray manifolds have been classified by Buitruille [3]. There are only four cases: \(S^6=\frac{G_2}{{\text {SU}}(3)}\), \(S^3\times S^3=\frac{{\text {SU}}(2)\times {\text {SU}}(2)\times {\text {SU}}(2)}{\varDelta {\text {SU}}(2)}\), \({{\mathbb {C}}}{{\mathbb {P}}}^3=\frac{{\text {Sp}}(2)}{{\text {Sp}}(1){\text {U}}(1)}=\frac{{\text {SO}}(5)}{{\text {U}}(2)}\) and the flag manifold \(F_{1,2}=\frac{{\text {SU}}(3)}{T^2}\), all of them equipped with the Killing form metric (up to scaling). In [23], Wang and Wang show instability of the latter three spaces. \(S^6\) carries the round metric and is thus strictly stable.
One aim of this article is to improve the coindex estimates from [18] to equalities for the homogeneous examples. Our first main result can be stated as follows.
Theorem 1.1
Let (M, g) be a homogeneous Gray manifold with standard metric g. The coindex of the Einstein metric g is
-
equal to 2 if \(M=S^3\times S^3=\frac{{\text {SU}}(2)\times {\text {SU}}(2)\times {\text {SU}}(2)}{\varDelta {\text {SU}}(2)}\),
-
equal to 1 if \(M={{\mathbb {C}}}{{\mathbb {P}}}^3=\frac{{\text {SO}}(5)}{{\text {U}}(2)}\),
-
equal to 2 if \(M=F_{1,2}=\frac{{\text {SU}}(3)}{T^2}\).
The destabilizing directions, i.e., contributions to the coindex, can be viewed as arising from harmonic 3-forms in the first and from harmonic 2-forms in the second and third case via the construction in [18]. For the last two cases, there is an additional geometric explanation: consider the Riemannian submersions given by the twistor fibrations
In both cases, the canonical variation (scaling the base against the fiber) yields a destabilizing direction by [23, Prop. 4.4]. For the flag manifold, there are actually three such fibrations whose canonical variations give rise to a two-dimensional space of tt-tensors, explaining the coindex of 2 (see Remark 4.10).
Also worth noting is the G-invariant stability problem, in which the Einstein–Hilbert functional S is restricted to the class of G-invariant metrics on a fixed homogeneous space \(M=G/H\). Since the destabilizing directions on all three cases in Theorem 1.1 are G-invariant (as explained in Remark 3.5), it follows that their metrics are G-unstable. In fact, they are even G-strongly unstable, i.e.,all G-invariant tt-variations of the metric are destabilizing and hence these metrics are local minima of S among G-invariant metrics. For \({{\mathbb {C}}}{{\mathbb {P}}}^3\) [10, Table 1, 7a] and \(F_{1,2}\) [11, Table 2], this was already known from the results of Lauret and Lauret. For \(S^3\times S^3\),see Remark 4.4.
Let us return to the general setting of a compact manifold M. An Einstein metric g on M is called rigid if it is isolated in the moduli space of Einstein structures (disregarding variation by homothetic scaling and action of diffeomorphisms). If an Einstein manifold (M, g) admits infinitesimal Einstein deformations, one naturally asks whether they are integrable into a curve of Einstein metrics on M. In fact, not every infinitesimally deformable Einstein must lie within a nontrivial curve of Einstein metrics. The first example of such a metric is the canonical symmetric metric on \({{\mathbb {C}}}{{\mathbb {P}}}^1\times {{\mathbb {C}}}{{\mathbb {P}}}^{2k}\) found by Koiso [9], who started the investigation of stability and infinitesimal deformatibility of symmetric spaces [8]. Another recent example due to Batat et al. is the bi-invariant metric on \({\text {SU}}(2n+1)\) [2]. We add one more example to this list by proving the following result.
Theorem 1.2
The Einstein metric on the Gray manifold \(F_{1,2}\) is rigid, that is, its infinitesimal Einstein deformations are not integrable.
In all of the above examples, integrability fails at an obstruction to second order (see the end of Sect. 2.2). We suspect that this phenomenon occurs generically. Given some infinitesimal Einstein deformation, i.e. an element of the null space of \(S_g''\), the obstruction polynomial (2) has no immediate compulsion to vanish and should do so only coincidentally—see for example the case \({\text {SU}}(2n)\) in [2].
Since Gray manifolds are Einstein, every infinitesimal deformation of the nearly Kähler structure corresponds to an infinitesimal Einstein deformation, but not necessarily vice versa [14]. Infinitesimal deformability of the nearly Kähler structure has been investigated by Moroianu et al. [12]. The question whether a given infinitesimal nearly Kähler deformation can be integrated into a curve of nearly Kähler structures has been studied by Foscolo [5], where a similar polynomial occurs as integrability obstruction to second order. In particular, he showed that the infinitesimal nearly Kähler deformations on \(F_{1,2}\) are all obstructed. One can view Theorem 1.2 as a generalization of this result to the Einstein picture. The Einstein metrics and thus nearly Kähler structures on homogeneous Gray manifolds other than \(F_{1,2}\) are automatically rigid since they possess no infinitesimal deformations [14].
This article is organized as follows. In Sect. 2, notation is fixed and the necessary preliminaries are recapitulated. Section 3 concerns itself with a description of eigenspaces of the Lichnerowicz Laplacian on tt-tensors on general Gray manifolds as well as a discussion of the homogeneous case, in which explicit calculations are possible by means of harmonic analysis. These results are applied in Sect. 4 to each of the unstable Gray manifolds \(S^3\times S^3\), \({{\mathbb {C}}}{{\mathbb {P}}}^3\) and \(F_{1,2}\) to obtain the results collected in Theorem 1.1. Finally, Sect. 5 recalls the description of the infinitesimal Einstein deformations on \(F_{1,2}\) given in [13] and proceeds to show the nonintegrability to second order, proving Theorem 1.2.
The author owes gratitude to Prof. Semmelmann for helpful exchanges about a gap in the argument given in the proof of [14, Thm. 5.1] (the corrected argument is the proof of Lemma 3.1, which includes the aforementioned as the special case \(\lambda =10\)). Furthermore, the author would like to thank Prof. Weingart for his useful suggestions regarding the rigidity argument.
2 Preliminaries
2.1 Nearly Kähler Manifolds
An almost Hermitian manifold (M, g, J) is an even-dimensional Riemannian manifold (M, g) with an almost complex structure J that is compatible with the metric, i.e.,
for any \(X,Y\in T_pM\). The Kähler form \(\omega \) is then defined by
Any almost Hermitian structure has an associated canonical Hermitian connection \({{\bar{\nabla }}}\) (see, for example, [4, Sect. 2] for a general definition). In particular, it satisfies \({{\bar{\nabla }}} g=0\) and \({{\bar{\nabla }}} J=0\).
Let \(\nabla \) denote the Levi-Civita connection of the Riemannian manifold (M, g). An almost Hermitian manifold (M, g, J) is called nearly Kähler if \(\nabla J\) is skew-symmetric, or equivalently, if
for all \(X\in T_pM\). In this case, the canonical Hermitian connection can be described by
for any two vector fields \(X,Y\in {\mathfrak {X}}(M)\). A nearly Kähler manifold is called strictly nearly Kähler if it is not Kähler. Gray manifolds are compact strict nearly Kähler manifolds of dimension 6.
As usual, the almost complex structure J defines a splitting of the complexified cotangent bundle \(T^*M^{\mathbb {C}}=\varLambda ^{1,0}M\oplus \varLambda ^{0,1}M\) and hence of the bundle of k-forms into (p, q)-forms with \(p+q=k\). The complex bundle of (p, q)-forms will be denoted with the prefix \(\varLambda ^{p,q}\), and the space of its smooth sections by \(\varOmega ^{p,q}\). The Kähler form \(\omega \) is of type (1, 1). A (p, q)-form \(\alpha \) is called primitive if it vanishes under contraction with the Kähler form, i.e.,if \(\omega \lrcorner \alpha =0\). We will denote the bundle of primitive (p, q)-forms by \(\varLambda ^{p,q}_0\). Furthermore, let \(\varLambda ^{p,q}_{\mathbb {R}}\) denote the projection of the complex bundle \(\varLambda ^{p,q}\) to the real bundle \(\varLambda ^{p+q}\).
Likewise, the bundle \({\text {Sym}}TM\) of g-symmetric endomorphisms of the tangent bundle splits into a direct sum \({\text {Sym}}^+TM\oplus {\text {Sym}}^-TM\), where the elements of \({\text {Sym}}^\pm TM\) commute (resp. anticommute) with J. We further denote by \({\text {Sym}}^+_0TM\) the subbundle of trace-free endomorphisms in \({\text {Sym}}^+TM\), and with \({\mathscr {S}}^\pm \), \({\mathscr {S}}^+_0\) the spaces of smooth sections in the respective bundles.
Let \({\mathscr {S}}^k=\varGamma ({\text {Sym}}^kT^*M)\) denote the space of symmetric k-tensor fields. Note that the metric yields a natural identification \({\text {Sym}}^2T^*M\cong {\text {Sym}}TM\). The subspace of tt-tensors in \({\mathscr {S}}^2\) (i.e., \(h\in {\mathscr {S}}^2\) satisfying \({\text {tr}}_gh=0\) and \(\delta h=0\)) will be denoted by \({\mathscr {S}}^2_{\mathrm {tt}}\).
If (M, g, J) is nearly Kähler, then the tensor \(\varPsi ^+:=\nabla \omega \) is totally skew-symmetric and in fact the real part of a \({{\bar{\nabla }}}\)-parallel complex volume form \(\varPsi ^++\mathrm {i}\varPsi ^-\). The imaginary part \(\varPsi ^-\) can be described by \(X\lrcorner \varPsi ^-=J\circ (\nabla _XJ)\) for all \(X\in TM\). The strict nearly Kähler case is characterized by the nonvanishing of \(\varPsi ^+\).
Let (M, g, J) be a strict nearly Kähler manifold of dimension 6. There are \({{\bar{\nabla }}}\)-parallel isomorphisms
of vector bundles with structure group \({\text {SU}}(3)\), each arising from an equivalence of \({\text {SU}}(3)\)-representations. Here, \(h_*\) denotes the extension of the endomorphism \(h\in {\text {End}}TM\) to tensor bundles as a derivation.
2.2 Stability and Rigidity
The Lichnerowicz Laplacian \(\varDelta _L\) of a Riemannian manifold (M, g) is an operator that generalizes the Hodge Laplacian \(\varDelta \) on differential forms to tensor fields of any rank. It is defined by
where q(R) is the curvature endomorphism acting on tensors by
for some local orthonormal frame \((e_i)\) of TM. The asterisk denotes the natural action of \(\varLambda ^2T\cong \mathfrak {so}(T)\). In particular, \(q(R)={\text {Ric}}\) on 1-forms.
On an almost Hermitian manifold, we analogously define the Hermitian Laplace operator \({{\bar{\varDelta }}}\) by replacing the Levi-Civita connection \(\nabla \) in the above definition by the canonical Hermitian connection \({{\bar{\nabla }}}\), i.e.,
Here, R and \({{\bar{R}}}\) denote the curvature tensors of the connections \(\nabla \) and \({{\bar{\nabla }}}\), respectively. Both \(\varDelta _L\) and \({{\bar{\varDelta }}}\) are instances of the standard Laplacian of a given connection (see [19]), an operator with several neat properties—for example, it commutes with parallel bundle maps. Comparison formulas for the two Laplace operators in the setting of 6-dimensional nearly Kähler manifolds can be found in [13] and [14]. For our purposes, it is important to note that \(\varDelta \) and \({{\bar{\varDelta }}}\) coincide on coclosed primitive (1, 1)-forms, as well as on coclosed (2, 1)- and (1, 2)-forms, which follows from combining Cor. 3.5 and Cor. 4.4 of [14].
Consider a fixed compact orientable smooth manifold M of dimension \(n>2\). On the set of all Riemannian metrics on M, the total scalar curvature functional (or Einstein–Hilbert action) is defined by
Einstein metrics on M are then precisely the critical points of the restriction of S to metrics of a fixed total volume. Let (M, g) be an Einstein manifold with \({\text {Ric}}=Eg\). If (M, g) not isometric to the standard sphere, there is a well-known decomposition
that is orthogonal with respect to the second variation \(S''_g\) (see [1]). Furthermore,
On the latter space, \(S''_g\) has finite coindex and nullity, i.e., the maximal subspace of \({\mathscr {S}}^2_{\mathrm {tt}}\) on which \(S''_g\) is nonnegative is finite-dimensional. The sum \(L_{\mathfrak {X}}g\oplus {\mathscr {S}}^2_{\mathrm {tt}}=T_g{\mathfrak {S}}\) can also be regarded as formal tangent space to the set \({\mathfrak {S}}\) of metrics with constant scalar curvature and fixed total volume.
The stability problem is to decide whether an Einstein metric g is a local maximum or a saddle point of \(S\big |_{{\mathfrak {S}}}\). We are primarily concerned with the linearized version, considering only the second variation of S at g. An Einstein metric g is called (linearly) stable if \(S''_g\big |_{{\mathscr {S}}^2_{\mathrm {tt}}}\le 0\), or, equivalently, if \(\varDelta _L\ge 2E\) on \({\mathscr {S}}^2_{\mathrm {tt}}\). If strict inequality holds, we call g strictly stable. On the other hand, g is called (linearly) unstable if there exists \(h\in {\mathscr {S}}^2_{\mathrm {tt}}\) such that \(S_g''(h,h)>0\), or, equivalently, if \((\varDelta _Lh,h)_{L^2}<2E\Vert h\Vert ^2_{L^2}\). The dimension of the maximal subspace of \({\mathscr {S}}^2_{\mathrm {tt}}\) on which \(S_g''>0\) is called the coindex of g.
A closely related notion is that of rigidity. An Einstein metric g is called rigid if it is isolated in the moduli space, i.e., the space of Einstein metrics modulo diffeomorphisms and homotheties. Since the moduli space is locally arcwise connected [1, Cor. 12.52], rigidity of g is equivalent to the nonexistence of a smooth curve \((g_t)\) of Einstein metrics through \(g=g_0\) with nonvanishing first-order jet \(\dot{g}_0\in {\mathscr {S}}^2_{\mathrm {tt}}\).
Denote by \(\varepsilon (g)=\{h\in {\mathscr {S}}^2_{\mathrm {tt}}\,|\,\varDelta _Lh=2Eh\}\) the null space of \(S''_g\), also called the space of infinitesimal Einstein deformations (IED). If \(\varepsilon (g)\ne 0\), we call g infinitesimally deformable. A metric with \(\varepsilon (g)=0\) is automatically rigid [1, Cor. 12.66]—in particular, strict stability implies rigidity.
In general, an IED needs not be integrable into a curve of Einstein metrics. On the set of unit volume Riemannian metrics, define the Einstein operator \({\mathcal {E}}\) by
Then a metric g is Einstein if and only if \({\mathcal {E}}(g)=0\). An IED \(h\in \varepsilon (g)\) is called formally integrable to order k if there exist \(h_2,\ldots ,h_k\in {\mathscr {S}}^2\) such that
A classical result [1, Cor. 12.50] is that an IED \(h\in \varepsilon (g)\) can be integrated into a curve \((g_t)\) of Einstein metrics with \(\dot{g}_0=h\) if and only if it is formally integrable to all orders \(k\ge 2\).
The integrability criterion to each order can be expressed in terms of derivatives of \({\mathcal {E}}\). By a result of Koiso [9, Lem. 4.7], \(h\in \varepsilon (g)\) is integrable to order 2 if and only if \({\mathcal {E}}''_g(h,h)\perp \varepsilon (g)\) in the \(L^2\) sense. Also due to Koiso [9, Lem. 4.3] is the formula
for the second-order obstruction, where we implicitly sum over a local orthonormal frame \((e_i)\) of TM. The vanishing of the quantity in (2) is a necessary condition for the integrability of h.
2.3 Harmonic Analysis
Let \((M=G/H,g)\) be a Riemannian homogeneous space, where G is some Lie group and H is a closed subgroup. We will always denote the corresponding Lie algebras by \({\mathfrak {g}}\) and \({\mathfrak {h}}\), respectively. The homogeneous space M is called reductive if there exists an \({\text {Ad}}\big |_H\)-invariant complement \({\mathfrak {m}}\) of \({\mathfrak {h}}\subset {\mathfrak {g}}\). This is always the case if H is compact (in particular, if G is compact). Through the canonical projection \(\pi : G\rightarrow M\), the reductive complement \({\mathfrak {m}}\subset {\mathfrak {g}}\cong T_eG\) is canonically identified with the tangent space \(T_oM\) at the base point \(o=eH\).
The G-invariant metric on a reductive Riemannian homogeneous space (M, g) is determined by an \({\text {Ad}}(H)\)-invariant inner product on \({\mathfrak {m}}\). Suppose that Q is an \({\text {Ad}}(G)\)-invariant inner product on \({\mathfrak {g}}\). Then \({\mathfrak {m}}:={\mathfrak {h}}^\perp \) is an \({\text {Ad}}(H)\)-invariant subspace. We call (M, g) a normal homogeneous space if the metric is induced by the restriction Q to \({\mathfrak {m}}\), i.e., \(g_o=Q\big |_{{\mathfrak {m}}\times {\mathfrak {m}}}\). If G is compact and semisimple, then the Killing form \(B_{\mathfrak {g}}\) is negative-definite. In this case, the standard metric is defined by \(g_o=-B_{\mathfrak {g}}\big |_{{\mathfrak {m}}\times {\mathfrak {m}}}\).
A normal homogeneous space is in particular naturally reductive, i.e.,
(where \(X_{\mathfrak {m}}\) denotes the projection of X to \({\mathfrak {m}}\)) holds for all \(X,Y,Z\in {\mathfrak {m}}\).
Let \(\rho : H\rightarrow {\text {Aut}}V\) be a finite-dimensional (real or complex) representation. Denote by \(VM=G\times _\rho V\) the associated homogeneous vector bundle over M. Its sections can be viewed as H-equivariant smooth V-valued functions on G—the isomorphism is explicitly given by
where \(s(xH)=[x,{{\hat{s}}}(x)]\) for any \(x\in G\). Left-translation on sections of VM gives rise to the left-regular representation on \(C^\infty (G,V)^H\), explicitly given by
for \(x,y\in G\).
If M is reductive, we can write every tensor bundle as an associated bundle of some tensor power of the reductive complement. For example,
(note that the H-representations \({\mathfrak {m}}\) and \({\mathfrak {m}}^*\) are equivalent via the Riemannian metric).
For a compact Lie group G, denote by \({\hat{G}}\) the set of dominant integral weights of G (after choosing a suitable maximal torus \(T\subset G\)). Recall that the elements of \({{\hat{G}}}\) are in one-to-one correspondence with equivalence classes of irreducible complex representations of G. Any representative of such a class with highest weight \(\gamma \in {{\hat{G}}}\) will be denoted by \((V_\gamma ,\rho _\gamma )\). Let V be a unitary representation of H. The homogeneous version of the Peter–Weyl theorem [22, Thm. 5.3.6] states that the left-regular representation decomposes into
Here, \({\text {Hom}}_H(V_\gamma ,V)\) simply counts the multiplicity of \(V_\gamma \) inside \(L^2(G,V)^H\) and is called the space of Fourier (matrix) coefficients. The equivalence in (3) is made explicit by
Let V, W be unitary representations of H and \({\mathfrak {D}}: \varGamma (VM)\rightarrow \varGamma (WM)\) be a G-invariant differential operator. Combining (3) with Schur’s Lemma, the operator \({\mathfrak {D}}\) acts as a linear mapping
for each fixed \(\gamma \in {{\hat{G}}}\). We call this mapping the prototypical differential operator associated with \({\mathfrak {D}}\) and \(\gamma \) (as introduced by Semmelmann and Weingart [20]).
On a reductive homogeneous space, a choice of reductive complement \({\mathfrak {m}}\) determines a G-invariant connection \(\nabla ^{\mathrm {red}}\) on VM, called the canonical reductive (or Ambrose–Singer) connection, by stipulating that
for all \(X\in TM\), \(s\in \varGamma (VM)\), where the horizontal lift \({{\tilde{X}}}\in TG\) is the unique vector in the canonical horizontal distribution \({\mathcal {H}}=\bigcup _{x\in G}dl_x({\mathfrak {m}})\) such that \(d\pi ({{\tilde{X}}})=X\). This connection has the important property that all G-invariant sections of VM are parallel. If (M, g) is naturally reductive, \(\nabla ^{\mathrm {red}}\) is a metric connection with parallel totally skew torsion tensor \(\tau \), given (at the base point) by
On any representation \(\rho : G\rightarrow {\text {Aut}}V\) of a compact Lie group G, the Casimir operator with respect to a fixed \({\text {Ad}}(G)\)-invariant inner product on \({\mathfrak {g}}\) is the equivariant endomorphism of V defined by
where \((e_i)\) is an orthonormal basis of \({\mathfrak {g}}\) with respect to Q. We omit the superscript Q if the choice of inner product is clear from context. For \(\gamma \in {{\hat{G}}}\), the Casimir operator on \(V_\gamma \) acts as multiplication with the Casimir constant
by Freudenthal’s formula, cf. [6]. Here, \(\langle \cdot ,\cdot \rangle _{{\mathfrak {t}}^*,Q}\) is the inner product induced by Q on the dual \({\mathfrak {t}}^*\) of the Lie algebra \({\mathfrak {t}}\) of the torus \(T\subset G\), while \(\delta _{\mathfrak {g}}\) denotes the half-sum of positive roots of \({\mathfrak {g}}\).
A crucial fact [13, Lem. 5.2] is that on a normal homogeneous space with Riemannian metric induced by an \({\text {Ad}}(G)\)-invariant inner product Q on \({\mathfrak {g}}\), the standard Laplacian of \(\nabla ^{\mathrm {red}}\) is precisely the Casimir operator of G acting on the left-regular representation, i.e.,
In particular, the prototypical differential operator associated with \(\varDelta ^{\mathrm {red}}\) and \(\gamma \) is simply multiplication by the Casimir constant. In other words, the eigenspaces of \(\varDelta ^{\mathrm {red}}\) are the isotypical components
in the Peter–Weyl decomposition (3). The eigenvalues are readily computable by means of Freudenthal’s formula (6).
It should be noted that in the symmetric case, the torsion of \(\nabla ^{\mathrm {red}}\) vanishes. Hence,\(\nabla ^{\mathrm {red}}\) coincides with the Levi-Civita connection \(\nabla \). It follows that \(\varDelta _L=\varDelta ^{\mathrm {red}}\), so the spectrum of the Lichnerowicz Laplacian on any tensor bundle is easily computable, facilitating the foundational work by Koiso on the stability of symmetric spaces [8].
2.4 3-Symmetric Spaces
A homogeneous space \(M=G/H\) is called 3-symmetric if there exists an automorphism \(\sigma \in {\text {Aut}}G\) of order 3 such that \(G_0^\sigma \subset H\subset G^\sigma \), where \(G^\sigma \) is the fixed point set of \(\sigma \) and \(G^\sigma _0\) is the connected component of the identity in \(G^\sigma \).
The complexified Lie algebra \({\mathfrak {g}}^{\mathbb {C}}\) decomposes into eigenspaces of the differential at the base point \(\sigma _*: {\mathfrak {g}}\rightarrow {\mathfrak {g}}\) as
The eigenvalues of \(\sigma _*\) are 1 on \({\mathfrak {h}}^{\mathbb {C}}\), \(\mathrm {j}:=e^{\frac{2\pi \mathrm {i}}{3}}\) on \({\mathfrak {m}}^+\) and \(\mathrm {j}^2={{\bar{\mathrm {j}}}}=e^{\frac{4\pi \mathrm {i}}{3}}\) on \({\mathfrak {m}}^-\), respectively. M then carries a natural G-invariant almost complex structure J with \(\pm \mathrm {i}\)-eigenspaces \({\mathfrak {m}}^\pm \), given by
at the base point, where \({\mathfrak {m}}^{\mathbb {C}}={\mathfrak {m}}^+\oplus {\mathfrak {m}}^-\). Furthermore, M is a reductive homogeneous space, since \({\mathfrak {m}}\) is invariant under the adjoint action of \(H\subset G^\sigma \).
When endowed with a G-invariant Riemannian metric g compatible with J, (M, g) is called a Riemannian 3-symmetric space. In particular, (M, g, J) is almost Hermitian. Furthermore, the almost Hermitian structure (M, g, J) is nearly Kähler if and only if (M, g) is naturally reductive [4, Prop. 3.8].
For an extensive treatment of 3-symmetric spaces, see [4]. The final thing we need for our purposes is the key observation [4, Prop. 3.5] that on a Riemannian 3-symmetric space, the canonical reductive connection \(\nabla ^{\mathrm {red}}\) associated with \({\mathfrak {m}}\) coincides with the canonical Hermitian connection \({{\bar{\nabla }}}\) defined in Sect. 2.1.
3 Small Lichnerowicz Eigenvalues on Gray Manifolds
Throughout what follows, let (M, g, J) be a Gray manifold. In order to classify destabilizing directions for the Einstein–Hilbert functional, we need to find all tt-eigentensors of the Lichnerowicz Laplacian to eigenvalues smaller than the critical eigenvalue \(2E\). That is, we want to solve the system
in \(h\in {\mathscr {S}}^2_0\) for some \(\lambda <2E=10\). We follow the discussion in [14, Sect. 5] to transform (L1) into an eigenvalue problem for the more familiar Hodge-deRham Laplacian. Viewed as a section of \({\text {Sym}}TM\), the tensor h splits into \(h=h^++h^-\) with \(h^+\in {\mathscr {S}}^+_0\) and \(h^-\in {\mathscr {S}}^-\). By applying the bundle isomorphisms given in (1), we obtain tensors \(\varphi :=h^+\circ J\in \varOmega ^{1,1}_{0,{\mathbb {R}}}\) and \(\sigma :=h^-_*\varPsi ^+\in \varOmega ^{2,1}_{0,{\mathbb {R}}}\) carrying the information of h.
Lemma 3.1
Under the isomorphisms \(h^+\mapsto \varphi \) and \(h^-\mapsto \sigma \) above, if \(\lambda <16\), the system of equations (L1) is equivalent to
Proof
Using the formulae from Prop. 3.4 and Cor. 4.4 of [14], the first equation of (L1) can be rewritten as
with \(s\in {\mathscr {S}}^-\) defined by \(s_*\varPsi ^+=2\delta h^+\wedge \omega +4d\varphi \). We note that \((\delta h^-\lrcorner \varPsi ^++\delta \sigma )\circ J\) is necessarily a traceless symmetric 2-tensor, hence automatically \(\delta h^-\lrcorner \varPsi ^++\delta \sigma \in \varOmega ^{1,1}_{0,{\mathbb {R}}}\), or, equivalently,
Using that \({{\bar{\nabla }}}^*{{\bar{\nabla }}}+q({{\bar{R}}})\) preserves the spaces \({\mathscr {S}}^\pm \) and \(\varOmega ^{2,0}_{\mathbb {R}}\), we can write (L1) equivalently as
Let now \(\eta \in \varOmega ^1\) such that \((\delta \sigma )_{2,0}=\eta \lrcorner \varPsi ^+\). Since \(\delta h^+=-J\delta \varphi \) and \({{\bar{\nabla }}}^*{{\bar{\nabla }}}+q({{\bar{R}}})\) commutes with the bundle isomorphisms from (1), we can apply them to obtain
Using the remaining formulae in Cor. 3.5 and Cor. 4.4 of [14], we see that this is equivalent to
Suppose now that \(\lambda <16\). By applying \(\delta \) to the first line of the above, it follows that
The Lichnerowicz estimate \(\varDelta \ge 2q(R)=2E=10\) on coclosed 1-forms now implies that \(\delta \varphi =0\) and hence \(\eta =0\), simplifying the above to (L2).Footnote 1\(\square \)
The space of solutions is described by the following lemma, which is a generalization of [14, Lem. 5.2].
Lemma 3.2
Suppose that \(\lambda =10-\varepsilon \) in system (L2) for some \(\varepsilon >0\). Denote with \(E(\mu ):=\ker (\varDelta -\mu )\big |_{\varOmega ^{(1,1)}_{0,{\mathbb {R}}}}\cap \ker \delta \) the \(\mu \)-eigenspace of \(\varDelta \) on coclosed primitive (1, 1)-forms.
-
(i)
Suppose that \(\varepsilon <\frac{25}{4}\) and \(\varepsilon \ne 6\). Then the space of solutions to system (L2) is isomorphic to the direct sum \(E(\mu _1)\oplus E(\mu _2)\oplus E(\mu _3)\) where \(\mu _{1,2}=7-\varepsilon \pm \sqrt{25-4\varepsilon }\) and \(\mu _3=6-\varepsilon \). The isomorphism is given by
$$\begin{aligned} \varPsi :\ (\varphi ,\sigma )\mapsto ((3-\sqrt{25-4\varepsilon })\varphi +\delta \sigma ,(3+\sqrt{25-4\varepsilon })\varphi +\delta \sigma ,*d\sigma ). \end{aligned}$$The inverse is given by
$$\begin{aligned} \varPhi :\ (\alpha ,\beta ,\gamma )\mapsto \left( \frac{\beta -\alpha }{2\sqrt{25 -4\varepsilon }},\frac{d\beta -d\alpha }{2(6-\varepsilon )\sqrt{25-4\varepsilon }} +\frac{d\alpha +d\beta }{2(6-\varepsilon )}-\frac{*d\gamma }{6-\varepsilon }\right) . \end{aligned}$$If \(6<\varepsilon <\frac{25}{4}\), then \(E(\mu _3)\) becomes trivial and thus \(\gamma =*d\sigma =0\).
-
(ii)
If \(\varepsilon =6\), then the space of solutions to (L2) is isomorphic to \(E(2)\oplus \ker \varDelta \big |_{\varOmega ^3}\), with isomorphism given by
$$\begin{aligned} \varPsi _6:\ (\varphi ,\sigma +\tau )\mapsto (\varphi ,\tau )=\left( -\frac{1}{4} \delta \sigma ,\tau \right) \end{aligned}$$for any \(\sigma \in {\text {im}}\varDelta \big |_{\varOmega ^3}\) and \(\tau \in \ker \varDelta \big |_{\varOmega ^3}\), and inverse
$$\begin{aligned} \varPhi _6:\ (\varphi ,\tau )\mapsto \left( \varphi ,-2d\varphi +\tau \right) . \end{aligned}$$ -
(iii)
If \(\varepsilon =\frac{25}{4}\), then the space of solutions to (L2) is isomorphic to \(E(\frac{3}{4})\), with isomorphism given by
$$\begin{aligned} \varPsi _\frac{25}{4}:\ (\varphi ,\sigma )\mapsto \varphi =-\frac{1}{3}\delta \sigma \end{aligned}$$and inverse
$$\begin{aligned} \varPhi _\frac{25}{4}:\ \varphi \mapsto \left( \varphi ,-4d\varphi \right) . \end{aligned}$$ -
(iv)
If \(\varepsilon >\frac{25}{4}\), then the space of solutions to (L2) is trivial.
Proof
The proof of the first part works completely analogously to the one of [14, Lem. 5.2]. We observe that if \((\varphi ,\sigma )\) is a solution to (L2), then
The eigenvalues of the matrix A are \(\mu _{1,2}=7-\varepsilon \pm \sqrt{25-4\varepsilon }\) with corresponding eigenvectors \(v_{1,2}=({\begin{matrix}3\mp \sqrt{25-4\varepsilon }\\ 1\end{matrix}})\), if \(\varepsilon \ne \frac{25}{4}\).
Let \((\alpha ,\beta ,\gamma ):=\varPsi (\varphi ,\sigma )\). Then
If \(\varepsilon <\frac{25}{4}\), then \(25-4\varepsilon >0\) and we can always recover the data \((\varphi ,\delta \sigma )\) from \((\alpha ,\beta ,\gamma )\). We also have
and thus
Hence if \(\varepsilon \ne 6\), we can also recover \(\sigma \). In total, \(\varPsi \) is invertible, and one can check that its inverse is given by \(\varPhi \).
If \(6<\varepsilon <\frac{25}{4}\), then \(\mu _3<0\) and since \(\varDelta \) is nonnegative, \(E(\mu _3)\)=0.
Let \(\varepsilon =6\). If \((\varphi ,\sigma )\) is a solution to (L2), then
i.e. \(d\sigma \) is harmonic. But this implies that \(\delta d\sigma =0\) and hence \(d\sigma =0\). Also, \(2\varphi +\delta \sigma \in E(2)\) and \(4\varphi +\delta \sigma \in E(0)\). At the same time,
Since both E(2) and \({\text {im}}\delta \) are orthogonal to E(0), it follows that \(4\varphi +\delta \sigma =0\). Now
Furthermore \(d\sigma =0\) and \(\sigma \bot \ker \varDelta \big |_{\varOmega ^3}\) imply that \(\sigma \in {\text {im}}d\big |_{\varOmega ^2}\). Since
for any \(\eta \in \varOmega ^2\), it follows that \(\sigma =-2d\varphi \). One can check that \((\varphi ,-2d\varphi +\tau )\) solves (L2) for any \(\varphi \in E(2)\) and \(\tau \in \ker \varDelta \big |_{\varOmega ^3}\). Note that \(\ker \varDelta \big |_{\varOmega ^3}\subset \varOmega ^{2,1}_{0,{\mathbb {R}}}\) by a theorem of Verbitsky [21, Thm. 6.2].
Let \(\varepsilon =\frac{25}{4}\). In this case, \(\mu _1=\mu _2\) and the matrix A is not diagonalizable. If we set
then we obtain the system
For any \(\beta \in E(\frac{3}{4})\), we therefore need to solve \((\varDelta -\frac{3}{4})\alpha '=\beta '\). But \({\text {im}}(\varDelta -\frac{3}{4})\bot \ker (\varDelta -\frac{3}{4})\), hence there can only exist a solution if \(\beta '=0\). In this case, we obtain \(\varDelta \alpha '=\frac{3}{4}\alpha '\). Furthermore, \(*d\sigma \in E(-\frac{1}{4})\). Since \(\varDelta \) is nonnegative, this implies that \(d\sigma =0\). We can recover \(\sigma \) from \(\alpha '\) by
and \(\varphi \) by \(3\varphi +\delta \sigma =\beta '=0\). In total, the space of solutions is isomorphic to \(E(\frac{3}{4})\).
Let \(\varepsilon >\frac{25}{4}\). Then \(\mu _1,\mu _2\) are imaginary and \(\mu _3<0\). Since \(\varDelta \) is nonnegative, \(E(\mu _i)=0\) for \(i=1,2,3\). The mapping \(\varPsi \) is still an isomorphism, hence the space of solutions is trivial. \(\square \)
Remark 3.3
Note that in case (i) of the above lemma, the eigenvalues \(\mu _i\) are subject to the bounds \(\mu _1<12\), \(\mu _2<2\) and \(\mu _3<6\) if we assume that \(\varepsilon >0\). In the critical case where we set \(\varepsilon =0\), we recover the description
by Moroianu and Semmelmann [14, Thm. 5.1].
Recall from Sect. 2.4 that a naturally reductive Riemannian 3-symmetric space (G/H, g) carries a nearly Kähler structure whose Hermitian connection \({{\bar{\nabla }}}\) coincides with the canonical reductive connection \(\nabla ^{\mathrm {red}}\) of the homogeneous structure. In fact, these assumptions hold for all of the homogeneous Gray manifolds, namely \(S^6\), \(S^3\times S^3\), \({{\mathbb {C}}}{{\mathbb {P}}}^3\) and the flag manifold \(F_{1,2}\).
The Hermitian Laplace operator \({{\bar{\varDelta }}}\) therefore coincides with the standard Laplacian \(\varDelta ^{\mathrm {red}}\). In light of Sect. 2.3, this enables us to describe the eigenspaces of \({{\bar{\varDelta }}}\) in terms of irreducible complex representations of G.
However, the statement of Lemma 3.2 involves the eigenspaces of \(\varDelta \) restricted to the subspace of coclosed forms in \(\varOmega ^{1,1}_{0,{\mathbb {R}}}\). We note that \({{\bar{\varDelta }}}=\varDelta \) on \(\varOmega ^{1,1}_0\cap \ker \delta \) follows from combining Cor. 3.5 and Cor. 4.5 of [14]. Thus it suffices to first search for small eigenvalues of \({{\bar{\varDelta }}}\) on \(\varOmega ^{1,1}_0\). To single out the coclosed elements in an eigenspace, we are going to perform explicit computations utilizing the following lemma. A similar formula for the divergence on symmetric tensors has already been employed to decide the stability of certain symmetric spaces, cf. [17, Lem. 3.3] and [20, Sect. 2].
Lemma 3.4
Let \((M=G/H,g,J)\) be a homogeneous Gray manifold with reductive decomposition \({\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {m}}\) as in Sect. 2.4. Let \(\delta : \varOmega ^{1,1}_0\rightarrow \varOmega ^1_{\mathbb {C}}\) denote the codifferential. Its prototypical differential operator is given by
for any orthonormal basis \((e_i)\) of \({\mathfrak {m}}\).
Proof
By (5), the Ambrose–Singer connection \(\nabla ^{\mathrm {red}}={{\bar{\nabla }}}\) translates into a directional derivative on \(C^\infty (G,\varLambda ^{1,1}_0{\mathfrak {m}})\). Fix some \(\gamma \in {{\hat{G}}}\), \(v\in V_\gamma \) and homomorphism \(F\in {\text {Hom}}_H(V_\gamma ,\varLambda ^{1,1}_0{\mathfrak {m}})\), and let \(\alpha \in \varOmega ^{1,1}_0\) be associated with \(v\otimes F\) via the equivalence (3). Differentiating the smooth function
defined in (4), we find that, for any \(x\in G\) and \(X\in {\mathfrak {m}}\cong T_oM\),
Let \((e_i)\) denote some local orthonormal frame of TM. It follows from [14, Lem. 4.2] that
(essentially using that \({{\bar{\nabla }}}\) has skew torsion). Combining the above, we obtain
at the base point. The assertion now follows from the G-invariance of \(\delta \). \(\square \)
Remark 3.5
Recall that plugging harmonic 2- and 3-forms into the bundle isomorphisms (1) yields destabilizing directions for any Gray manifold [18]. Moreover, on a homogeneous Gray manifold \(M=G/H\), harmonic 2- and 3-forms are always G-invariant. Indeed, \(\ker \varDelta \big |_{\varOmega ^2}\subset \varOmega ^{1,1}_{0,{\mathbb {R}}}\) and \(\ker \varDelta \big |_{\varOmega ^3}\subset \varOmega ^{2,1}_{0,{\mathbb {R}}}\) by [21, Thm. 6.2]. Since harmonic forms are coclosed, Cor. 3.5 and Cor. 4.5 of [14] imply that these are also harmonic for the Hermitian Laplace operator \({{\bar{\varDelta }}}\). In the homogeneous case, this means they lie in the kernel of \({\text {Cas}}^G_\ell \), thus showing G-invariance. Since the isomorphisms (1) are G-equivariant, it follows that the destabilizing directions obtained from this construction are themselves G-invariant. As we will see in the following section, there are no other destabilizing directions, thus the coindex coincides with the G-invariant coindex in each case.
4 Case-By-Case Stability Analysis
4.1 Nearly Kähler \(S^3\times S^3\)
Let \(K={\text {SU}}(2)\) with Lie algebra \({\mathfrak {k}}=\mathfrak {su}(2)\), let \(G=K\times K\times K\) with Lie algebra \({\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {k}}\oplus {\mathfrak {k}}\) and let \(H=\varDelta K\subset G\) be the diagonal, with Lie algebra \({\mathfrak {h}}\cong {\mathfrak {k}}\). We consider the homogeneous space \(M=G/H\). Let \(B_{\mathfrak {k}}\) denote the Killing form of \({\mathfrak {k}}\). The inner product on \({\mathfrak {g}}\) that is given by \(-\frac{1}{12}(B_{\mathfrak {k}}\oplus B_{\mathfrak {k}}\oplus B_{\mathfrak {k}})\) defines a normal Riemannian metric g on M, which has scalar curvature \({\text {scal}}_g=30\). The automorphism
that cyclically permutes the factors is of order three, fixes H and hence gives M the structure of a Riemannian 3-symmetric space.
We denote by \(E={\mathbb {C}}^2\) the standard representation of \(K={\text {SU}}(2)\). Furthermore, we label the irreducible complex representations of K by \(k\in {\mathbb {N}}_0\), where \(V_k={\text {Sym}}^kE\) is the unique \((k+1)\)-dimensional irreducible complex representation of K.
Lemma 4.1
Let \(V_{\gamma }\) be an irreducible complex representation of G with \({\text {Cas}}^G_\gamma <12\) and
Then \(V_\gamma \) is equivalent to one of the representations \(E\otimes E\otimes {\mathbb {C}}\), \(E\otimes {\mathbb {C}}\otimes E\) and \({\mathbb {C}}\otimes E\otimes E\) of G. In any of those cases,
and the Casimir eigenvalue is \({\text {Cas}}_\gamma ^G=9\).
Proof
Since irreducible representations of \(G=K\times K\times K\) are precisely the threefold tensor products of irreducible representations of K, we can label them by
where \(a,b,c\in {\mathbb {N}}_0\). Restricting the representation \(V_{(a,b,c)}\) to the diagonal \(H\subset G\) simply yields the tensor product \(V_a\otimes V_b\otimes V_c\) as a representation of K. The Clebsch–Gordan rules allow us to decompose these into irreducible summands. From [13, (31)], we know that the Casimir eigenvalues of G with respect to the inner product \(-\frac{1}{12}(B_{\mathfrak {k}}\oplus B_{\mathfrak {k}}\oplus B_{\mathfrak {k}})\) are given by
for \(a,b,c\in {\mathbb {N}}_0\). The results for the first few Casimir eigenvalues are listed in Table 1.
By [13, Lem. 5.5], we know that \(\varLambda ^{1,1}_0{\mathfrak {m}}\cong V_4\oplus V_2\) as a representation of K. Comparing summands now yields that the only irreducible complex representations \(V_\gamma \) of G with Casimir eigenvalue smaller than 12 and nontrivial \({\text {Hom}}_H(V_\gamma ,\varLambda ^{1,1}_0{\mathfrak {m}})\) are \(V_{(1,1,0)}\), \(V_{(1,0,1)}\) and \(V_{(0,1,1)}\). \(\square \)
Since \({{\bar{\varDelta }}}\) acts as the Casimir operator, this means that 9 is the only eigenvalue smaller than 12 in the spectrum of \({{\bar{\varDelta }}}\) on \(\varOmega ^{1,1}_0\). The eigenvalue 12 itself does also occur on \(\varOmega ^{1,1}_0\), but in [13] it is shown that the corresponding eigenforms are not coclosed, hence proving that (M, g) has no infinitesimal Einstein deformations. It now remains to check whether this is the case for the eigenforms to the eigenvalue 9.
Lemma 4.2
The eigenspace of \({{\bar{\varDelta }}}\) on \(\varOmega ^{1,1}_{0,{\mathbb {R}}}\) to the eigenvalue 9 contains no nontrivial coclosed forms.
Proof
We explicitly calculate the codifferential on the summands in question using the formula from Lemma 3.4.
Lemma 4.1 tells us that the relevant summands of the left-regular representation on \(\varOmega ^{1,1}_0\) are \(V_{(1,1,0)}\), \(V_{(1,0,1)}\) and \(V_{(0,1,1)}\). First, the representation \(V_{(1,1,0)}=E\otimes E\otimes {\mathbb {C}}\) is given by
for any \(k_1,k_2,k_3\in K\) and \(v_1,v_2\in E\). Recall that \({\mathfrak {m}}^{\mathbb {C}}={\mathfrak {m}}^+\oplus {\mathfrak {m}}^-\), where \({\mathfrak {m}}^+\) is the eigenspace of \(\sigma _*\) to the eigenvalue \(\mathrm {j}=-\frac{1}{2}+\frac{3}{2}\mathrm {i}\), and \({\mathfrak {m}}^-\) is the eigenspace to \(\mathrm {j}^2=-\frac{1}{2}-\frac{3}{2}\mathrm {i}\). Explicitly,
Let \((Y_1,Y_2,Y_3)\) be an orthonormal basis of \({\mathfrak {k}}\) with respect to the inner product \(-B_{{\mathfrak {k}}}\). Then
constitute an orthonormal basis of \({\mathfrak {m}}^{\mathbb {C}}\) with respect to \(-\frac{1}{12}(B_{\mathfrak {k}}\oplus B_{\mathfrak {k}}\oplus B_{\mathfrak {k}})\). With respect to the basis
we can represent \(\rho _*(X_i)\) by the \(4\times 4\)-matrices
This works similarly for \(\overline{X_i}\) by means of simply replacing the symbol \(\mathrm {j}\) with \(\mathrm {j}^2\).
Turning to the decomposition of \(\varLambda ^{1,1}{\mathfrak {m}}\) and \(E\otimes E\) into K-irreducible summands, we have
with common summand \(\varLambda ^2{\mathfrak {k}}^{\mathbb {C}}\cong {\mathfrak {k}}^{\mathbb {C}}\cong V_2={\text {Sym}}^2E\). If we choose the basis of the image of \(\varLambda ^2{\mathfrak {k}}^{\mathbb {C}}\) in \(\varLambda ^{1,1}{\mathfrak {m}}\) as
then a generator of \({\text {Hom}}_K(E\otimes E,\varLambda ^{1,1}_0{\mathfrak {m}})\) is represented by the matrix
with respect to \({\mathcal {B}}\) and \({\mathcal {B}}'\). Taking
as a basis of \({\mathfrak {m}}^{\mathbb {C}}\), we compute \(\delta (F)\) according to Lemma 3.4:
with respect to the bases \({\mathcal {B}}\) and \({\mathcal {B}}''\). We have thus shown that
does not vanish. For the other two representations \(V_{(1,0,1)}\) and \(V_{(0,1,1)}\) modeled on the vector space \(E\otimes E\) with the same Casimir eigenvalue, the only thing that changes is the action of G, amounting to cyclic permutations of the factors \(1,\mathrm {j},\mathrm {j}^2\) in (8). The computations work out analogously. We conclude that the eigenspace of \({{\bar{\varDelta }}}\) on \(\varOmega ^{1,1}_0\) to the eigenvalue 9 contains no nonzero coclosed forms. \(\square \)
It remains to apply Lemma 3.2 in order to finally obtain the desired result.
Proposition 4.3
On the nearly Kähler manifold \(S^3\times S^3\), the space of destabilizing directions for its Einstein metric g consists solely of the 2-dimensional \(\varDelta _L\)-eigenspace to the eigenvalue 4, the latter arising from harmonic 3-forms. In total, the coindex of g is 2.
Proof
We recall the bounds \(\mu _1<12\), \(\mu _2<2\) and \(\mu _3<6\) from Lemma 3.2. Lemmas 4.1 and 4.2 imply that \(E(\mu )\) is trivial for all \(\mu <12\). However, 3.2, (ii) yields a space of solutions isomorphic to the space of harmonic 3-forms in the case \(\varepsilon =6\). Since \(b_3=2\), this gives us a 2-dimensional subspace of \({\mathscr {S}}^2_{\mathrm {tt}}\) such that
for any element h. \(\square \)
Remark 4.4
As seen in Remark 3.5, the destabilizing directions, which come from harmonic 3-forms, are G-invariant. Conversely, all G-invariant traceless variations of the metric are destabilizing. Since \({\mathfrak {m}}\cong \mathfrak {su}(2)\oplus \mathfrak {su}(2)\), Schur’s Lemma implies that
Thus,\({\mathscr {S}}^2_0(M)^G\) is already exhausted by the two-dimensional space of destabilizing directions, meaning that \(S^3\times S^3\) is G-strongly unstable in the sense of [10].
4.2 Nearly Kähler \({{\mathbb {C}}}{{\mathbb {P}}}^3\)
Let \(G={\text {SO}}(5)\) and \(H={\text {U}}(2)\). We consider H embedded into G via the natural inclusions \({\text {U}}(2)\subset {\text {SO}}(4)\subset {\text {SO}}(5)\). The normal Riemannian metric g induced by \(-\frac{1}{12}B_{{\mathfrak {g}}}\) on the homogeneous space \(M=G/H\) is the nearly Kähler metric on \({{\mathbb {C}}}{{\mathbb {P}}}^3\), normalized to \({\text {scal}}_g=30\). It should be noted that (M, g) is, again, naturally reductive and 3-symmetric, with reductive complement \({\mathfrak {m}}={\mathfrak {h}}^\perp \).
Let \({\mathfrak {t}}=\{{\text {diag}}(\mathrm {i}\theta _1,\mathrm {i}\theta _2)\,|\,\theta _1,\theta _2\in {\mathbb {R}}\}\subset {\mathfrak {h}}\subset {\mathfrak {g}}\) be the maximal torus Lie algebra. The positive roots \(\alpha _i\in {\mathfrak {t}}^*\) of G can then be expressed as
In terms of root spaces of G, we have
The almost complex structure J can be defined by specifying its \(\pm \mathrm {i}\)-eigenspaces
In passing, we note that the standard complex structure on \({{\mathbb {C}}}{{\mathbb {P}}}^3\) has \({\mathfrak {m}}^+={\mathfrak {g}}^{\alpha _1}\oplus {\mathfrak {g}}^{\alpha _2}\oplus {\mathfrak {g}}^{\alpha _3}\).
We label the irreducible complex representations \(V_\gamma \) of G by their highest weights \(\gamma =(a,b)\), where \(a,b\in {\mathbb {N}}_0\), \(a\ge b\). For example, \(V_{(1,0)}={\mathbb {C}}^5\) is the complexified standard representation of G, while \(V_{(1,1)}=\mathfrak {so}(5)^{\mathbb {C}}\) is the complexified adjoint representation.
Again, let \(E={\mathbb {C}}^2\) be the standard representation of \({\text {SU}}(2)\). Let furthermore \({\mathbb {C}}_k\) denote the representation of \({\text {U}}(1)\) on \({\mathbb {C}}\) defined by
for any \(k\in {\mathbb {Z}}\). The irreducible complex representations of \(H\cong ({\text {SU}}(2)\times {\text {U}}(1))/{\mathbb {Z}}_2\) are then given by \(E^a_b:={\text {Sym}}^aE\otimes {\mathbb {C}}_b\) for \(a\in {\mathbb {N}}_0\) and \(b\in {\mathbb {Z}}\), \(a\equiv b\mod 2\).
Lemma 4.5
Let \(V_{\gamma }\) be an irreducible complex representation of G with \({\text {Cas}}^G_\gamma <12\) and
Then \(V_\gamma \) is equivalent to either the trivial representation \(V_{(0,0)}\) or the standard representation \(V_{(1,0)}\). In both cases,
and the Casimir eigenvalues are \({\text {Cas}}_{(0,0)}^G=0\) and \({\text {Cas}}_{(1,0)}^G=8\), respectively.
Proof
We first work out how to decompose the restriction of \(V_{(1,0)}\) to H into irreducible summands. We know that \(V_{(1,0)}={\mathbb {C}}^5\) is the complexified standard representation of G. The inclusion \({\text {U}}(2)\subset {\text {SO}}(4)\) can be understood as realification \((E^1_1)^{\mathbb {R}}\) of the defining representation \(E^1_1\) of \({\text {U}}(2)\). Furthermore, the inclusion \({\text {SO}}(4)\subset {\text {SO}}(5)\) defines a five-dimensional real representation \({\mathbb {R}}^4\oplus {\mathbb {R}}\) of \({\text {SO}}(4)\), where the group acts as on its defining representation on the first summand and trivially on the second. In total, the restriction of the real standard representation of \(G={\text {SO}}(5)\) to \(H={\text {U}}(2)\) is given by \((E^1_1)^{\mathbb {R}}\oplus {\mathbb {R}}\). Complexifying then yields the decomposition
For the branching of \(V_{(1,1)}\) under the restriction to H, we refer to [13, Lem. 5.9]. Using the decomposition
of G-representations, the known branchings of \(V_{(1,1)}\) and \(V_{(1,0)}\) as well as the Clebsch–Gordan rules, we can also work out the branching of \(V_{(2,0)}\) to H, although it will not be needed hereafter.
By [13, (33)], we know that on the irreducible representation \(V_{(a,b)}\) of G, the Casimir eigenvalue of G with respect to the inner product \(-\frac{1}{12}B_{{\mathfrak {g}}}\) is given by
Table 2 lists the results for the smallest few Casimir eigenvalues of G.
[13, Lem. 5.8] tells us that
as a representation of \(H={\text {U}}(2)\). By comparing summands, we conclude that \(V_{(0,0)}\) and \(V_{(1,0)}\) are the only irreducible complex representations \(V_\gamma \) of G with Casimir eigenvalue smaller than 12 and nontrivial \({\text {Hom}}_H(V_\gamma ,\varLambda ^{1,1}_0{\mathfrak {m}})\). \(\square \)
Again, \({{\bar{\varDelta }}}\) acts as the Casimir operator. We therefore know that 0 and 8 are the only eigenvalues smaller than 12 in the spectrum of \({{\bar{\varDelta }}}\) on \(\varOmega ^{1,1}_0\). As in the case \(S^3\times S^3\), the eigenvalue 12 does occur on \(\varOmega ^{1,1}_0\), but the corresponding eigenforms are not coclosed (see [13]). Hence, (M, g) has no infinitesimal Einstein deformations. It remains to check whether the eigenforms to the eigenvalues 0 and 8 are coclosed.
Lemma 4.6
The eigenspace of \({{\bar{\varDelta }}}\) on \(\varOmega ^{1,1}_{0,{\mathbb {R}}}\) to the eigenvalue 0 consists of coclosed forms, while the eigenspace to the eigenvalue 8 contains no nontrivial coclosed forms.
Proof
The eigenspace of \(\varDelta \) to the eigenvalue 0 corresponds to the trivial summand in the left-regular representation, i.e. to G-invariant elements of \(\varOmega ^{1,1}_0\). But these are parallel with respect to the Ambrose–Singer connection (which equals the canonical Hermitian connection \({{\bar{\nabla }}}\)). Recall that
by [14, Lem. 4.2]. It follows any element in the 0-eigenspace of \({{\bar{\varDelta }}}\) on \(\varOmega ^{1,1}_0\) is coclosed.
For the eigenspace to the eigenvalue 8, we again make use of the formula from Lemma 3.4 in an explicit calculation. Lemma 4.5 tells us that the relevant summand of the left-regular representation on \(\varOmega ^{1,1}_0\) is \(V_{(1,0)}\). A generator F of the one-dimensional space \({\text {Hom}}_H(V_{(1,0)},\varLambda ^{1,1}_0{\mathfrak {m}})\) must map the trivial summand \(E^0_0\) in the H-representation
to the trivial summand in
The former is spanned by \(v_5\), where \((v_1,\ldots ,v_5)\) is the standard basis of \(V_{(1,0)}={\mathbb {C}}^5\). To describe the latter, we remark that the root spaces in decomposition (9) can be written as
in terms of the basis
of \({\mathfrak {m}}\subset \mathfrak {so}(5)\), where \(E_{ij}\) denotes the \(5\times 5\)-matrix with 1 at position (i, j) and zero at all other entries. Note that under the inner product \(-\frac{1}{12}B_{{\mathfrak {g}}}\), the basis \((\sqrt{2}e_i,f_j)\) is orthonormal. It follows from the definition of J in terms of \({\mathfrak {m}}^\pm \) that the Kähler form can be written as
By [13, (32)], the root space \({\mathfrak {g}}^{-\alpha _3}\subset {\mathfrak {g}}^{\mathbb {C}}\) is H-invariant and equivalent to \(E^0_{-2}\). By conjugation, \({\mathfrak {g}}^{\alpha _3}\cong E^0_2\). Hence, H acts trivially on \({\mathfrak {g}}^{-\alpha _3}\otimes {\mathfrak {g}}^{\alpha _3}\cong E^0_{-2}\otimes E^0_2=E^0_0\). Recall that \({\mathfrak {g}}^{-\alpha _3}\subset {\mathfrak {m}}^+\) and \({\mathfrak {g}}^{\alpha _3}\subset {\mathfrak {m}}^-\). It follows that \(f_{12}=\frac{\mathrm {i}}{2}(f_1+\mathrm {i}f_2)\wedge (f_1-\mathrm {i}f_2)\) spans a trivial subspace of \(\varLambda ^{1,1}{\mathfrak {m}}={\mathfrak {m}}^+\wedge {\mathfrak {m}}^-\). Since \(\varLambda ^{1,1}_0{\mathfrak {m}}\) is the orthogonal complement of \(\omega \) in \(\varLambda ^{1,1}{\mathfrak {m}}\), the remaining trivial summand in \(\varLambda ^{1,1}_0{\mathfrak {m}}\) must be spanned by
Having found the H-trivial subspaces of \(V_{(1,0)}\) and \(\varLambda ^{1,1}_0{\mathfrak {m}}\), we see that the space \({\text {Hom}}_H(V_{(1,0)},\varLambda ^{1,1}_0{\mathfrak {m}})\) is spanned by
Observe that \(f_jv_k\perp v_5\) for all j, k and \(e_iv_k\perp v_5\) for all i, k, except for
Hence, by Lemma 3.4,
for any \(X\in {\mathfrak {m}}^{\mathbb {C}}\). We have thus shown that
is not the zero map—in fact, it maps \(\delta (F)=F'\), where
Thus, the eigenspace of \({{\bar{\varDelta }}}\) on \(\varOmega ^{1,1}_0\) to the eigenvalue 8 contains no nonzero coclosed forms. \(\square \)
As before, the desired result follows from an application of Lemma 3.2.
Proposition 4.7
On the nearly Kähler manifold \({{\mathbb {C}}}{{\mathbb {P}}}^3\), the space of destabilizing directions for its Einstein metric g consists solely of the 1-dimensional \(\varDelta _L\)-eigenspace to the eigenvalue 6, arising from harmonic 2-forms. Consequently, the coindex of g is 1.
Proof
Lemmas 4.5 and 4.6 imply that \(E(\mu )\) from Lemma 3.2 is trivial for all \(\mu <12\) except if \(\mu =0\). We have already seen that E(0) consists of harmonic forms. In fact, [21, Thm. 6.2] implies that all harmonic 2-forms on M lie in E(0). If we solve
for \(\varepsilon \), we obtain \(\varepsilon =5\pm 1\); from \(\mu _3=6-\varepsilon =0\) we obtain \(\varepsilon =6\). If \(\varepsilon =6\), i.e. in case (ii) of Lemma 3.2, the space of solutions to (L2) is isomorphic to \(E(2)\oplus \ker \varDelta \big |_{\varOmega ^3}\). Since 2 does not appear in the spectrum on \(\varOmega ^{1,1}_0\) and \(b_3({{\mathbb {C}}}{{\mathbb {P}}}^3)=0\), this space is trivial. The only possibility in which E(0) contributes to the space of solutions of (L2) is in case (i) of Lemma 3.2 with \(\varepsilon =4\). Since \(b_2=1\), we obtain a 1-dimensional subspace of \({\mathscr {S}}^2_{\mathrm {tt}}\) on which
for any of its elements h. \(\square \)
4.3 The Flag Manifold \(F_{1,2}\)
Let \(G={\text {SU}}(3)\) and \(H=T^2\), the latter embedded into G via
The homogeneous space \(M=G/H\) is a description of the manifold \(F_{1,2}\) of flags in \({\mathbb {C}}^3\). As in the previous examples, we endow M with the normal Riemannian metric g induced by \(-\frac{1}{12}B_{\mathfrak {g}}\), which has scalar curvature \({\text {scal}}_g=30\). The Riemannian homogeneous space (M, g) is naturally reductive, 3-symmetric and hence nearly Kähler. For more information on the nearly Kähler structure, see [13, Sect. 5.6].
Denote by \(E={\mathbb {C}}^3\) the standard representation of G. Any irreducible complex representation of G can then be described as the Cartan summand \(V_{(k,l)}\) of the tensor product \({\text {Sym}}^kE\otimes {\text {Sym}}^l{{\bar{E}}}\) for some \(k,l\in {\mathbb {N}}_0\). For example, \(V_{(1,1)}\) is equivalent to the complexified adjoint representation \(\mathfrak {su}(3)^{\mathbb {C}}\) of G.
Lemma 4.8
Let \(V_{\gamma }\) be an irreducible complex representation of G with \({\text {Cas}}^G_\gamma <12\) and
Then \(V_\gamma \) is the trivial representation and \(\dim {\text {Hom}}_H({\mathbb {C}},\varLambda ^{1,1}_0{\mathfrak {m}})=2\).
Proof
It follows from [13, (35)] that the Casimir eigenvalue on the irreducible representation \(V_{(k,l)}\) of G with respect to the inner product \(-\frac{1}{12}B_{\mathfrak {g}}\) is given by
By analyzing the weights of the respective representations, it has already been checked in [13, Sect. 5.6] that
and \(\dim {\text {Hom}}_H(V_{(1,1)},\varLambda ^{1,1}_0{\mathfrak {m}})=4\).
The maximal subspace of \(\varOmega ^{1,1}_0\) on which G acts trivially is precisely the kernel of the Casimir operator. We can argue exactly as in the proof of Lemma 4.6 that the elements of this space are coclosed. Furthermore, we recall that \(\varDelta ={{\bar{\varDelta }}}\) on coclosed primitive (1, 1)-forms. Since \({{\bar{\varDelta }}}\) acts as the Casimir operator, we are simply talking about the subspace of harmonic forms in \(\varOmega ^{1,1}_0\). By Verbitsky’s theorem [21, Thm. 6.2], all harmonic 2-forms lie in \(\varOmega ^{1,1}_0\). Thus, the multiplicity of the trivial representation \({\mathbb {C}}\) in \(\varOmega ^{1,1}_0\) is \(b_2(F_{1,2})=2\). By Frobenius reciprocity, this is also the dimension of \({\text {Hom}}_H({\mathbb {C}},\varLambda ^{1,1}_0{\mathfrak {m}})\).
The results of the above discussion are listed in Table 3. \(\square \)
We have thus shown that 0 is the only eigenvalue smaller than 12 in the spectrum of \({{\bar{\varDelta }}}\) on \(\varOmega ^{1,1}_0\). Besides, it has been proven in [13, Sect. 6] that an 8-dimensional subspace of the 32-dimensional eigenspace to the eigenvalue 12 consists of coclosed forms and hence yields infinitesimal Einstein deformations of (M, g). We will describe these more explicitly in Sect. 5.
Proposition 4.9
On the nearly Kähler manifold \(F_{1,2}\), the space of destabilizing directions for its Einstein metric g consists solely of a 2-dimensional \(\varDelta _L\)-eigenspace to the eigenvalue 6, arising from harmonic 2-forms. In total, the coindex of g is 2.
Proof
One last time, we want to apply Lemma 3.2. By Lemma 4.8 and the fact that harmonic forms are coclosed, we see that for \(\mu <12\), the eigenspace \(E(\mu )\) is only nontrivial if \(\mu =0\). With the same reasoning as in the proof of Prop. 4.7, we conclude that (L2) can be solved for \(\varepsilon =4\), yielding a 2-dimensional subspace of \({\mathscr {S}}^2_{\mathrm {tt}}\) such that
for all its elements h. \(\square \)
Remark 4.10
The space of destabilizing directions (or of harmonic 2-forms) can be described rather explicitly. The 3-dimensional space
of H-invariant elements of \(\varLambda ^{1,1}{\mathfrak {m}}\) corresponds to the space \((\varOmega ^{1,1})^G\) of G-invariant (1, 1)-forms on \(F_{1,2}\) and is spanned by
using the notation introduced in Sect. 5. The Kähler form \(\omega \) corresponding to the strict nearly Kähler structure on \(F_{1,2}\) is given by
The 2-dimensional space \(\varLambda ^{1,1}_0{\mathfrak {m}}^H\cong (\varOmega ^{1,1}_0)^G=\ker \varDelta \big |_{\varOmega ^2}\) responsible for the destabilizing directions is now the orthogonal complement of \({{\hat{\omega }}}\) in \(\varLambda ^{1,1}{\mathfrak {m}}^H\).
The twistor space over \({{\mathbb {C}}}{{\mathbb {P}}}^2\) can be identified with \(F_{1,2}\) in three distinct ways, giving rise to three fibrations \(F_{1,2}\rightarrow {{\mathbb {C}}}{{\mathbb {P}}}^2\), which in turn induce six almost complex structures on \(F_{1,2}\). Three of them are actually integrable with respective Kähler forms \(\omega _1,\omega _2,\omega _3\), described by
while the other three coincide with the almost complex structure with Kähler form \(\omega \). See [15, Sect. 3.2.3] for a detailed description.
Thus, in light of the construction in Sect. 3 using the isomorphism
the destabilizing directions of the nearly Kähler metric g on \(F_{1,2}\) can be viewed as coming from variations of \(\omega =g(J\cdot ,\cdot )\) in the directions of \(\omega _1,\omega _2,\omega _3\) while fixing J.
Alternatively, consider the canonical variation, i.e., change of scale of fiber against base, on each of the three aforementioned fibrations
In [23, Prop. 4.4], it is shown that these variations yield destabilizing tt-tensors. General destabilizing directions \(h\in {\mathscr {S}}^2_{\mathrm {tt}}\) are thus (at the base point) of the form
with \(t_1+t_2+t_3=0\), where each of the pairwise orthogonal subspaces \({\mathfrak {m}}_j\) is the vertical tangent space with respect to \(\pi _j\). Since \(\pi _j\) are Riemannian submersions with totally geodesic fibers, the destabilizing directions are Killing tensors by [7, Ex. 7.3], that is, they satisfy the Killing equationFootnote 2
5 Rigidity of \(F_{1,2}\)
5.1 The Infinitesimal Einstein Deformations of \(F_{1,2}\)
We will utilize the explicit description of the infinitesimal Einstein deformations of \(F_{1,2}\) given in [13, Sect. 6]. For this, it is helpful to represent \(M=F_{1,2}\) as a quotient of \(G={\text {U}}(3)\) by the diagonally embedded torus \(H=T^3\).
Denote by \(E_{ij}\) the \(3\times 3\)-matrix with a 1 at position (i, j) and zero entries elsewhere. Let \(\{h_1,h_2,h_3,e_1,\ldots ,e_6\}\) be the basis of \({\mathfrak {g}}={\mathfrak {u}}(3)\) given by
Note that \(\{h_1,h_2,h_3\}\) span the Lie algebra \({\mathfrak {h}}\subset {\mathfrak {g}}\), while the reductive complement \({\mathfrak {m}}\subset {\mathfrak {g}}\) is spanned by \(\{e_1,\ldots ,e_6\}\). We now define the inner product \(\langle \cdot ,\cdot \rangle \) on \({\mathfrak {g}}\) (and the induced bi-invariant metric on G) in such a way that \((e_i,\sqrt{2}h_j)\) is an orthonormal system. One easily checks that this coincides with \(-\frac{1}{12}B_{\mathfrak {su}(3)}\) when restricted to \(\mathfrak {su}(3)\subset {\mathfrak {g}}\), so we recover the same metric g on \(F_{1,2}\).
The space \(\varepsilon (g)\) of infinitesimal Einstein deformations of g is equivalent to \(\mathfrak {su}(3)\) via the following procedure. For a fixed element \(\xi \in \mathfrak {su}(3)\subset {\mathfrak {g}}\), let \(\xi ^*\in C^\infty (G,{\mathfrak {g}})\) be given by \(\xi ^*(x)={\text {Ad}}(x)\xi \). This defines smooth, real-valued functions \(x_1,x_2,x_3,v_1,\ldots ,v_6\) on G via
As before, we identify sections in a tensor bundle \(E=G\times _H V\) over M with H-equivariant functions on G with values in V and denote this by
As seen in Remark 4.10, the Kähler form \(\omega \in \varOmega ^2\) corresponds to the (constant) function
(we write \(e_{ij}=e_i\wedge e_j\) to shorten notation). Define a real-valued function \({\hat{\varphi \in }} C^\infty (G,\varLambda ^2{\mathfrak {m}})\) by
Using the description of the Kähler form via \({{\hat{\omega }}}\) and the fact that \(v_1+v_2+v_3=0\), it is easy to check that \({\hat{\varphi _\xi }}\) is in fact \(\varLambda ^{1,1}_0\)-valued.
The functions \(v_i\in C^\infty (G)\) are H-invariant since
and \({\text {ad}}(h_j)h_i=[h_j,h_i]=0\), hence \(dv_i(h_j)=0\). Using the commutator relations of \({\mathfrak {u}}(3)\), one can check that the 2-forms \(e_{12},e_{34},e_{56}\in \varLambda ^2{\mathfrak {m}}\) are H-invariant as well. This implies that the function \({\hat{\varphi _\xi }}\) is H-equivariant. In total, \({\hat{\varphi _\xi \in }} C^\infty (G,\varLambda ^{1,1}_{0,{\mathbb {R}}}{\mathfrak {m}})^H\) and thus \({\hat{\varphi _\xi }}\) projects to a primitive (1, 1)-form \(\varphi _\xi \) on M. The coclosedness of \(\varphi _\xi \) has also been checked in [13, Sect. 6].
It is worth noting that in the language of harmonic analysis on homogeneous spaces, \(\varphi _\xi \) is associated with the element
where the Fourier coefficient F is given by
It is therefore no surprise that \({{\bar{\varDelta }}}\varphi _\xi =12\varphi _\xi \), since 12 is the eigenvalue of the Casimir operator on \(V_{(1,1)}=\mathfrak {su}(3)^{\mathbb {C}}\) (see Table 3). The fact that each tensor \(\varphi _\xi \) thus obtained is coclosed amounts to \(\delta (F)=0\), where \(\delta \) also denotes the prototypical differential operator
associated with the invariant differential operator \(\delta : \varOmega ^{1,1}_{0,{\mathbb {R}}}\rightarrow \varOmega ^1\).
The corresponding symmetric 2-tensor, which is the actual infinitesimal Einstein deformation, is now given as \(h_\xi =-J\circ \varphi _\xi \). By composing \({\hat{\varphi _\xi }}\) with \({\hat{\omega _\xi }}\), we obtain
In this way, each \(\xi \in \mathfrak {su}(3)\) determines a unique element \(\varphi _\xi \in \varOmega ^{1,1}_{0,{\mathbb {R}}}\) and hence a unique \(h_\xi \in \varepsilon (g)\).
In passing, we note that the infinitesimal Einstein deformations of \(F_{1,2}\) in fact coincide with the infinitesimal deformations of the nearly Kähler structure [13, Cor. 5.12]. Their nonintegrability in the nearly Kähler sense was already established [5]. We now turn to the question whether the integrability in the Einstein sense is also obstructed.
5.2 The Obstruction Against Integrability
We first note that via the equivalence \(\varepsilon (g)\cong \mathfrak {su}(3)\) constructed in Sect. 5.1, the integrability obstruction to second order
can be viewed as a G-equivariant multilinear map that is symmetric in the first two entries, i.e. \({\mathcal {I}}\in ({\text {Sym}}^2\mathfrak {su}(3)^*\otimes \mathfrak {su}(3)^*)^G\). Both of the spaces
turn out to be one-dimensional and hence equal—in particular, \({\mathcal {I}}\) must be totally symmetric. Hence \(\left( {\mathcal {E}}_g''(h,h),k\right) _{L^2}\) can be recovered from expressions of the type \({\mathcal {I}}(h,h,h)\) via polarization. Concretely,
for \(h,k\in \varepsilon (g)\).
The space \({\text {Sym}}^3\mathfrak {su}(3)^G\) is generated by the G-invariant cubic homogeneous polynomial \(\mathrm {i}\det \). We therefore know that \({\mathcal {I}}(h_\xi ,h_\xi ,h_\xi )=c\cdot \mathrm {i}\det (\xi )\) for some \(c\in {\mathbb {R}}\). Next, we proceed to show that \(c\ne 0\).
Introducing the notation \(\alpha ^\sigma (X_1,\ldots ,X_r):=\alpha (X_{\sigma (1)},\ldots ,X_{\sigma (r)})\) for any permutation \(\sigma \in {\mathfrak {S}}_r\) and any tensor \(\alpha \) of rank r, we can rewrite formula (2) as
Integrating by parts and computing
with some local orthonormal frame \((f_i)\) of TM, we obtain
for \(h\in \varepsilon (g)\), where \(I_0,I_1,I_2\in C^\infty (M)\) are defined by
The functions \(I_0,I_1,I_2\) on M give rise to H-invariant functions \({{\hat{I}}}_0,{{\hat{I}}}_1,{{\hat{I}}}_2\in C^\infty (G)^H\), the first of which can already be easily computed:
using that \(v_1+v_2+v_3=0\). In order to obtain the other two terms, we have to compute derivatives of h. Recall that the canonical Hermitian connection \({{\bar{\nabla }}}\) and the Levi-Civita connection \(\nabla \) are related by
where \(A_X=J\circ (\nabla _XJ)\) on TM and then extended as a derivation to tensors of arbitrary rank. Identifying 2-forms with skew-symmetric endomorphisms of TM, we can also write \(A_X=X\lrcorner \varPsi ^-\), where \(\varPsi ^-\in \varOmega ^3\) is the imaginary part of the complex volume form of M, which is G-invariant and at the base point given by
(see also [13, Sect. 6]).
The canonical horizontal distribution \({\mathcal {H}}\subset TG\) is spanned by the left-invariant vector fields \(e_1,\ldots ,e_6\). For any vector \(X\in TM\), let \({\tilde{X}}\in {\mathcal {H}}\) denote its horizontal lift. Since \({{\bar{\nabla }}}\) is the Ambrose–Singer connection of the homogeneous space \(M=G/H\), it follows from (5) that
We compute
Secondly, it follows from the G-invariance of A thatFootnote 3
Using the above expression for \(\varPsi ^-\), we compute
To obtain \(\nabla h\), we simply combine:
The coefficients of \(\widehat{\nabla h}\in C^\infty (G,{\mathfrak {m}}^{\otimes 3})\) with respect to the basis \((e_i)\) are listed in Table 4.
Now, we can finally tackle the terms \(I_1\) and \(I_2\) in the integrability obstruction. Let \((f_i)\) be a local orthonormal frame of TM. Then
By the G-invariance of the Riemannian metric on M, it follows that
Note that \(({{\hat{f}}}_i(x))\) forms an orthonormal basis of \({\mathfrak {m}}\) at each point \(x\in G\). Since the above expression is independent of the choice of orthonormal basis, we can substitute in the orthonormal basis \((e_i)\) of \({\mathfrak {m}}\). Hence the above is equal to
Similarly, we have
Plugging in the coefficients from Table 4, we obtain
One can check that these functions are indeed H-invariant and thus project to functions \(I_0,I_1,I_2\) on M. Recall that \({\text {scal}}_g=30\), whence \(E=5\). Subsuming the above results, the full integrability obstruction is given by
Integrating over G amounts to projecting the integrand to its G-invariant, i.e., constant, part. If we view \(v_1,\ldots ,x_6\) as linear forms on \(\mathfrak {su}(3)\), the integrand I is an H-invariant cubic homogeneous polynomial in \(\mathfrak {su}(3)^*\). Recall that the inner product on \(\mathfrak {su}(3)\) is induced by \(-\frac{1}{12}B\), and
We therefore have the relations \(\langle x_i,x_j\rangle =\langle e_i,e_j\rangle =\delta _{ij}\) as well as \(\langle x_i,v_j\rangle =\langle e_i,h_j\rangle =0\), while
The generator \(\mathrm {i}\det \) of \({\text {Sym}}^3\mathfrak {su}(3)^G\) can be written as
The inner product on \({\text {Sym}}^k\mathfrak {su}(3)\) is induced by the inner product on \(\mathfrak {su}(3)\) via
We therefore see that
and hence \({\mathcal {I}}(h,h,h)=c\cdot \mathrm {i}\det (h)\) for some \(c\ne 0\).
Suppose now that \(\det (\xi )=0\) for some nonzero \(\xi \in \mathfrak {su}(3)\). By equation (10), \({\mathcal {E}}_g''(h_\xi ,h_\xi )\) is orthogonal to \(\varepsilon (g)\) if and only if \(\xi \) is a critical point of \(\det \), i.e.,if
for all \(\eta \in \mathfrak {su}(3)\). Equivalently, the rank of the complex \(3\times 3\)-matrix \(\xi \) is equal to 1. However, no such element of \(\mathfrak {su}(3)\) exists, since nonzero skew-Hermitian matrices have rank at least 2.
This concludes the proof of Theorem 1.2.
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Notes
This bridges the gap in the proof of [14, Thm. 5.1], where \(\eta =0\) was assumed without justification.
This can be verified directly, using the description of \(\nabla _X-{{\bar{\nabla }}}_X\) in Sect. 5.2 and the fact that h is G-invariant and hence \({{\bar{\nabla }}}\)-parallel.
We write \(\alpha \odot \beta =\alpha \otimes \beta +\beta \otimes \alpha \) for \(\alpha ,\beta \in {\mathfrak {m}}\).
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Schwahn, P. Coindex and Rigidity of Einstein Metrics on Homogeneous Gray Manifolds. J Geom Anal 32, 302 (2022). https://doi.org/10.1007/s12220-022-01061-4
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DOI: https://doi.org/10.1007/s12220-022-01061-4