Bochner Formulae for Orthogonal G-structures on Compact Manifolds

Gil Bor & Luis Hernández Lamoneda ^{∗} Centro de Investigación en Matemáticas (CIMAT), A.P. 402, Guanajuato 36000, Gto., Mexico

September 13, 2000

Abstract A general technique is introduced for deriving Bochner type formulae on a compact riemannian manifold, relating its curvature tensor with the intrinsic torsion of a compatible (orthogonal) G-structure. The technique is illustrated for the groups G = U_{n}, SU_{n}, G_{2} and Spin_{7}, with various applications of the derived formulae in these cases.

1 Introduction

Given an n-dimensional smooth manifold M and a subgroup G ⊂ GL_{n}(ℝ), a G-structure on M consists of a reduction of its structure group to G. For example, for G = O_{n} (the orthogonal group), this amounts to giving a riemannian metric on M. If we further reduce to a subgroup G ⊂ O_{n}, we then say that we have an orthogonal G-structure (a structure which is compatible with the riemannian metric). For example, for G = U_{n} ⊂ O_{2n} (the unitary group), this amounts to an almost-hermitian structure (an hermitian inner product at each tangent space).

An important invariant of a G-structure, playing a key role in Cartan’s “method of equivalence” (classification of G-structures), is its intrinsic torsion tensor. It is a first order invariant measuring “flatness” (localintegrability) of the structure and its vanishing is equivalent to the existence of a torsionless G-connection on T M. See for example [Bryant2] for further information.

In this article we study a general scheme for obtaining a curvature obstruction to the existence of orthogonal G-structures on compact riemannian manifolds. This obstruction comes in the form of an integral formula relating the G-irreducible components of the intrinsic torsion tensor of the G-structure with G-invariants of the curvature tensor of the associated riemannian metric. We illustrate this technique for the groups G = U_{n} (n ≥ 2), SU_{n} (n ≥ 3), G_{2} and Spin_{7} (on manifolds of dimensions 2n, 2n, 7 and 8 resp.; SU_{2} “belongs” to the Sp_{n} series).

For the first two groups (U_{n} and SU_{n}) our technique recaptures and extends several known results about almost-hermitian structures [Falcitelli-Farinola-Salamon], [Gauduchon], [Sekigawa-Vanhecke]. For the other two groups (G_{2} and Spin_{7}) the results are apparently new.

We have also applied the method to the groups Sp_{k}Sp_{1} and Sp_{k}, but the calculations are more complicated and we leave these cases for a forthcoming article.

Here are some typical applications of the formulas derived in this paper:

• Let (M, g) be a compact, conformally flat manifold with a compatible (orthogonal) almost-complex structure J, let ω = g(J·, ·) be the corresponding “Kähler form” and denote by s the scalar curvature. Then

(a) If J is integrable, then ∫_{M} s ≥ 0, with equality if and only if d^{∗}ω = 0.

(b) If ω is symplectic (dω = 0) then ∫_{M} s ≤ 0, with equality if and only if J is integrable (i.e. the structure is Kähler).

∗The authors acknowledge support from grant 28491-E of CONACyT

Moreover, in dimension 4 the result remains true for anti-self-dual manifolds (now the vanishing of the total scalar curvature is equivalent, in both cases, to the structure being Kähler).

See Proposition 2 (section 3.6 below). Part (a) was proved in [Gauduchon].

• Let M = Γ\G/K be a compact locally symmetric space with an orthogonal almost-complex structure J. Suppose that M is of non-compact type and that J is integrable, or that M is of compact type and the structure is symplectic. Then, in either case, the structure is in fact Kähler and descends from one of the 2^{k} G-invariant Kähler structures on G/K, where G is a product of k simple groups.

See Proposition 4 (section 3.6 below). Again, this extends results of [Gauduchon] and [Hernández].

• Let (M, g) be a compact riemannian manifold of dimension ≥ 6 with a compatible complex structure. Let iκ be the curvature of the Levi-Civita connection on the canonical bundle of M and s^{∗} the ∗-scalar curvature. Then

∫_{M} ∥∇ω∥^{2} = ∫_{M} (2⟨κ, ω⟩ − s^{∗}).

In particular, such a manifold is Kähler if and only if ∫ s^{∗} = 2 ∫ ⟨κ, ω⟩.

See Corollary 1 (section 4.8 below). Although we have not found an explicit reference to this result in the literature, it might follow from results in [Wood].

• Let M be a compact manifold of dimension 7 with a G_{2}-structure. Then its intrinsic torsion decomposes into four G_{2}-irreducible components, τ = τ_{1} + τ_{7} + τ_{14} + τ_{27} (indexed by the dimension of the irreducible subspace they belong to), satisfying

∫_{M} 6∥τ_{1}∥^{2} + 5∥τ_{7}∥^{2} − ∥τ_{14}∥^{2} − ∥τ_{27}∥^{2} = \frac{2}{3} ∫ s,

where s is the scalar curvature. Thus, for example, a compact riemannian manifold with a calibrated G_{2}-structure (dϕ = 0, where ϕ is the “fundamental” 3-form, or equivalently, τ_{1} = τ_{7} = τ_{27} = 0) has non-positive total scalar curvature.

See Corollary 3 (section 5.5 below).

The rest of the article is organized as follows. The next section describes the technique we use, then each section treats in turn one of the four groups G = U_{n}, SU_{n}, G_{2} and Spin_{7}, with a few applications of the formulae obtained in each case.

2 The Technique

Let G ⊂ O_{n} be the stabilizer subgroup of a p-form ϕ ∈ Λ^{p}(V^{∗}) on V = ℝ^{n} (p = 2, m, 3, 4 for G = U_{m}, SU_{m} (n = 2m), G_{2}, Spin_{7} resp.). Define a map Λ^{2} ⊗ Λ^{p} → Λ^{p}, α ⊗ ϕ ↦ α · ϕ, given by

(θ ∧ θ′) · ϕ = θ′ ∧ [int(θ ⊗ ϕ)] − θ ∧ [int(θ′ ⊗ ϕ)], θ, θ′ ∈ Λ^{1}, ϕ ∈ Λ^{p},

where int : Λ^{1} ⊗ Λ^{p} → Λ^{p−1} is “interior product” (contraction). Considering Λ^{2} as the Lie algebra 𝔬_{n} of O_{n}, this map is just the infinitesimal pull-back action of 𝔬_{n} on Λ^{p}; thus, the Lie algebra 𝔤 of G is the kernel of ·ϕ : α ↦ α · ϕ. Note also that for α, β ∈ Λ^{2}, α · β = −β · α, reflecting the fact that · corresponds to the Lie bracket under the identification of Λ^{2} with 𝔬_{n}.

Now let M be an n-dimensional manifold with a G-structure and denote also by ϕ the associated p-form on M. The intrinsic torsion of such a G-structure can be identified with the covariant derivative ∇ϕ ∈ Λ^{1} ⊗ Λ^{p}. In fact,

LEMMA 1 ∇ϕ lies in the sub-bundle W = Λ^{1} ⊗ (𝔤^{⊥} · ϕ) ⊂ Λ^{1} ⊗ Λ^{p}. In particular, W ≅ Λ^{1} ⊗ 𝔤^{⊥}.

This is well-known. See for example [Salamon].

Remark: We will use repeatedly the natural correspondence between G-representations and their associated bundles on a manifold with a G-structure. Thus, V = ℝ^{n} corresponds to the tangent bundle, V^{∗} to the cotangent bundle, Λ^{p}(V^{∗}) to Λ^{p}M (so we can abbreviate safely both by Λ^{p}), an invariant subspace of a Grepresentation corresponds to a sub-bundle, a fixed element corresponds to a section of the associated bundle, etc.

Next, we decompose orthogonally the intrinsic torsion space W = Λ^{1} ⊗ (𝔤^{⊥} · ϕ), and consequently ∇ϕ, into G-irreducible components. These components carry interesting geometric information about the G-structure. For example, for G = U_{n} (almost-hermitian structure), ∇ϕ has 4 irreducible components, the sum of certain two of them measuring the integrability of the associated almost-complex structure (the Nijenhuis tensor), the sum of certain three of them representing the local triviality of the corresponding almost-symplectic structure (the exterior derivative of the Kähler form), etc. See [Gray-Hervella] for a complete description of the “16 types of U_{n}-structures”.

Now we apply to ϕ the following integral formula, holding for any p-form on a compact Riemannian manifold (this follows, for example, from formulae WF I and WF II of [Wu, pp. 305-306]):

∫_{M} ∥dϕ∥^{2} + ∥d^{∗}ϕ∥^{2} − p!∥∇ϕ∥^{2} = ∫_{M}⟨R̃ϕ, ϕ⟩.  (1)

Here R̃ is the endomorphism on p-forms induced by the riemann curvature tensor R ∈ Λ^{2} ⊗ Λ^{2} by applying (twice) the action of Λ^{2} on Λ^{p}; thus, in components, with respect to a local frame of 1-forms {θ_{j}},

R̃ϕ = ∑_{i<j} R_{ij} · [(θ_{i} ∧ θ_{j}) · ϕ].

Remark: We use the convention for the curvature tensor R = ∑_{i<j} R_{ij} ⊗ (θ_{i} ∧ θ_{j}) with R_{ij} = [∇_{e_{i}}, ∇_{e_{j}}] −

∇_{[e_{i},e_{j}]} (an anti-symmetric endomorphism interpreted as a 2-form), where {e_{i}} is a frame dual to the θ_{i}.

Note that if ∇ϕ vanishes (a “torsion-free” structure, or G−holonomy) one has R ∈ 𝔤 ⊗ 𝔤, hence R̃ϕ = 0.

We will also make use of a “twisted” version of this formula (in the U_{n} case) which looks exactly the same except that ϕ is a p-form on M with values in some vector bundle E with connection, where all objects (∇, d, R, …) are substituted by their twisted version ([Wu, p. 430]).

Next, dϕ and d^{∗}ϕ can be expressed in terms of ∇ϕ, dϕ = alt(∇ϕ) and d^{∗}ϕ = −int(∇ϕ), where alt : Λ^{1} ⊗ Λ^{p} → Λ^{p+1} is alternation (extrior product). These are G-equivariant operations, so on the left hand side of formula (1) we obtain some quadratic form in the G-irreducible components of ∇ϕ. On the right hand side, the integrand is a G-invariant of the curvature tensor, so can be expressed in terms of the standard invariants, like the scalar curvature.

In executing the above plan for a given G, we make use of some simple techniques of representation theory, summarized in the following two lemmas.

LEMMA 2 Suppose that the intrinsic torsion space W, as a G-representation, is multiplicity-free, i.e., does not contain G-isomorphic irreducible subspaces (this occurs for all G except SU_{n}, in this article) and let W = W_{1}⊕W_{2}⊕… be the (unique) decomposition of W into mutually orthogonal, G-irreducible subspaces. Then there exist non-negative constants a_{i}, b_{i} such that for any w_{i} ∈ W_{i}, ∥alt(w_{i})∥^{2} = a_{i}∥w_{i}∥^{2} and ∥int(w_{i})∥^{2} = b_{i}∥w_{i}∥^{2}. Hence, if we decompose ∇ϕ into its irreducible components, ∇ϕ = ∑_{i}(∇ϕ)_{i} then

∥dϕ∥^{2} = ∑_{i} a_{i}∥(∇ϕ)_{i}∥^{2} and ∥d^{∗}ϕ∥^{2} = ∑_{i} b_{i}∥(∇ϕ)_{i}∥^{2}.

This is essentially Schur’s Lemma.

For the next lemma, recall that on a Riemannian manifold, the curvature operator R is just the interpretation of the curvature tensor R as an endomorphism on 2-forms,

R(α) = − ∑_{i<j}⟨R_{ij}, α⟩θ_{i} ∧ θ_{j}, α ∈ Λ^{2},

(with the sign convention so as to make sure that R is a positive operator for the round sphere…).

LEMMA 3 Assume that the G-representation 𝔤^{⊥} ⊂ Λ^{2} is multiplicity-free (this occurs for all four G studied in this article) and let 𝔤^{⊥} = V_{1} ⊕ V_{2} ⊕ … be the (unique) decomposition into G-irreducible mutually orthogonal subspaces. Let c_{j} > 0 be the homothety factors of ·ϕ : V_{j} → W and denote by tr (R, V_{j}) the trace of the “jj-block of R”, i.e. the trace of the endomorphism given by the restriction to V_{j} of the curvature operator R, followed by orthogonal projection onto V_{j}. Then the integrand on the right-hand side of formula (1) is given by

⟨R̃ϕ, ϕ⟩ = ∑_{j} c_{j} tr (R, V_{j}).

Proof. The decomposition Λ^{2} = V_{0} ⊕ V_{1} ⊕ V_{2} ⊕ …, V_{0} = 𝔤, induces a “block” decomposition of the curvature tensor, R = ∑_{i,j} R_{ij}, R_{ij} ∈ V_{i} ⊗ V_{j}. We claim that all but the R_{jj} terms, j ≥ 1, are in the kernel of the map R ↦ ⟨R̃ϕ, ϕ⟩. Indeed, if R = α ⊗ β, with either α or β ∈ 𝔤, then ⟨R̃ϕ, ϕ⟩ = ⟨α · (β · ϕ), ϕ⟩ = −⟨β · ϕ, α · ϕ⟩ = 0, since the map α· : Λ^{p} → Λ^{p} is skew-symmetric (being the derivative of an isometric O_{n}-action), and 𝔤 is the kernel of ·ϕ : Λ^{2} → Λ^{p}. Similarly, if R = α ⊗ β with α ∈ V_{i}, β ∈ V_{j}, with i and j distinct and ≥ 1, then again ⟨R̃ϕ, ϕ⟩ = −⟨β·ϕ, α·ϕ⟩ = 0, since V_{i}·ϕ and V_{j}·ϕ, being irreducible and non-isomorphic, are orthogonal subspaces of Λ^{p}. Finally, if R = α ⊗ β with α, β ∈ V_{j}, j ≥ 1, then ⟨R̃ϕ, ϕ⟩ = −⟨β · ϕ, α · ϕ⟩ = −c_{j}⟨β, α⟩ = c_{j} tr (R, V_{j}).

In the next four sections we illustrate the technique explained above for the groups G = U_{n}, SU_{n}, G_{2} and Spin_{7}: decomposition of W and 𝔤^{⊥}, calculation of the constants a_{i}, b_{i} and the curvature invariants c_{j} tr (R, V_{j}), thus computing formula (1). The SU_{n} case is somewhat of an exception because Lemma 2 does not apply to it.

3 U_{n}

U_{n} is the stabilizer in O_{2n} of the “Kähler form” ω = ⟨J·, ·⟩ ∈ Λ^{1,1}, where J is the standard almost-complex structure on ℂ^{n} ≅ ℝ^{2n} (scalar multiplication by i). In terms of a unitary basis z_{1}, …, z_{n} for Λ^{1,0}, ω = i ∑_{j} z_{j} ∧ z̄_{j}. A manifold with a U_{n}-structure is often called an almost-hermitian manifold.

3.1 The decomposition of ∇ω

This was done in [Gray-Hervella] and will be reviewed here briefly.

In the decomposition Λ^{2} ⊗ ℂ = Λ^{1,1} ⊕ Λ^{2,0} ⊕ Λ^{0,2}, we have 𝔲_{n} ⊗ ℂ = Λ^{1,1} and 𝔲_{n}^{⊥} ⊗ ℂ = Λ^{2,0} ⊕ Λ^{0,2}, so that 𝔲_{n}^{⊥} is irreducible. Note that ω is the 2-form corresponding to J under the standard identification of anti-symmetric endomorphisms with 2-forms, hence for any (p, q)-form α we have

ω · α = i(q − p)α,  α ∈ Λ^{p,q}.  (2)

In particular, Λ^{2,0} and Λ^{0,2} are invariant under ω·, hence also 𝔲_{n}^{⊥}, the real part of Λ^{2,0} ⊕ Λ^{0,2}. We thus get

W ⊗ ℂ = Λ^{1} ⊗ (𝔲_{n}^{⊥} · ω) ⊗ ℂ = (Λ^{1,0} ⊕ Λ^{0,1}) ⊗ (Λ^{2,0} ⊕ Λ^{0,2}) = (Λ^{1,0} ⊗ Λ^{2,0}) ⊕ Λ^{1,2} + conj.

Now

Λ^{1,0} ⊗ Λ^{2,0} + conj. = (W_{1} ⊕ W_{2}) ⊗ ℂ,  Λ^{1,2} + conj. = (W_{3} ⊕ W_{4}) ⊗ ℂ

where

• W_{1} = real part of Λ^{3,0};

• W_{2} = real part of the image of (Λ^{1,0} ⊗ Λ^{1,0} ⊗ Λ^{1,0}) under the Young symmetrizer (1 − (23))(1 + (12));

• W_{3} = real part of the “primitive” part of Λ^{1,2} (kernel of the contraction in the first and second entries);

• W_{4} = Λ^{1} ∧ ω.

Note that for n = 1: W = 0; for n = 2: W_{1} = W_{3} = 0 but W_{2} and W_{4} are non-zero and non-isomorphic, hence orthogonal; and that for n ≥ 3 all four summands are non-zero and non-isomorphic, hence mutually orthogonal.

Denote by ∇ω = (∇ω)_{1} + (∇ω)_{2} + (∇ω)_{3} + (∇ω)_{4} the decomposition of ∇ω, where (∇ω)_{i} ∈ W_{i}. In [Gray-Hervella] one can get more information about the geometric meaning of the 4 components (∇ω)_{i} ∈ W_{i} and the vanishing of certain subgroups of the 4 components. We list here some of the better known possibilities:

Table 1: Types of U_{n}-structures

Vanishing

components of ∇ω

Name Geometric meaning

all  Kähler  local holonomy ⊂ U_{n}

1, 2 Hermitian Integrability of the almost-complex structure

1,3,4  Almost-Kähler or symplectic  dω = 0
2,3,4  Nearly Kähler  ∇ω = dω
4  Cosymplectic or balanced  d^{∗}ω = 0

3.2 The left-hand side of Formula (1)

The W_{i} are mutually distinct (as U_{n}-representations), hence, according to Lemma 2, there are constants a_{i} such that

∥dω∥^{2} = ∥alt(∇ω)∥^{2} = ∑_{i} ∥alt(∇ω)_{i}∥^{2} = ∑_{i} a_{i}∥(∇ω)_{i}∥^{2},

where a_{i} is the homothety factor of alt : W_{i} → Λ^{3}, i.e. ∥alt(w_{i})∥^{2} = a_{i}∥w_{i}∥^{2} for every w_{i} ∈ W_{i}. Similarly,

∥d^{∗}ω∥^{2} = ∑_{i} ∥int(∇ω)_{i}∥^{2} = ∑_{i} b_{i}∥(∇ω)_{i}∥^{2},

where the b_{i} are the homothety factors of int : W_{i} → Λ^{1}.

The next table summarizes the result of the computation of the homothety factors a_{i}, b_{i} and the elements w_{i} ∈ W_{i} ⊗ ℂ we used for computing them.

Table 2: Calculation of the homothety factors a_{i} and b_{i} for U_{n}

\begin{matrix}Summand & w_{i} ∈ W_{i} ⊗ ℂ & ∥w_{i}∥^{2} & ∥alt(w_{i})∥^{2} & ∥int(w_{i})∥^{2} & a_{i} & b_{i} \\ W_{1} & z_{1} ∧ z_{2} ∧ z_{3} & 1/6 & 1 & 0 & 6 & 0 \\ W_{2} & – & – & 0 & 0 & 0 & 0 \\ W_{3} & z_{1} ⊗ (z̄_{2} ∧ z̄_{3}) & 1/2 & 1 & 0 & 2 & 0 \\ W_{4} & ∑_{j=2}^{n} z_{j} ⊗ (z̄_{j} ∧ z̄_{1}) & (n − 1)/2 & n − 1 & (n − 1)^{2} & 2 & 2(n − 1) \end{matrix}

Remarks:

• To explain the zeros in the table, note that int : W → Λ^{1}, but Λ^{1} is irreducible and non-isomorphic to W_{1}, W_{2} or W_{3}, hence int(W_{1}) = int(W_{2}) = int(W_{3}) = 0. Similarly, Λ^{3} does not contain a subspace isomorphic to W_{2}, hence alt(W_{2}) = 0.

• For the remaining entries, we pick elements w_{i} in W_{i} ⊗ ℂ, i = 1, 3, 4, and compute alt(w_{i}) and int(w_{i}) and their norms. We use throughout a unitary basis z_{1}, …, z_{n} for Λ^{1,0}. Thus for example, we take w_{1} = z_{1} ∧ z_{2} ∧ z_{3}. Then alt(w_{1}) = w_{1}, hence ∥w_{1}∥^{2} = 1/6 (as a tensor), ∥alt(w_{1})∥^{2} = 1 (as a 3-form) and so a_{1} = 6.

It follows from this calculation that

∥dω∥^{2} = 6∥(∇ω)_{1}∥^{2} + 2∥(∇ω)_{3}∥^{2} + 2∥(∇ω)_{4}∥^{2}, ∥d^{∗}ω∥^{2} = 2(n − 1)∥(∇ω)_{4}∥^{2}.

3.3 The right-hand side of Formula (1) (the curvature term)

According to Lemma 3, since 𝔲_{n}^{⊥} is irreducible, ⟨R̃ω, ω⟩ = c tr (R, 𝔲_{n}^{⊥}), where c > 0 is the homothety factor of ·ω : 𝔲_{n}^{⊥} → 𝔲_{n}^{⊥}. Since for z_{1} ∧ z_{2} ∈ 𝔲_{n}^{⊥} ⊗ ℂ, (z_{1} ∧ z_{2}) · ω = 2iz_{1} ∧ z_{2}, one concludes that c = 4.

On the other hand, it is possible to express ⟨R̃ω, ω⟩ in terms of the standard U_{n}-invariants, the scalar curvature s = 2 tr (R) and the ∗−scalar curvature s^{∗} = 2 tr (JR).

LEMMA 4 ⟨R̃ω, ω⟩ = s − s^{∗}.

Proof. The splitting Λ^{2} = 𝔲_{n} ⊕ 𝔲_{n}^{⊥} implies

tr (R) = tr (R, 𝔲_{n}) + tr (R, 𝔲_{n}^{⊥}),
tr (JR) = tr (JR, 𝔲_{n}) + tr (JR, 𝔲_{n}^{⊥}).

Since J = 1 on Λ^{1,1} = 𝔲_{n} ⊗ ℂ and J = −1 on Λ^{2,0} ⊕ Λ^{0,2} = 𝔲_{n}^{⊥} ⊗ ℂ, we have that

tr (JR) = tr (RJ) = tr (R, 𝔲_{n}) − tr (R, 𝔲_{n}^{⊥}).

Subtracting, we get s − s^{∗} = 2 tr (R) − 2 tr (JR) = 4 tr (R, 𝔲_{n}^{⊥}) = ⟨R̃ω, ω⟩.

3.4 The U_{n} formula

Now denote by E_{i} the L_{2}-norm of the component (∇ω)_{i}, so that ∫ ∥∇ω∥^{2} = E_{1} + ⋯ + E_{4}. Plugging into Formula (1) the information gathered above, the formula for the U_{n} case reduces to

2E_{1} − E_{2} + (n − 1)E_{4} = \frac{1}{2} ∫ s − s^{∗}.  (3)

Note that E_{3} does not appear and that E_{1} = E_{3} = 0 for n = 2 (i.e. real 4-manifolds).

3.5 Two Homogeneous Examples

3.5.1. The 6-sphere. The sphere S^{6} with the round metric admits a homogeneous orthogonal almost-complex structure (G_{2}-invariant) which is “nearly-Kähler” (∇ω = dω, see [Gray2]). Since s = 30 and s^{∗} = 6, formula (1) gives E_{1} = 6 V ol(S^{6}); by homogeneity, ∥dω∥^{2} ≡ 6 identically.

3.5.2. Product of odd spheres. S^{2m+1} × S^{2n+1} ⊂ ℂ^{m+1} × ℂ^{n+1} with the product of standard metrics admits a compatible (U_{m+1} × U_{n+1})-homogeneous complex structure due to Hopf and Calabi-Eckmann. In fact, for n = 0 (so-called Hopf manifolds) (∇ω)_{3} = 0 as well (see [Gray-Hervella]).

Working with the definition of J, we get s − s^{∗} = 4(m^{2} + n^{2}), thus (m + n)E_{4} = 2(n^{2} + m^{2})V ol(M); by homogeneity, ∥(∇ω)_{4}∥^{2} ≡ 2(n^{2} + m^{2})/(m + n) identically .

3.6 Applications of the U_{n} formula

Some of the following applications appeared in [Gauduchon] and [Hernández], using similar techniques to ours (representation theory and integral formulae), although from a less general point of view. The literature on almost-hermitian manifolds is quite vast so it is possible that we have missed some other relevant references.

3.6.1. Complex structures and negative curvature

PROPOSITION 1 On a compact riemannian manifold of dimension ≥ 4 with a negative-definite curvature operator (R < 0) any orthogonal almost-complex structure satisfies E_{2} ≠ 0. In particular, such a manifold cannot admit an orthogonal complex structure.

Proof. Observe that the integrand on the right hand side of Formula (3) is (s − s^{∗})/2 = 2 tr (R, 𝔲_{n}^{⊥}) < 0.

In particular, this result holds for a hyperbolic manifold, since its curvature operator is −Id_{Λ^{2}} . This was also proved in [Gauduchon] and [Hernández].

Note that the compactness assumption is essential, because for the unit ball in ℂ^{n} the standard hyperbolic and complex structures are compatible.

In dimension 4 it is known that the orthogonality assumption is superfluous, i.e. a compact hyperbolic 4-manifold does not admit a complex structure (orthogonal or not), see [Kotschick]. For higher dimensions the problem is open.

Another question, open as far as we know, is that of the existence of orthogonal almost-complex structures on compact hyperbolic manifolds. The only result in this direction that we are aware of is an unpublished proof of Kotschick that such examples do exist in dimension 4.

3.6.2. Conformally flat manifolds. A conformally flat manifold (i.e. locally conformal to euclidean space) is characterized by the vanishing of its Weyl tensor (see for example the book [Besse], p. 60).

LEMMA 5 On a conformally flat almost-hermitian 2n-dimensional manifold s = (2n − 1)s^{∗}.

Proof. The space of curvature-like tensors on ℝ^{2n} with vanishing Weyl tensor is (O_{2n}-equivariantly) isomorphic to the space of quadratic forms on ℝ^{2n} (essentially the space of Ricci tensors). The latter has a 1-dimensional space of U_{n}-invariants (by the Schur lemma, since ℝ^{2n} = ℂ^{n} is an irreducible U_{n}-representation), hence s^{∗} and s are proportional. It is thus sufficient to verify the identity for a single curvature operator of the said type, say for the identity operator on Λ^{2}(ℝ^{2n}) (the curvature operator of the sphere). We leave this simple verification to the reader.

Consequently, on a conformally flat almost-hermitian compact manifold Formula (3) reduces to

2E_{1} − E_{2} + (n − 1)E_{4} = \frac{n − 1}{2n − 1} ∫_{M} s.

An immediate consequence of this is the following result.

PROPOSITION 2 Let M be a compact, conformally flat almost-hermitian manifold. Then

(a) If the structure is hermitian (i.e. J is integrable), then ∫_{M} s ≥ 0, with equality if and only if d^{∗}ω = 0;

(b) If the structure is symplectic (i.e. dω = 0) then ∫_{M} s ≤ 0, with equality if and only if the structure is Kähler.

Proof. The assumption of part (a) is that E_{1} = E_{2} = 0, thus the inequality. Vanishing of the total scalar curvature further implies that E_{4} = 0, and since ∥(∇ω)_{4}∥^{2} is a multiple of ∥d^{∗}ω∥^{2} the result follows. Part (b) is proved similarly.

Part (a) was proved in [Gauduchon]. Observe that this again shows that on a compact real hyperbolic manifold there can be no orthogonal complex structure, since a hyperbolic manifold is conformally flat with s < 0.

In dimension 4 we get a stronger result, because the last lemma remains true for the larger class of anti-self-dual manifolds. Recall that an oriented riemannian 4-manifold is said to be anti-self-dual (ASD) if W^{+} ≡ 0, where W = W^{+} ⊕ W^{−} is the decomposition of the Weyl tensor relative to the eigenspaces of the Hodge-* operator. Furthermore, in dimension 4, an hermitian manifold with d^{∗}ω = 0 is in fact Kähler (recall that E_{3} ≡ 0 in dimension 4). Hence,

PROPOSITION 3 Let M be a compact, ASD almost-hermitian 4-manifold. Then

(a) If the structure is hermitian then ∫_{M} s ≥ 0, with equality if and only if the structure is Kähler.

(b) If the structure is symplectic then ∫_{M} s ≤ 0, with equality if and only if the structure is Kähler.

Further information on the relation between hermitian geometry and the Weyl tensor may be found in [Gauduchon] and the references therein.

3.6.3. Locally symmetric spaces. Some symmetric spaces G/K admit a G-invariant orthogonal complex structure (which is in fact Kähler), unique up to a sign if G is simple. These are called the hermitian symmetric spaces. If G is a product of k simple groups then we get 2^{k} G-invariant orthogonal complex structures on G/K. We show next that on compact quotients of symmetric spaces of non-compact type (all irreducible factors are non-compact and non-euclidean) these are the only orthogonal complex structures. A dual statement for compact type spaces also holds.

PROPOSITION 4 Let M be an almost-hermitian manifold of dimension ≥ 4 which is a compact connected quotient of a symmetric space Z = G/K with k irreducible factors. Suppose that either

(a) the structure on M is hermitian and Z is of non-compact type; or,

(b) the structure on M is symplectic and Z is of compact type.

Then Z is hermitian symmetric and the structure on M descends from one of the 2^{k} G−invariant Kähler structures on Z.

Note that once again, this result implies the impossibility of an hyperbolic hermitian structure on a compact manifold of dimension ≥ 4, since (1) a hyperbolic manifold is a locally symmetric space of non-compact type, and (2) hyperbolic space is not hermitian symmetric in dimension > 2.

Part (a) was proved by [Gauduchon] when G/K is irreducible and in [Hernández] with the superfluous hypothesis that the irreducible factors of G/K should be different from the real hyperbolic plane.

For the proof given here we need, in addition to the Bochner formula (3), the following basic algebraic facts from the theory of symmetric spaces. We use the following standard notation: G/K is a symmetric space, 𝔤 and 𝔨 the Lie algebras of G and K (resp.) and 𝔭 = 𝔨^{⊥} (with respect to the Killing form of 𝔤).

LEMMA 6 Suppose J ∈ End(𝔭) is an orthogonal complex structure such that its i-eigenspace 𝔭^{1,0} ⊂ 𝔭 ⊗ ℂ is an abelian subalgebra of 𝔤 ⊗ ℂ. Then J is K-invariant, hence G/K is a hermitian symmetric space and J coincides with the value at T_{[K]}G/K ≅ 𝔭 of one of the G-invariant orthogonal complex structures on G/K.

This is essentially lemma 4.8 of [Carlson-Toledo].

LEMMA 7 Let R be the curvature operator of a symmetric space G/K of compact or non-compact type. Then for any X, Y, X′, Y′ ∈ 𝔭 ≅ T_{[K]}G/K,

• ⟨R(X ∧ Y), X′ ∧ Y′⟩ = −⟨[X, Y], [X′, Y′]⟩ in the non-compact type case;

• ⟨R(X ∧ Y), X′ ∧ Y′⟩ = ⟨[X, Y], [X′, Y′]⟩ in the compact type case.

This follows from [Helgason], theorem 4.2, p. 215.

Proof of Proposition 4. Fix a point x ∈ M and identify T_{x}M ≅ 𝔭. Next pick a unitary basis {z_{i}} for T_{x}^{1,0}M ≅ 𝔭^{1,0} and use lemma 7 above to calculate the integrand on the right hand side of Formula (3) at x:

(s − s^{∗})/2 = 2 tr (R, 𝔲_{n}^{⊥}) = 2 ∑_{i<j} ⟨R(z_{i} ∧ z_{j}), \overline{z_{i} ∧ z_{j}}⟩ = ±2 ∑_{i<j} ∥[z_{i}, z_{j}]∥^{2},

with the sign “–” in case (a) (non-compact type) and “+” in case (b) (compact type). In either case, we get different signs on the two sides of Formula (3) and conclude that 𝔭^{1,0} is an abelian subalgebra of 𝔤⊗ℂ. Lemma 6 above then implies that Z = G/K is hermitian and the value of J at x comes from one of the 2^{k} G-invariant complex structures on G/K. By continuity, J corresponds to one of the G-invariant complex structures on Z.

We single out two immediate consequences of the last proposition:

• Compact type: Proposition 4 determines all symplectic structures on ℂP^{n} (or products of them), compatible with the standard (Fubini-Study) metric.

• Non-compact type: products of compact hyperbolic manifolds of odd dimension do not admit an orthogonal complex structure (i.e. there is no negatively curved version of the Hopf-Calabi-Eckmann hermitian manifolds).

4 SU_{n}, n ≥ 3.

4.1 Definition of an SU_{n}-structure

The group SU_{n}, n ≥ 2, is the stabilizer in U_{n} of the complex volume form ψ = z_{1} ∧ … ∧ z_{n} ∈ Λ^{n,0}(V^{∗}), V = ℂ^{n} = ℝ^{2n}. Its (complexified) Lie algebra is the “primitive” part of Λ^{1,1}, i.e. the subspace orthogonal to ω. In fact,

LEMMA 8 For n ≥ 3, SU_{n} is the stabilizer of ψ in O_{2n}.

This must have appeared somewhere, but at any rate here is a sketch of a proof: first, check the claim on the Lie algebra level, i.e. α · ψ = 0 ⟹ α ∈ 𝔰𝔲_{n} (divide into cases, according to the (p, q)-type of α). Next, g^{∗}ψ = ψ ⟹ (g^{∗}ω) · ψ = ω · ψ hence g^{∗}ω − ω ∈ 𝔰𝔲_{n}. On the other hand, since 𝔰𝔲_{n} =Ker(·ψ), g^{∗}ψ = ψ ⟹ g^{∗}𝔰𝔲_{n} = 𝔰𝔲_{n} ⟹ g^{∗}𝔰𝔲_{n}^{⊥} = 𝔰𝔲_{n}^{⊥}. Since ω ∈ 𝔰𝔲_{n}^{⊥}, we get g^{∗}ω − ω ∈ 𝔰𝔲_{n}^{⊥}, hence g^{∗}ω = ω, so g ∈ U_{n}.

Furthermore, it is easy to see that SU_{n} is the stabilizer of any non-zero element in Λ^{n,0} ⊕ Λ^{0,n}; it turns out to be useful, for what follows, to choose this to be a real element, η = ψ + \overline{ψ}.

4.2 Decomposition of ∇η

Under SU_{n}, the Lie algebra of U_{n} (real part of Λ^{1,1}) decomposes as 𝔲_{n} = 𝔰𝔲_{n} ⊕ ℝω, hence 𝔰𝔲_{n}^{⊥} = ℝω ⊕ 𝔲_{n}^{⊥} (two SU_{n}-irreducible summands, distinct for n ≥ 3). It follows that

W = Λ^{1} ⊗ (𝔰𝔲_{n}^{⊥} · η) = Λ^{1} ⊗ (ω · η) ⊕ Λ^{1} ⊗ (𝔲_{n}^{⊥} · η).

Now W_{0} := Λ^{1} ⊗ (ω · η) ≅ Λ^{1} is irreducible, where

ω · η = ω · (ψ + \overline{ψ}) = −in(ψ − \overline{ψ}) = nη^{⊥}, η^{⊥} := (ψ − \overline{ψ})/i.

As for the decomposition of Λ^{1} ⊗ (𝔲_{n}^{⊥} · η) into irreducibles, we use the U_{n}-decomposition of Λ^{1} ⊗ 𝔲_{n}^{⊥} into 4 irreducibles W_{1}, …, W_{4}, and give their corresponding images the same names here (irreducible U_{n} representations remain irreducible upon restriction to SU_{n}). We thus obtain the decomposition

W = Λ^{1} ⊗ (𝔰𝔲_{n}^{⊥} · η) = W_{0} ⊕ ⋯ ⊕ W_{4}.

Remark: Note that we have two isomorphic summands: W_{0} ≅ W_{4} ≅ Λ^{1}. Thus, alt(∇η)_{0} and alt(∇η)_{4} need not be orthogonal (indeed they are not, in general), which complicates the expression for the left-hand side of Formula (1) (appearance of “mixed terms”).

4.3 The left-hand side of Formula (1)

First, note that ∗η = ±η, hence ∥dη∥ = ∥d^{∗}η∥. Thus,

∥dη∥^{2} = ∥d^{∗}η∥^{2} = ∥int(∇η)∥^{2} = ∑_{j=0}^{4} ∥int(∇η)_{j}∥^{2} + 2⟨int(∇η)_{0}, int(∇η)_{4}⟩ =

= ∑_{i=0}^{4} a_{i}∥(∇η)_{i}∥^{2} + 2⟨int(∇η)_{0}, int(∇η)_{4}⟩,

with a_{i} the homothety factor of int (or alt) in W_{j}. The next table summarizes the calculation of the a_{i}. As a shorthand notation, we use z_{1¯2} for z_{1} ∧ z̄_{2}, etc.

Table 3: Calculation of the homothety factors a_{i} for SU_{n}, n ≥ 3

\begin{matrix}Summand & w_{i} ∈ W_{i} ⊗ ℂ & ∥w_{i}∥^{2} & ∥int(w_{i})∥^{2} & a_{i} \\ W_{0} & z_{1} ⊗ η & 2/n! & 1 & n!/2 \\ W_{1} & {z_{1} ⊗ (z_{23} · η) + z_{2} ⊗ (z_{31} · \atop η) + z_{3} ⊗ (z_{12} · η)} & 6/n! & 12 & 2 · n! \\ W_{2} & z_{1} ⊗ (z_{12} · η) & 2/n! & 1 & n!/2 \\ W_{3} & z_{1} ⊗ (z¯2¯3 · η) & − & 0 & 0 \\ W_{4} & ∑_{k=2}^{n} z_{k} ⊗ (zk̄¯1 · η) & 2(n − 1)/n! & (n − 1)^{2} & (n − 1) · n!/2 \end{matrix}

Clearly, an SU_{n}-structure induces an U_{n}-structure with its Kähler form ω, thus one expects to express ∇ω in terms of ∇η. This is done in the next lemma.

LEMMA 9 For any SU_{n}-structure with Kähler form ω and volume-forms η and η^{⊥} as above, we have the orthogonal decomposition ∇η = (∇η)_{0} + … + (∇η)_{4}, with

(a) (∇η)_{0} = −θ ⊗ η^{⊥}, where θ = −⟨∇η, η^{⊥}⟩/2 = ⟨∇ψ, \overline{ψ}⟩/i.

(b) (∇η)_{i} = −(∇ω)_{i} ·η^{⊥}/2 ∈ W_{i}, i = 1, …, 4, where ·η^{⊥} : Λ^{1} ⊗Λ^{2} → Λ^{1} ⊗Λ^{n} is given by σ ⊗α ↦ σ ⊗(α·η^{⊥}).

(c) ∥(∇η)_{i}∥^{2} = ∥(∇ω)_{i}∥^{2}/n!, i = 1, …, 4.

(d) ⟨int(∇η)_{0}, int(∇η)_{4}⟩ = ⟨θ, d^{∗}ω⟩/2,

Remark: The 1-form iθ = ⟨∇ψ, \overline{ψ}⟩ has the interpretation of the connection form of the canonical bundle Λ^{n,0}, relative to its section ψ.

Proof.

(a) This is the usual projection formula on η^{⊥} (note that ∥η∥^{2} = 2).

(b) Note first that by our definition of η = ψ + ψ̄, (∇η)_{1} + … + (∇η)_{4} = (∇ψ)^{n−1,1} + conj., where (∇ψ)^{n−1,1} is the component of ∇ψ in Λ^{1} ⊗ Λ^{n−1,1}. Now start from ω · ψ = −inψ (using equation 2) and apply ∇ to get (∇ω) · ψ + ω · (∇ψ) = −in(∇ψ). By taking the (p, q)-decomposition of the last equation and applying again equation 2 to the (p, q)-components of ∇ψ, we get (∇ω) · ψ = −2i(∇ψ)^{n−1,1}. Subtracting from the last equation its conjugate, we get (∇ω) · η^{⊥} = −2[(∇ψ)^{n−1,1} + conj.], as claimed.

(c) The map ·η^{⊥} : 𝔲_{n}^{⊥} → Λ^{n} is a homothety onto its image (by Schur’s lemma, since the domain is irreducible) and the homothety factor can be calculated by checking the effect on a single element (calculation omitted).

(d) We need to make use of the identity

int(w · η^{⊥}) = −int[int(w) ⊗ η^{⊥}],  w ∈ W_{4} ⊂ Λ^{1} ⊗ Λ^{2}.  (∗)

Proof: by the O_{2n}-equivariance of int : Λ^{1}⊗Λ^{n} → Λ^{n−1}, α·int(σ⊗η^{⊥}) = int[(α·σ)⊗η^{⊥}]+int[σ⊗(α·η^{⊥})], for all σ ∈ Λ^{1}, α ∈ Λ^{2}. Now we claim that the map defined on Λ^{1} ⊗ Λ^{2} by the left-hand side of the last equation vanishes when restricted to W_{4}; by U_{n}-equivariance, it is enough to check this on a single element in W_{4} ⊗ ℂ, say Σ_{k=2}^{n}z_{k} ⊗ zk̄¯1. Indeed, Σ_{k=2}^{n}zk̄¯1 · int(z_{k} ⊗ η^{⊥}) = 0, since int(z_{k} ⊗ η^{⊥}) ∈ Λ^{0,n−1} and Λ^{0,2} acts trivially on forms of type (0, p). Now use the fact that α · σ = int(σ ⊗ α), for σ ⊗ α ∈ Λ^{1} ⊗ Λ^{2} and the identity (∗) follows. Now using int[(∇ω)_{4}] = −d^{∗}ω (see the U_{n} section) and item (b) (for i = 4), we conclude that int[(∇η)_{4}] = −int(d^{∗}ω ⊗ η^{⊥})/2. Combined with item (a), we have

⟨int(∇η)_{0}, int(∇η)_{4}⟩ = \frac{1}{2}⟨int(θ ⊗ η^{⊥}), int(d^{∗}ω ⊗ η^{⊥})⟩.

Now the map Λ^{1} → Λ^{n−1}, σ ↦ int(σ ⊗ η^{⊥}) is SU_{n}−equivariant with an irreducible domain, hence a homothety onto its image, so it is enough to check its effect on a single element, say z_{1}, giving ∥int(z_{1} ⊗ η^{⊥})∥ = ∥int(z_{1} ⊗\overline{ψ})∥ = ∥z̄_{2} ∧…∧z̄_{n}∥ = 1, i.e. we have an isometry, and so ⟨int(θ⊗η^{⊥}), int(d^{∗}ω ⊗η^{⊥})⟩ = ⟨θ, d^{∗}ω⟩, as required.

LEMMA 10

∫ ⟨int(∇η)_{0}, int(∇η)_{4}⟩ = \frac{1}{2} ∫ ⟨κ, ω⟩,

where iκ = d⟨∇ψ, \overline{ψ}⟩ is the curvature 2-form of the connection induced by the Levi-Civita connection on Λ^{n,0}.

Proof. Integration by parts of item (d) of the previous lemma,

4.4 The right hand side of Formula (1) (the curvature term)

LEMMA 11 ⟨R̃η, η⟩ = (s + s^{∗})/2.

Proof. According to lemma 3 of the introduction,

⟨R̃η, η⟩ = c_{1} tr (R, ℝω) + c_{2} tr (R, 𝔲_{n}^{⊥}),

with c_{1} = ∥ω · η∥^{2}/∥ω∥^{2} and c_{2} = ∥α · η∥^{2} for a unitary α ∈ Λ^{2,0}. We compute the terms appearing in this formula:

• c_{1}: we have ω · η = −in(ψ − \overline{ψ}), so ∥ω · η∥^{2} = 2n^{2}, ∥ω∥^{2} = n, hence c_{1} = 2n.

• c_{2}: for α = z_{1} ∧ z_{2}, ∥α∥^{2} = 1 and α · η = α · \overline{ψ} = (α · ᾱ) ∧ z̄_{3} ∧ … ∧ z̄_{n} = (z_{1} ∧ z̄_{1} + z_{2} ∧ z̄_{2}) ∧ z̄_{3} ∧ … ∧ z̄_{n}, so ∥α · η∥^{2} = 2 hence c_{2} = 2.

• tr (R, 𝔲_{n}^{⊥}): in the U_{n} section we found tr (R, 𝔲_{n}^{⊥}) = (s − s^{∗})/4.

• tr (R, ℝω): using the Bianchi identity,

⟨Rω, ω⟩ = ∑_{j,k}⟨Rz_{jk̄}, z_{j̄k}⟩ + ∑_{j,k}⟨Rz_{jk}, zk̄j̄⟩ = ∑_{j,k}⟨RJz_{jk̄}, \overline{z_{jk̄}}⟩ + ∑_{j,k}⟨RJz_{jk}, \overline{z_{jk}}⟩ =
= tr (RJ, Λ^{1,1}) + 2 tr (RJ, Λ^{2,0}) = tr (RJ) = s^{∗}/2,

hence tr (R, ℝω) = ⟨Rω, ω⟩/∥ω∥^{2} = s^{∗}/2n.

Combining all the above we get the desired result.

4.5 The SU_{n} formula

Putting together all the information gathered so far in this section we arrive at:

3E_{1} − E_{3} + (n − 2)E_{4} + 2 ∫_{M}⟨κ, ω⟩ = \frac{1}{2} ∫_{M} s + s^{∗}.  (4)

where E_{j} = ∫_{M} ∥(∇ω)_{j}∥^{2} and iκ is the curvature of the Levi-Civita connection on the canonical bundle.

4.6 Extension to U_{n}-structures

Observe that Formula (4), as it now appears, makes sense for any U_{n}-structure, and that the only information eventually used about the SU_{n}-structure is that it exists, i.e. that we have a U_{n}-structure for which c_{1} = 0 (the canonical bundle Λ^{n,0} is topologically trivial). It is thus tempting to guess that the formula holds for any U_{n}-structure. Indeed,

PROPOSITION 5 Formula (4) holds for any U_{n}-structure.

Proof. This follows from the twisted version of formula (1) for the case of an n−form ϕ with values in the vector bundle E =real part of Λ^{n,0} ⊕Λ^{0,n}, equipped with the connection induced by the Levi-Civita connection (by projection). We take ϕ = η ⊗ η + η^{⊥} ⊗ η^{⊥} (basically the inclusion map of E in Λ^{n}), then this section is defined globally, although η is defined only locally, and one checks easily that Formula (1) applied to ϕ yields Formula (4). Details of this calculation are routine and are left to the reader.

4.7 The U_{n} examples revisited

The S^{6} example. Since it is of type W_{1} the (∇η)_{4} term vanishes hence also the crossed term. Since s = 30, s^{∗} = 6, the equation is 3E_{1} = (1/2)36 V ol(S^{6}) so E_{1} = 6 V ol(S^{6}), consistent with the calculation with the U_{n} formula.

The S^{2m+1} × S^{1} example: We have already remarked that S^{2m+1} × S^{1} admits a U_{n}-structure of type W_{4}, from which we deduced that

m ∫_{M} ∥∇ω∥^{2} = \frac{1}{2} ∫_{M} s − s^{∗} = 2m^{2}vol(M).

For this example the formula derived in this section reads

(m − 1) ∫_{M} ∥∇ω∥^{2} = \frac{1}{2} ∫_{M} s + s^{∗} − 2 ∫_{M} ⟨κ, ω⟩.

From which we deduce that,

∥∇ω∥^{2} = 2m = ⟨κ, ω⟩.

4.8 Applications

For easy reference, let us collect both Formulas (3) and (4), taking their sum and difference:

PROPOSITION 6 Let M be a compact almost-hermitian manifold of (real) dimension 2n, n ≥ 3. Then,

5E_{1} − E_{2} − E_{3} + (2n − 3)E_{4} = ∫_{M} [s − 2⟨κ, ω⟩]

and

E_{1} + E_{2} − E_{3} − E_{4} = ∫_{M} [s^{∗} − 2⟨κ, ω⟩],

where E_{j} = ∫_{M} ∥(∇ω)_{j}∥^{2} and iκ is the curvature of the canonical bundle.

One can use these formulas to characterize, via curvature conditions, different types of U_{n}-structures. For example,

COROLLARY 1 Let M be a compact hermitian manifold of (real) dimension ≥ 6. Then

∫_{M} ∥∇ω∥^{2} = ∫_{M} 2⟨κ, ω⟩ − s^{∗};

in particular, such a manifold is Kähler iff ∫ s^{∗} = 2 ∫ ⟨κ, ω⟩.

COROLLARY 2 Let M be an almost-hermitian compact manifold of (real) dimension ≥ 6, with d^{∗}ω = 0 and c_{1} = 0. Then

5E_{1} − E_{2} − E_{3} = ∫_{M} s,

and

E_{1} + E_{2} − E_{3} = ∫_{M} s^{∗}.

In particular,

• [Gray1] if M is Nearly-Kähler and c_{1} = 0, then ∫_{M} ∥∇ω∥^{2} = \frac{1}{5} ∫_{M} s = ∫_{M} s^{∗};

• [Sekigawa-Vanhecke] if M is almost-Kähler and c_{1} = 0, then ∫_{M} ∥∇ω∥^{2} = − ∫_{M} s = ∫_{M} s^{∗};

• [Sekigawa-Vanhecke] if M is hermitian, d^{∗}ω = 0 and c_{1} = 0, then ∫_{M} ∥∇ω∥^{2} = − ∫_{M} s = − ∫_{M} s^{∗}.

Proof. c_{1} = 0 implies that κ is exact, hence, since d^{∗}ω = 0, integration by parts gives ∫_{M} ⟨κ, ω⟩ = 0. Since ∥(∇ω)_{4}∥ is proportional to ∥d^{∗}ω∥, the result follows from the above proposition.

5 G_{2}

5.1 Definition of a G_{2}-structure

Let V = ℝ^{7} with its standard euclidean structure, an orthonormal basis {e_{1}, …, e_{7}} and a dual basis {θ_{1}, …, θ_{7}}. Denote by Λ^{k} := Λ^{k}(V^{∗}) and basis elements by θ_{ij} := θ_{i} ∧ θ_{j}, θ_{ijk} = θ_{i} ∧ θ_{j} ∧ θ_{k}, … etc..The group G_{2} ⊂ O_{7} is the stabilizer of the 3-form

ϕ = θ_{124} + θ_{235} + θ_{346} + θ_{457} + θ_{561} + θ_{672} + θ_{713}.

(In fact, according to [Bryant1], G_{2} is the stabilizer of ϕ in GL_{7}). A good way to remember this formula is to note that the 7 terms are obtained by cyclic permutations (mod 7) of the first term θ_{124}.

5.2 Decomposition of ∇ϕ

In the following decompositions the subscripts on irreducible G_{2}-representations denote their dimensions, omitting the detailed description of some irreducible spaces when their nature (other than their dimension) is immaterial for us. Fortunately, the G_{2} irreducible representations that we meet here are distinguished by their dimensions alone. The following lemma is taken almost verbatim from [Bryant1] (see also [Fernandez-Gray] where the decomposition was first obtained).

LEMMA 12 Under G_{2}, we have the following decompositions into irreducible subspaces:

• Λ^{2} = Λ_{(14)}^{2} ⊕ Λ_{(7)}^{2}, corresponding to 𝔰𝔬_{7} = 𝔤_{2} ⊕ 𝔤_{2}^{⊥}, where Λ_{(14)}^{2} = 𝔤_{2} is the kernel of ·ϕ : Λ^{2} → Λ^{3}, α ↦ α · ϕ, and Λ_{(7)}^{2} = 𝔤_{2}^{⊥} ≅ Λ^{1} is the image of θ ↦ int(θ ⊗ ϕ), θ ∈ Λ^{1}.

• Λ^{3} = Λ_{(1)}^{3} ⊕ Λ_{(7)}^{3} ⊕ Λ_{(27)}^{3}, where Λ_{(1)}^{3} = ℝϕ and Λ_{(7)}^{3} ≅ Λ^{1} is the image of θ ↦ int[θ ⊗ (∗ϕ)], θ ∈ Λ^{1}. Hence, 𝔤_{2}^{⊥} · ϕ = Λ_{(7)}^{3} is irreducible.

Next, we decompose

Λ^{1} ⊗ Λ_{(7)}^{3} ≅ Λ^{1} ⊗ Λ^{1} = Λ^{2} ⊕ S^{2} = Λ_{(14)}^{2} ⊕ Λ_{(7)}^{2} ⊕ S_{(1)}^{2} ⊕ S_{(27)}^{2},

where we use the decomposition of S^{2} (quadratic forms on V) into {multiples of the inner-product}⊕{traceless}.

It is easy to see that S_{(1)}^{2} consists of multiples of the invariant 4-form ∗ϕ, using the following argument: since Λ^{3} contains a single isomorphic copy of Λ^{1}, it follows (using Schur’s lemma) that the subspace of fixed vectors in Λ^{1} ⊗ Λ^{3} is 1-dimensional. On the other hand, Λ^{4} ⊂ Λ^{1} ⊗ Λ^{3}, hence the fixed subspace in Λ^{1} ⊗ Λ^{3}, and hence in Λ^{1} ⊗ Λ_{(7)}^{3}, must consist precisely of the multiples of ∗ϕ.

If we name the irreducible subspaces of W by their dimension, then we have

W = Λ^{1} ⊗ (𝔤_{2}^{⊥} · ϕ) = W_{1} ⊕ W_{7} ⊕ W_{14} ⊕ W_{27}.

5.3 The left-hand side of Formula (1)

Denote by ∇ϕ = (∇ϕ)_{1} + (∇ϕ)_{7} + (∇ϕ)_{14} + (∇ϕ)_{27} the decomposition of ∇ϕ, where (∇ϕ)_{i} ∈ W_{i}. The W_{i} are mutually distinct G_{2}-representations, hence, according to Lemma 2 of the Introduction, there are constants a_{i} such that

∥dϕ∥^{2} = ∥alt(∇ϕ)∥^{2} = ∑_{i} ∥alt(∇ϕ)_{i}∥^{2} = ∑_{i} a_{i}∥(∇ϕ)_{i}∥^{2},

where a_{i} is the homothety factor of alt : W_{i} → Λ^{4}, i.e. ∥alt(w_{i})∥^{2} = a_{i}∥w_{i}∥^{2} for every w_{i} ∈ W_{i}. Similarly,

∥d^{∗}ϕ∥^{2} = ∑_{i} ∥int(∇ϕ)_{i}∥^{2} = ∑_{i} b_{i}∥(∇ϕ)_{i}∥^{2},

where the b_{i} are the homothety factors of int : W_{i} → Λ^{2}.

The next table summarizes the result of the computation of the homothety factors a_{i} and b_{i}. We use the map T : Λ^{1} ⊗ Λ^{1} → Λ^{1} ⊗ Λ_{(7)}^{3}, θ ⊗ θ′ ↦ θ ⊗ int[θ′ ⊗ (∗ϕ)].

Table 4: Calculation of the homothety factors a_{i}, b_{i} for G_{2}

Summand element w_{i} ∈ W_{i}  ∥w_{i}∥^{2} ∥alt(w_{i})∥^{2} ∥int(w_{i})∥^{2} a_{i} b_{i}
W_{1}  ∗ϕ  7/24  7  0  24 0
W_{7}  T [int(θ_{1} ⊗ ϕ)]  4  36  48  9 12
W_{14}  T (θ_{12} + θ_{36})  8/3  0  8  0 3
W_{27}  T (θ_{1} ⊗ θ_{1} − θ_{2} ⊗ θ_{2}) 4/3  4  0  3 0

We thus have

∥dϕ∥^{2} = 24∥(∇ϕ)_{1}∥^{2} + 9∥(∇ϕ)_{7}∥^{2} + 3∥(∇ϕ)_{27}∥^{2}, ∥d^{∗}ϕ∥^{2} = 12∥(∇ϕ)_{7}∥^{2} + 3∥(∇ϕ)_{14}∥^{2}.

Remark: It follows from these formulas that ϕ is parallel (∇ϕ = 0) if it is harmonic (dϕ = d^{∗}ϕ = 0).

5.4 The right-hand side of Formula (1) (the curvature term)

The curvature term turns out to be particularly simple in the G_{2} case.

LEMMA 13 Let M be a 7-dimensional manifold with a G_{2}-structure, a 3-form ϕ, a Riemann curvature tensor R and scalar curvature s. Then ⟨R̃ϕ, ϕ⟩ = 2s.

Proof. Let us prove first that the space of G_{2}-fixed curvature-type tensors is 1-dimensional: since Λ^{2} decomposes into two non-isomorphic irreducible subspaces, the Schur lemma implies that the space of G_{2}−fixed elements in S^{2}(Λ^{2}) is 2-dimensional. Now the G_{2}−fixed subspace of Λ^{4} is 1-dimensional (multiples of ∗ϕ), hence the G_{2}−fixed subspace of its orthogonal complement in S^{2}(Λ^{2}) (this is exactly the space of curvaturetype tensors) is also 1-dimensional, as required. It is therefore enough to verify the identity on a particular curvature-type tensor (for which the outcome is non-zero), say R = θ_{12} ⊗ θ_{12}. This (simple) verification is left to the reader.

5.5 The G_{2} formula

Denote by E_{1}, …, E_{27} the L_{2} norms of (∇ϕ)_{1}, …, (∇ϕ)_{27} (resp.), then substituting all of the above into Formula (1), we get

6E_{1} + 5E_{7} − E_{14} − E_{27} = \frac{2}{3} ∫ s  (5)

COROLLARY 3 Let M be a compact, calibrated (i.e. dϕ = 0) G_{2}-manifold, then ∫ s ≤ 0 with equality if and only if the local holonomy of M is contained in G_{2}.

5.6 Examples

Hypersurfaces in ℝ^{8}. Consider ℝ^{8} with its standard Spin_{7}-structure Φ ∈ Λ^{4}((ℝ^{8})^{∗}) (see next section). Then on any oriented hypersurface M^{7} ⊂ ℝ^{8} there is a G_{2}-structure defined by ϕ = int_{N}Φ, where N is the unit normal on M given by the orientation. For example if M is a linear subspace (say x_{0} = 0) then we get the standard ϕ on ℝ^{7}. If M is the unit sphere S^{7} ⊂ ℝ^{8} we get the homogeneous space Spin_{7}/G_{2} with its (essentially unique) Spin_{7}−invariant G_{2}-structure.

For any hypersurface in ℝ^{8} the above defined G_{2}-structure is of type W_{1} ⊕ W_{27}, see [Fernandez-Gray], and therefore the formula for them reduces to

6E_{1} − E_{27} = \frac{2}{3} ∫_{M} s.

An Aloff-Wallach space. Such a space is a homogeneous space of the form SU_{3}/U_{1} equipped with a left invariant metric. In [Cabrera-Monar-Swan] it is shown that, for a certain choice of subgroup U_{1}, the 7-manifold SU_{3}/U_{1} admits (a non-parallel) SU_{3}-invariant G_{2}-structure of type W_{27}. For it, our formula immediately implies that ∥∇ϕ∥^{2} = −\frac{2}{3}s; in particular, this G_{2}-structure is associated to a left-invariant metric on SU_{3} with s < 0.

Integrable G_{2}-structures. In [Fernandez-Ugarte] G_{2}-structures of type W_{1} ⊕ W_{7} ⊕ W_{27} are studied; it is discussed there why these structures should be considered the G_{2} analogues of integrable almost hermitian structures. A few examples are given.

Further examples and properties may be found in [Fernandez-Gray].

6 Spin_{7}

6.1 Definition of a Spin_{7}-structure.

Let V = ℝ^{8} with the standard euclidean structure, an orthonormal basis e_{0}, e_{1}, …, e_{7}, and dual basis θ_{0}, …, θ_{7}. Denote by Λ^{k} := Λ^{k}(V^{∗}) and basis elements by θ_{ij} := θ_{i} ∧ θ_{j}, θ_{ijk} = θ_{i} ∧ θ_{j} ∧ θ_{k}, … etc..

The group Spin_{7} ⊂ SO_{8} is the stabilizer of the 4-form Φ = θ_{0} ∧ϕ+∗ϕ. Here we are thinking of ℝ^{8} = ℝ⊕ℝ^{7}, with ϕ and ∗ϕ on ℝ^{7} (pulled-back to ℝ^{8}) as defined before for G_{2}.

6.2 Decomposition of ∇Φ

Like in the G_{2} case, we denote representations by their dimensions and omit further information whenever is not used here. We have LEMMA 14 We have the following Spin_{7} decompositions:

• Λ^{2} = Λ_{(21)}^{2}⊕Λ_{(7)}^{2}, corresponding to so_{8} = 𝔰𝔭𝔦𝔫_{7}⊕𝔰𝔭𝔦𝔫_{7}^{⊥}, where Λ_{(21)}^{2} = 𝔰𝔭𝔦𝔫_{7} is the kernel of ·Φ : Λ^{2} → Λ^{4}, α ↦ α · Φ.

• Λ^{3} = Λ_{(8)}^{3} ⊕ Λ_{(48)}^{3}, where Λ_{(48)}^{3} is the kernel of Φ∧ : Λ^{2} → Λ^{6}, α ↦ Φ ∧ α.

• Λ^{4} = Λ_{(1)}^{4} ⊕ Λ_{(7)}^{4} ⊕ Λ_{(27)}^{4} ⊕ Λ_{(35)}^{4}, where Λ_{(7)}^{4} is the image of ·Φ : Λ_{(7)}^{2} → Λ^{4}, α ↦ α · Φ.

For the proof, see e.g. [Bryant1]. It follows from this lemma that W := Λ^{1} ⊗ (𝔰𝔭𝔦𝔫_{7}^{⊥} · Φ) = Λ^{1} ⊗ Λ_{(7)}^{4}, so we decompose

W ≅ Λ^{1} ⊗ Λ_{(7)}^{2} = W_{8} ⊕ W_{48},

corresponding to the kernel and co-kernel (image of the adjoint) of the interior product map int : Λ^{1}⊗Λ_{(7)}^{2} → Λ^{1}.

6.3 The left-hand side of Formula (1)

The 4-form Φ is self-dual. It follows, like in the case of SU_{n}, that ∥dΦ| = ∥d^{∗}Φ∥, so that

∥dΦ∥^{2} = ∥d^{∗}Φ∥^{2} = ∥int(∇Φ)∥^{2} = a_{8}∥(∇Φ)_{8}∥^{2} + a_{48}∥(∇Φ)_{48}∥^{2},

for some homothety factors a_{8}, a_{48}.

Table 5: Calculation of the homothety factors a_{i} for Spin_{7}

Summand w_{i} ∈ W_{i}  ∥w_{i}∥^{2} ∥int(w_{i})∥^{2} a_{i}
W_{8}  ∑_{i=1}^{7} θ_{i} ⊗ (θ_{0i} · Φ)  7/3  16 · 7  48
W_{48}  θ_{0} ⊗ [(θ_{12} + θ_{36}) · Φ] + θ_{1} ⊗ [θ_{20} · Φ]+  4/3  8  6

+θ_{2} ⊗ [θ_{01} · Φ] + θ_{3} ⊗ [θ_{60} · Φ] + θ_{6} ⊗ [θ_{03} · Φ]

Details of the calculation:

• To pick a w_{8} ∈ W_{8}, we define an isomorphism Λ^{1} → W_{8} by considering the composition

Λ^{1} → Λ^{1} ⊗ Λ^{2} → Λ^{1} ⊗ Λ^{4}, where the first map is int^{∗} (the adjoint of interior product) and the second is given by the map Λ^{2} → Λ^{4}, α ↦ α · Φ. Starting with θ_{0}, we have that int^{∗}(θ_{0}) is, up to a constant, ∑_{i=1}^{7} θ_{i} ⊗ θ_{0i}, and so we get (after some moderate calculation)

w_{8} = ∑_{i=1}^{7} θ_{i} ⊗ (θ_{0i} · Φ) = ∑_{i=1}^{7} θ_{i} ⊗ [θ_{i} ∧ ϕ − θ_{0} ∧ int(θ_{i} ⊗ (∗ϕ))],
int(w_{8}) = ∑_{i=1}^{7} int[θ_{i} ⊗ (θ_{i} ∧ ϕ − θ_{0} ∧ int_{(}θ_{i} ⊗ (∗ϕ)))] = ∑_{i=1}^{7} int[θ_{i} ⊗ (θ_{i} ∧ ϕ)] = 4ϕ.

• To pick a w_{48} ∈ W_{48} we define an isomorphism Λ_{(48)}^{3} → W_{48} via the composition

Λ_{(48)}^{3} → Λ^{3} → Λ^{1} ⊗ Λ^{2} → Λ^{1} ⊗ Λ^{4},

where the first map is inclusion, the second is alt^{∗} (the adjoint of alternation, or exterior product), and the third is given as before by the Λ^{2}-action on Φ.

To pick an element in Λ_{(48)}^{3} = {α | Φ ∧ α = 0}, we try an α of the form α = θ_{0} ∧ α_{0}, α_{0} ∈ Λ^{2}(ℝ^{7}). Then Φ ∧ α = 0 iff ∗ϕ ∧ α_{0} = 0. The last equation, by a Schur lemma type argument, is equivalent to α_{0} ∈ Λ_{(14)}^{2}, i.e. α_{0} is in the stabilizer of ϕ (or ∗ϕ), from which one obtains easily by inspection a solution such as α_{0} = θ_{12} + θ_{36}, so that α = θ_{012} + θ_{036} ∈ Λ_{(48)}^{3}. Now

alt^{∗}(α) = θ_{0} ⊗ (θ_{12} + θ_{36}) + θ_{1} ⊗ θ_{20} + θ_{2} ⊗ θ_{01} + θ_{3} ⊗ θ_{60} + θ_{6} ⊗ θ_{03},

and

θ_{0i} · Φ = θ_{i} ∧ ϕ − θ_{0} ∧ int[θ_{i} ⊗ (∗ϕ)],

so that

w_{48} = θ_{2} ⊗ [θ_{1} ∧ ϕ − θ_{0} ∧ int(θ_{1} ⊗ (∗ϕ))] − θ_{1} ⊗ [θ_{2} ∧ ϕ − θ_{0} ∧ int(θ_{2} ⊗ (∗ϕ))]
+θ_{6} ⊗ [θ_{3} ∧ ϕ − θ_{0} ∧ int(θ_{3} ⊗ (∗ϕ))] − θ_{3} ⊗ [θ_{6} ∧ ϕ − θ_{0} ∧ int(θ_{6} ⊗ (∗ϕ))],

(note that the θ_{0} ⊗ α_{0} term in alt^{∗}(α) maps to 0, since α_{0} · ϕ = 0).

Thus int(w_{48}) = … = −2α, where “… ” denotes a moderately tedious, yet straightforward, calculation.

We thus have,

∥dΦ∥^{2} = ∥d^{∗}Φ∥^{2} = 48∥(∇Φ)_{8}∥^{2} + 6∥(∇Φ)_{48}∥^{2}.

Remark: It follows from this formula that Φ is parallel if it is closed.

6.4 The right-hand side of Formula (1) (the curvature term)

The proof of the following lemma is very similar to the G_{2}-case and is left to the reader:

LEMMA 15 Let M be an 8-dimensional manifold with a Spin_{7}-structure, a 4-form Φ, a Riemann curvature tensor R and scalar curvature s. Then ⟨R̃Φ, Φ⟩ = 2s

6.5 The Spin_{7} formula

Denote by E_{8}, E_{48} the L_{2} norms of (∇Φ)_{8}, (∇Φ)_{48} (resp.). Then the above information into Formula (1) gives:

6E_{8} − E_{48} = \frac{1}{6} ∫ s.

6.6 Examples

S^{7} × S^{1}. In [Cabrera] it is shown that S^{7} × S^{1} admits an Spin_{7}-structure of type W_{8}.

The reader may find there a few more examples.

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