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

January 28, 2000

Abstract

A general technique is introduced for deriving Bochner type formulae on a compact riemannian manifold, relating the curvature with the intrinsic torsion of a compatible (orthogonal) G-structure. The technique is ilustrated 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

The idea of a “geometric structure” is captured, for a large class of examples, by the notion of a G-structure. A G-structure on an n-dimensional differential manifold consists of a reduction of the structure group of the manifold to a subgroup G ⊂ GL_{n}(ℝ) (i.e. a principal G−subbundle of the frame bundle). For example, a riemannian metric amounts to an O_{n}-structure, i.e. a reduction to the orthogonal group O_{n}, given by the orthonormal frame bundle. Another example: an almost complex structure on a 2n-dimensional manifold corresponds to a reduction to a GL_{n}(ℂ)-structure, given by the bundle of complex frames.

In this article we are concerned with orthogonal G-structures, i.e. with reductions to subgroups G ⊂ O_{n}. One standard context where these occur is on riemannian manifolds for which the holonomy group of the associated Levi-Civita connection is contained in G. This happens for example on a Kähler manifold, where the holonomy is contained in the unitary group U_{n}, thus giving a reduction to U_{n} ⊂ O_{2n}.

The list of groups that may appear as holonomy groups of complete, simply connected, irreducible, nonsymmetric riemannian n-manifold was found by Berger in [1] ( see [2] for a good introduction and also the survey article of Bryant [4]). These are: SO_{n}, U_{m}, SU_{m} (n = 2m), Sp_{k} · Sp_{1}, Sp_{k} (n = 4k), G_{2} (n = 7) and Spin_{7} (n = 8). Examples of compact manifolds with the last two holonomies have been found only quite recently [16], [17].

From the general point of view of the theory of G-structures, orthogonal G-structures arising from a holonomy reduction are the 1-flat structures and are characterized by the vanishing of their intrinsic torsion. We will not give here the precise definitions of these notions (and will not really use them; see for example [4] for more details). However, we hope the following will suffice as a motivation:

A G-structure is said to be flat if it is locally diffeomorphic to the trivial structure ℝ^{n} × G. For example, the condition for a riemannian manifold to be flat (locally isometric to euclidean space) is equivalent to the flatness of the corresponding O_{n}-structure, and the condition for an almost-complex structure to be integrable is equivalent to the flatness of the corresponding GL_{m}(ℂ)-structure.

Associated with a G-structure on a manifold is its intrinsic torsion tensor, which measures the extent to which the structure differs (to first-order) from a flat one. The G-structures with vanishing intrinsic torsion are distinguished by the existence of a torsionless G-connection and are called 1-flat. For example, any O_{n}- structure is automatically 1-flat (existence of the Levi-Civita connection) but not necessarily flat, and any 1-flat GL_{n}(ℂ) or Sp_{2n}(ℝ)-structure is automatically flat (the Nierenberg-Newlander theorem and the Darboux lemma resp.).

Although most orthogonal G-structures one encounters in the differential geometry literature are the 1-flat ones of the Berger list, orthogonal G-structures which are not 1-flat are also important. The study of such G-structures has attracted attention in connection with symplectic geometry, twistor geometry and the recent discoveries of compact manifolds with holonomies G_{2} and Spin_{7}.

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

In this article we study obstructions coming from the curvature to the existence of orthogonal G-structures on compact riemannian manifolds. This obstruction comes in the form of a Bochner type formula relating the intrinsic torsion and the curvature. We do this for G-structures associated to the groups G = U_{m}, SU_{m}, G_{2} and Spin_{7}. The methods we use apply also to the remaining groups on the Berger list, Sp_{k} · Sp_{1} and Sp_{k}, but the calculations are more complicated and we leave these cases for a forthcoming article.

In fact, if one considers G-structures which are not 1-flat, as we do here, there is no good reason to restrict oneself to the groups on the Berger list, but at any rate these are the first groups that occurred to us as natural candidates for testing the effectiveness of the Bochner type formulas of this article.

Let us describe now briefly our technique.

Let G ⊂ O_{n} be the stabilizer subgroup of a p-form ϕ on ℝ^{n} (p = 2, m, 3, 4 for G = U_{m}, SU_{m}, G_{2}, Spin_{7} resp.) If M is an n-dimensional manifold with a G-structure then its intrinsic torsion can be identified with the covariant derivative ∇ϕ of the associated p-form on M (denoted by ϕ as well). Thus, the (local) holonomy of the Levi-Civita connection on T M is contained in G if and only if ϕ is parallel, ∇ϕ = 0. If this does not happen, one can decompose ∇ϕ into G-irreducible components, ∇ϕ = (∇ϕ)_{1} ⊕ (∇ϕ)_{2} ⊕ . . .. These components carry interesting geometric information about the G-structure. For example, for G = U_{n}, ∇ϕ has 4 irreducible components, the sum of two of them measuring the intrinsic torsion of the associated almost-complex structure (the Nijenhuis tensor), the sum of certain three of them representing the intrinsic torsion of the corresponding almost symplectic structure (the exterior derivative of the Kähler form), etc.

Using the classical Bochner-Weitzenbock formulas for the laplacian on p-forms and certain assumptions on G (holding for all cases on the Berger list except SU_{n} and Sp_{n}), we get for a compact manifold M an integral formula of the form

∑_{i} c_{i} ∫_{M} | (∇ϕ)_{i} |^{2} = ∑_{j} ∫_{M} s_{j}(R),

where the s_{j} are certain G-invariants of the curvature and the c_{i} are constants that can be explicitly determined by representation theoretic calculations. For those groups G where the above mentioned assumptions are not satisfied the formula becomes more complicated (appearance of “off diagonal” terms; see the section on SU_{n}). The explicit calculation of the constants c_{i} and the invariants s_{j}, elementary in principle, can be quite complicated for some G and takes up the bulk of this paper.

The main strength of the formulas presented in this work is their full generality: they apply to any Gstructure on a compact manifold (for the groups considered), contain as special cases several results previously proved by ad hoc methods as well as yielding some new and interesting results.

The paper is arranged as follows. The next two sections treat the definitions, conventions and general setup for all groups G. Subsequent sections treat independently each group; in each of these sections we compute the above mentioned decomposition of the intrinsic torsion space into irreducible submodules, the coefficients c_{i} and the curvature invariants s_{j}, finishing each case with a few examples and applications that show how to use the formulas.

2 Some generalities and notation

2.1 Algebra:

2.1.1. V : an n-dimensional euclidean vector space with inner product ⟨ · , · ⟩ and norm | · |.

2.1.2. V^{p} = V ⊗ … ⊗ V : the p-th tensor product, with an inner product such that e_{i_{1}} ⊗ e_{i_{2}} ⊗ … ⊗ e_{i_{p}} is an orthonormal basis, where {e_{1}, e_{2}, …, e_{n}} is an orthonormal basis for V. Sometimes write ⟨·, ·⟩_{T} and | · |_{T} for this “tensorial” inner product and norm, if in doubt.

2.1.3. Λ^{p}(V) ⊂ V^{p} : the p-th exterior product, i.e. alternating p-tensors, σ · α = sign(σ)α, σ ∈ S_{p}.

2.1.4. alt : V^{p} → V^{p}, orthogonal projection onto Λ^{p}(V) (alternation) :

alt(α) = \frac{1}{p!} (∑_{σ} sign(σ)σ · α) ,

so alt^{2} = alt and alt(α) = α iff α ∈ Λ^{p}V.

2.1.5. Wedge product: α ∧ β = alt(α ⊗ β).

2.1.6. The “standard” norm in Λ^{p}(V) (as distinct from the “tensorial” norm induced from V^{p}) is such that {e_{i_{1}} ∧ … ∧ e_{i_{p}} | i_{1} < … < i_{p}} is an orthonormal basis, so that

|α|^{2} = p!|α|_{T}^{2} .

2.1.7. Volume form: V ol = θ_{1} ∧ … ∧ θ_{n} ∈ Λ^{n}(V^{∗}), where {θ_{1}, …, θ_{n}} is the dual basis to an oriented orthonormal basis for V, so |V ol| = 1.

2.1.8. Interpretation of Λ^{p}(V^{∗}) (denoted also simply by Λ^{p}) as alternating multilinear p-forms on V :

α_{1} ∧ … ∧ α_{p}(v_{1}, …, v_{p}) = ∑_{σ} sign(σ)α_{σ(1)}(v_{1}) … α_{σ(p)}(v_{p}) = det(α_{i}(v_{j})).

So e.g. the volume of the unit cube is 1, V ol(e_{1}, …, e_{n}) = 1.

2.1.9. Interior product int : V ⊗ Λ^{p}(V^{∗}) → Λ^{p−1}V^{∗}, int(v ⊗ α) = int_{v}α, where int_{v} is defined for θ ∈ V^{∗} by int_{v}θ = θ(v), then extended as an anti-derivation to Λ^{p}(V^{∗}). So e.g. int (e_{1} ⊗ (θ_{1} ∧ … ∧ θ_{p})) = θ_{2} ∧ … ∧ θ_{p}. With the usual identification V^{∗} ≅ V, we also have int : V^{∗} ⊗ Λ^{p}(V^{∗}) → Λ^{p−1}V^{∗}.

2.1.10. Hodge ∗ operator : given by α ∧ ∗β = ⟨α, β⟩V ol, so e.g ∗(θ_{1} ∧ … ∧ θ_{p}) = θ_{p+1} ∧ … ∧ θ_{n}.

2.1.11. Identification of 2-forms with anti-symmetric endomorphisms: define Λ^{2}(V^{∗}) → End(V), α ↦ A, by α(v, w) = ⟨Av, w⟩. In orthonormal bases: θ_{i} ∧ θ_{j} ↦ θ_{i} ⊗ e_{j} − θ_{j} ⊗ e_{i}.

2.1.12. Action of End(V) on Λ^{p}(V^{∗}): for p = 1, define A · ϕ = −A^{∗}ϕ, A ∈ End(V), ϕ ∈ V^{∗}, where A^{∗} ∈ End(V^{∗}) is the usual adjoint of A, (A^{∗}ϕ)(v) = ϕ(Av). Now extend this action to Λ^{p}(V^{∗}) as a derivation. Using the identification End(V) = V^{∗} ⊗ V, this becomes (θ ⊗ v) · ϕ = −θ ∧ (int_{v}ϕ). This is the derivative of the action of Aut(V) on p-forms by pull-back, ϕ ↦ (g^{−1})^{∗}ϕ. From this description it is also clear that End(V) should act by derivations of the exterior algebra on V^{∗}.

2.1.13. Combining the last two items we have the action of Λ^{2}(V^{∗}) on p−forms. It is convenient to express the outcome as follows: the action of Λ^{2}(V^{∗}) on V^{∗} is given by α · ϕ = int(ϕ ⊗ α), and the action on Λ^{p}(V^{∗}) is obtained by extenting this action as a derivation of the exterior algebra on V^{∗}.

Explicitely, for θ_{1}, θ_{2} ∈ V^{∗},

(θ_{1} ∧ θ_{2}) · ϕ = θ_{2}⟨θ_{1}, ϕ⟩ − θ_{1}⟨θ_{2}, ϕ⟩, ϕ ∈ V^{∗},

and

(θ_{1} ∧ θ_{2}) · ϕ = θ_{2} ∧ (int_{e_{1}}ϕ) − θ_{1} ∧ (int_{e_{2}}ϕ) , ϕ ∈ Λ^{p}(V^{∗}),

where e_{1}, e_{2} ∈ V are the vectors corresponding to θ_{1}, θ_{2} (resp.) under the isomorphism V ≅ V^{∗} given by the inner product.

Note that

α · β = −β · α, α, β ∈ Λ^{2}(V^{∗}),

reflecting the fact that the pull-back action of SO_{n} on Λ^{2}(V^{∗}) corresponds to the adjoint action (conjugation) under the identificaion 𝔰𝔬_{n} ≅ Λ^{2}, so that the action of Λ^{2}(V^{∗}) on Λ^{2}(V^{∗}) corresponds to Lie bracketing.

2.1.14. Complexification: if V is 2n-dimensional and carries an orthogonal almost complex structure J, then as usual, Λ^{k}(V^{∗} ⊗ ℂ) = ∑_{p+q=k} Λ^{p,q}(V^{∗}), where V^{∗} ⊗ ℂ = (V^{∗})^{1,0} ⊕ (V^{∗})^{0,1} is the decomposition into ±i eigenspaces of J (resp.) and Λ^{p,q}(V^{∗}) = Λ^{p,0}(V^{∗}) ⊗ Λ^{0,q}(V^{∗}). Here, Λ^{p,0}(V^{∗}) is the p−th complex exterior power of (V^{∗})^{1,0} and similarly for Λ^{0,q}(V^{∗}).

The inner product ⟨ , ⟩ extends complex-linearly to Λ^{k}(V ⊗ ℂ) and Λ^{k}(V^{∗} ⊗ ℂ). Thus, if {z_{1}, …, z_{n}} is a unitary basis for the (1, 0)-forms (i.e. ⟨z_{i}, z̄_{j}⟩ = δ_{ij}), then

{z_{i_{1}} ∧ … ∧ z_{i_{p}} ∧ z̄_{j_{1}} ∧ … ∧ z̄_{j_{q}} | 1 ≤ i_{1} < … < i_{p} ≤ n, 1 ≤ j_{1} < … < j_{q} ≤ n}

is a unitary basis for the (p, q)-forms Λ^{p,q}(V^{∗}).

2.2 Calculus on a riemannian manifold

2.2.1. Λ^{k}M, or sometimes simply Λ^{k}, is an abbreviation for Λ^{k}(T^{∗}M).

2.2.2. L_{2}-norm of a form: α ↦ ∫_{M} |α|^{2}V ol = ∫_{M} α ∧ ∗α.

2.2.3. Exterior derivative : determined by the standard axioms (anti-derivation, d^{2} = 0, d(function)= total derivative,…), given by d = alt ∘ ∇ = ∑_{i} θ_{i} ∧ ∇_{e_{i}}, where ∇ = ∑_{i} θ_{i} ⊗ ∇_{e_{i}} is the Levi-Civita connection on T M.

2.2.4. Coderivative : d^{∗} = ± ∗ d∗ = −int ∘ ∇ = − ∑_{i} int_{e_{i}}∇_{e_{i}}.

2.2.5. Laplacian : Δ = d^{∗}d + dd^{∗}

2.2.6. Curvature : for a vector bundle E → M with a connection ∇ : Γ(E) → Γ(T^{∗}M ⊗E), define R = d^{∇}∘d^{∇}, where d^{∇} : Γ(Λ^{p}M ⊗ E) → Γ(Λ^{p+1}M ⊗ E) is exterior derivative “twisted” by ∇, d^{∇}(α ⊗ s) = (dα) ⊗ s + (−1)^{p}α ∧ (∇s). Then R is a section of Λ^{2}M ⊗ End(E). Using frames, we have

R = ∑_{i<j}(θ_{i} ∧ θ_{j}) ⊗ R_{ij}, R_{ij} = [∇_{e_{i}}, ∇_{e_{j}}] − ∇_{[e_{i},e_{j}]}.

Proof. Let s be a local section of E, “normal” at a point x_{0} ∈ M, i.e. ∇s (x_{0}) = 0. Then at x_{0},

d^{∇}(d^{∇}(s)) = ∑_{i} d^{∇}(θ_{i} ⊗ ∇_{e_{i}}s) = ∑_{i,j}(dθ_{i}) ⊗ (∇_{e_{i}}s) − (θ_{i} ∧ θ_{j}) ⊗ ∇_{e_{j}}∇_{e_{i}}s =
= ∑_{i<j}(θ_{i} ∧ θ_{j})(∇_{e_{i}}∇_{e_{j}} − ∇_{e_{j}}∇_{e_{i}})s.

The claimed expession for R follows from the fact that it is C^{∞}(M)-linear in Γ(E) and coincides with d^{∇} ∘ d^{∇}, according to the above, for sections normal at a given point.

2.2.7. The Levi-Civita connection: denote by ∇ the Levi-Civita connection on T M (as well as associated tensor bundles.) The curvatute R = ∑_{i<j}(θ_{i} ∧ θ_{j}) ⊗ R_{ij} is then a section of Λ^{2}M ⊗ End(T M), with R_{ij} anti-symmetric, so using the above identification of Λ^{2}M with anti-symmetric endomorphisms, R becomes a tensor in Λ^{2}M ⊗ Λ^{2}M (in fact, in the symmetric square S^{2}(Λ^{2}M).)

2.2.8. For the curvature of Levi-Civita connection, it is also convenient to introduce the curvature operator R as a section of End(Λ^{2}M), defined by

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

(Here R_{ij} is considered as a 2−form, so it makes sense to take its dot product with α.)

2.2.9. Scalar curvature: For the Levi-Civita connection on T M, the scalar curvature s is defined by

s = 2 tr (R) = −2 ∑_{i<j}⟨θ_{i} ∧ θ_{j}, R_{ij}⟩.

For example, for M = S^{n} with the standard (round) metric, R = id_{Λ^{2}(S^{n})}, thus s = 2dim (Λ^{2}) = n(n − 1).

2.2.10. Two standard identities for the laplacian on p−forms (see [21], pp. 304-306):

Δϕ = −∇^{2}ϕ + R̃ϕ  Weitzenbock)  (1)
−\frac{1}{2}Δ|ϕ|^{2} = p!|∇ϕ|_{T}^{2} + ⟨∇^{2}ϕ, ϕ⟩,  (2)

where ϕ is any p-form, ∇^{2} = ∑_{i} ∇_{e_{i},e_{i}}^{2}, ∇_{X,Y}^{2} = ∇_{X}∇_{Y} − ∇_{∇_{X}Y}, and R̃ is the operator on p-forms induced by the curvature tensor R as follows: think of R as a section of Λ^{2}M ⊗ Λ^{2}M as above, and use (twice) the action of Λ^{2}M on p-forms, α ⊗ β : ϕ ↦ α · (β · ϕ). Thus

R̃(ϕ) = ∑_{i<j} (θ_{i} ∧ θ_{j}) · (R_{ij} · ϕ) =
= ∑_{i<j} [θ_{j} ∧ int_{e_{i}} − θ_{i} ∧ int_{e_{j}}] (R_{ij} · ϕ) =
= ∑_{i≠j} θ_{j} ∧ [int_{e_{i}} (R_{ij} · ϕ)] .

2.2.11. Combining equations (1) and (2) above,

⟨Δϕ, ϕ⟩ − p!|∇ϕ|_{T}^{2} − \frac{1}{2}Δ|ϕ|^{2} = ⟨R̃ϕ, ϕ⟩,  (3)

and integrating by parts over a compact manifold (passing to L^{2}-norms)

∫_{M} |dϕ|^{2} + |d^{∗}ϕ|^{2} − p!|∇ϕ|_{T}^{2} = ∫_{M}⟨R̃ϕ, ϕ⟩.  (4)

3 Orthogonal G-structures

As explained in the introduction, our idea is to apply formula (4) to the p−forms which determine orthogonal G-structures. Our candidates are the holonomy groups G ⊂ O_{n} on the Berger list. Each of these can be defined as the stabilizer subgroup of a certain p-form ϕ ∈ Λ^{p}(V^{∗}), V = ℝ^{n} (or a direct sum of such, in the Sp_{n} case). For the 4 examples in this article, we have:

• U_{n} ⊂ O_{2n} is the stabilizer of the standard symplectic 2-form, the imaginary part of the hermitian inner product on ℂ^{n}, (v, v′) ↦ ∑ v_{i}v̄_{i}′.

• SU_{n} ⊂ U_{n}, n ≥ 3, is the stabilizer (in O_{2n}) of any non-zero element in Λ^{n,0} ⊕ Λ^{0,n} ⊂ Λ^{n} ⊗ ℂ. (SU_{2} is better thought of as Sp_{1}.)

• G_{2} ⊂ O_{7} is the stabilizer of the 3-form ϕ = ∑ θ_{i} ∧ θ_{i+1} ∧ θ_{i+2} (cyclic sum over i = 1, …, 7 with i + 7 = i), as well as the 4-form ∗ϕ.

• Spin_{7} ⊂ O_{8} is the stabilizer of the 4-form Φ = θ_{0} ∧ π^{∗}(ϕ) + π^{∗}(∗ϕ), where π : ℝ^{8} → ℝ^{7} is the projection on the last 7 coordinates and ϕ is the same as for G_{2}.

Now let M be a riemannian manifold with a reduction of its structure group from O_{n} to such a G, i.e. G is the stabilizer of an element ϕ ∈ Λ^{p}(V^{∗}). Let ϕ also denote the associated p−form on M (a section of Λ^{p}M) and apply to it formula (4) above: on the left hand side we get some quadratic form in the intrinsic torsion tensor ∇ϕ (recall that both dϕ and d^{∗}ϕ can be expressed in terms of ∇ϕ) and on the right hand side a G-invariant of the curvature tensor. The idea then is to study both sides of the equation in terms of the representation theory of G. We begin with the left hand side. The following proposition can also be found also in [19]:

PROPOSITION 1 Let G ⊂ O_{n} be the stabilizer of a p-form ϕ ∈ Λ^{p}(V^{∗}), let M be an n-dimensional manifold with an orthogonal G-structure and denote by ϕ also the corresponding p-from on M. Then the covariant derivative ∇ϕ (w.r.t. the Levi-Civita connection) lies in the sub-bundle

W = Λ^{1} ⊗ (𝔤^{⊥} · ϕ) ⊂ Λ^{1} ⊗ Λ^{p},

considering Λ^{2} as the Lie algebra of SO_{n}, with 𝔤 ⊂ Λ^{2} the Lie algebra of G and 𝔤^{⊥} its orthogonal complement. In particular, W ≅ Λ^{1} ⊗ 𝔤^{⊥}.

Remark: We use repeatedly here and in the rest of the paper the natural identification between G-representations and their associated bundles on a manifold with a G-structure. Thus, V corresponds to the tangent bundle, V^{∗} to the contangent bundle, Λ^{p}(V^{∗}) to Λ^{p}M (so we can abbreviate safely both by Λ^{p}), a G-invariant subspace of a representation corresponds to a sub-bundle etc.

Proof. Pick a G-connection ∇_{0} (this always exists, but might have torsion), so ∇_{0}ϕ = 0. Hence ∇ϕ = (∇ − ∇_{0})ϕ = α · ϕ where α = ∇ − ∇_{0} is a Λ^{2}-valued 1-form acting on Λ^{p} by the derivative of the SO_{n}-action. Since ϕ is G−invariant, 𝔤 ⊂ Λ^{2} acts trivially, hence α · ϕ ∈ Λ^{1} ⊗ 𝔤^{⊥} · ϕ. Since G is the isotropy of ϕ, the kernel of ·ϕ is precisely 𝔤, and so ·ϕ : 𝔤^{⊥} → 𝔤^{⊥} · ϕ is an isomorphism.

Now consider the intrinsic-torsion space W = Λ^{1}⊗(𝔤^{⊥}·ϕ), and decompose it orthogonally into G-irreducible subspaces, W = W_{1} ⊕ W_{2} ⊕ …, so that the intrinsic torsion tensor decomposes accordingly, ∇ϕ = (∇ϕ)_{1} + (∇ϕ)_{2} + …, and |∇ϕ|^{2} = ∑_{i} | (∇ϕ)_{i} |^{2}.

Now consider the maps alt : W → Λ^{p+1} and int : W → Λ^{p−1}. Since these are G-equivariant maps, their restrictions to the irreducible subspaces W_{i} ⊂ W are homotethies onto their images, i.e. |alt(w_{i})|^{2} = a_{i}|w_{i}|^{2} and |int(w_{i})|^{2} = b_{i}|w_{i}|^{2} for all w_{i} ∈ W_{i} and some non-negative constants a_{i}, b_{i}.

Now let us make the assumption that the G-representation W is multiplicity-free, i.e. that the irreducible G-representations W_{i} are mutually non-isomorphic. This assumption holds for all the groups on the Berger list except SU_{n} and Sp_{n}. In these cases we can still proceed but we get a more complicated formula; see the section on SU_{n} below for details.

Under this assumption we have that the maps alt and int map the summands W_{i} onto mutually orthogonal subspaces of Λ^{p+1} and Λ^{p−1} (resp.), hence

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

and similarly

|d^{∗}ϕ|^{2} = |int(∇ϕ)|^{2} = ∑_{i} b_{i}|(∇ϕ)_{i}|_{T}^{2} .

Plugging all this into the left hand side of formula (4), we get an expression of the form ∑_{i} c_{i} ∫_{M} | (∇ϕ)_{i} |^{2}, for some real constants c_{i}.

To study the right hand side of (4) (the curvature expression), let us decompose Λ^{2} = 𝔤 ⊕ 𝔤^{⊥}, and further decompose 𝔤^{⊥} into irreducible G−subspaces, 𝔤^{⊥} = ⨁_{k} V_{k}.

It follows, by G−equivariance and the Schur lemma, that each of the maps ·ϕ : V_{k} → Λ^{p}, α ↦ α · ϕ, is a homotethy, i.e. there is some constant d_{k} > 0 such that |α · ϕ|^{2} = d_{k}|α|^{2} for all α ∈ V_{k}.

Let us denote by tr (R, V_{k}) the trace of the restriction of the curvature operator R to V_{k}, followed by orthogonal projection onto V_{k} (the trace of “the kk−block” of R).

Now let us assume that the G-representation 𝔤^{⊥} is multiplicity-free, i.e. that all the irreducible Grepresentations V_{k} are mutually non-isomorphic. This assumption holds for all the groups on the Berger list except for Sp_{n}; in particular, it holds for all the 4 groups considered in this article. Under this assumption, we get:

PROPOSITION 2 Let G ⊂ O_{n} be the stabilizer subgroup of a p-form ϕ ∈ Λ^{p}(V^{∗}) such that 𝔤^{⊥} ⊂ 𝔰𝔬_{n} = Λ^{2} is a multiplicity-free G−representation. Let M be an n−dimensional manifold with a G-structure, ϕ the associated p-form on M and R the curvature tensor of the associated riemannian metric. Then

⟨R̃ϕ, ϕ⟩ = ∑_{k} d_{k} tr (R, V_{k}),

with notation as above.

Proof. The decomposition Λ^{2} = 𝔤 ⊕ (⨁_{k} V_{k}) induces a “block” decomposition of the curvature tensor R ∈ Λ^{2} ⊗ Λ^{2},

R = R_{00} + ∑_{k} R_{0k} + ∑_{k≠k′} R_{kk′} + ∑_{k} R_{kk},

where R_{00} ∈ 𝔤 ⊗ 𝔤, R_{0k} ∈ (𝔤 ⊗ V_{k}) ⊕ (V_{k} ⊗ 𝔤), R_{kk′} ∈ V_{k} ⊗ V_{k′} and R_{kk} ∈ V_{k} ⊗ V_{k}. We claim that all but the R_{kk} terms are in the kernel of the map R ↦ ⟨R̃ϕ, ϕ⟩.

Indeed, if R = α ⊗ β ∈ Λ^{2} ⊗ Λ^{2}, where α ∈ 𝔤 or β ∈ 𝔤, then

⟨R̃ϕ, ϕ⟩ = ⟨α · (β · ϕ), ϕ⟩ = −⟨β · ϕ, α · ϕ⟩ = 0,

since the map α· : Λ^{k} → Λ^{k} is skew-symmetric (being the derivative of an isometric SO_{n}-action), and 𝔤 is in the kernel of ·ϕ : Λ^{2} → Λ^{p} since ϕ is G−invariant.

Similarly, if R = α ⊗ β with α ∈ V_{k} and β ∈ V_{k′} for k ≠ k′, then again ⟨R̃ϕ, ϕ⟩ = −⟨β · ϕ, α · ϕ⟩ = 0, since V_{k} · ϕ and V_{k′} · ϕ, being irreducible and non-isomorphic (or 0), are orthogonal subspaces of Λ^{p}.

Finally, if R = α ⊗ β with α, β ∈ V_{k}, then

⟨R̃ϕ, ϕ⟩ = −⟨β · ϕ, α · ϕ⟩ = −d_{k}⟨β, α⟩ = d_{k} tr (R, V_{k}).

Remark: Observe that if the holonomy of M is contained in G, i.e. ∇ϕ = 0, then ⟨R̃ϕ, ϕ⟩ = 0, since R takes values in 𝔤 (in the notation of the last proof, R = R_{00}). This is also evident from equation (3), since ∇ϕ = 0 implies Δϕ = 0 and Δ|ϕ|^{2} = 0.

4 Calculations

In each of the following examples we take a subgroup G ⊂ SO_{n} and do the following:

1. Define G as the stabilizer of a p−form ϕ.

2. Determine the decompositions Λ^{2} = 𝔤 ⊕ 𝔤^{⊥} and 𝔤^{⊥} = ⨁_{k} V_{k} under G, then further G−decompose W = Λ^{1} ⊗ (𝔤^{⊥} · ϕ) = ⨁_{k} W_{k}, thus obtaining a decomposition of ∇ϕ = ∑_{k}(∇ϕ)_{k}.

3. Determine the norms of dϕ = alt(∇ϕ) and d^{∗}ϕ = −int(ϕ) in terms of the components (∇ϕ)_{k}.

4. Interpret the curvature term ⟨R̃ϕ, ϕ⟩ by means of some standard curvature invariants.

5. Plug everything into formula (4) and obtain a formula relating the L_{2}-norms of the various components of ∇ϕ and curvature invariants.

6. Give some examples and applications of the formula obtained.

5 U_{n}

5.1 Definition of a U_{n} structure

U_{n} is the stabilizer in O_{2n} of the “Kahler 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 U_{n}-structure on a 2n-dimensional manifold is thus given by a riemannian metric together with a 2-form ω of the above type; i.e. one can find, for each x ∈ M, a linear (real) isometry ℂ^{n} → T_{x} which pulls-back the 2-form to the above standard example.

A manifold with a U_{n}-structure is often called an almost-hermitian manifold.

5.2 Decomposition of ∇ω

The decomposition of the intrinsic torsion space W = Λ^{1} ⊗ [(u_{n})^{⊥} · ω] into 4 irreducible subspaces under U_{n} was done in [13]. Here we review briefly the decomposition.

In order to decompose W = Λ^{1}⊗[(u_{n})^{⊥}·ω] we decompose its complexification W_{ℂ} = W ⊗_{ℝ}ℂ into (complex) irreducible subspaces, then take their real part. We use the shorthand notation Λ^{p,q} for Λ^{p,q}(V^{∗}).

Now

Λ^{2} ⊗ ℂ = Λ^{1,1} ⊕ Λ^{2,0} ⊕ Λ^{0,2},

with

𝔲_{n} ⊗ ℂ = Λ^{1,1},

so

(𝔲_{n})^{⊥} ⊗ ℂ = Λ^{2,0} ⊕ Λ^{0,2}.

Next, to determine (𝔲_{n})^{⊥} · ω we note that ω is the 2-form corresponding to J under our identification of anti-symmetric endomorphisms with 2-forms, hence for any (p, q)-form α we have immediately

ω · α = i(q − p)α,  α ∈ Λ^{p,q},  (5)

so 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^{1,0} ⊕ Λ^{0,1}) ⊗ (Λ^{2,0} ⊕ Λ^{0,2}) =
= (Λ^{1,0} ⊗ Λ^{2,0}) ⊕ (Λ^{1,0} ⊗ Λ^{0,2}) + conjugate.

Now

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

where

• W_{1} ⊗ ℂ = Λ^{3,0} + conjugate,

• W_{2} ⊗ ℂ = the image of all tensors in Λ^{1,0} ⊗ Λ^{1,0} ⊗ Λ^{1,0} + conjugate under the Young symmetrizer (1 − (23))(1 + (12)), or the kernel (in W ⊗ ℂ) of 1 + (123) + (132) (the “Bianchi identity” operator),

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

• W_{4} ⊗ ℂ = image of Λ^{1} ⊗ ℂ under wedging with ω ≃ Λ^{1} ⊗ ℂ.

We thus obtain that the intrinsic torsion tensor ∇ω of a U_{n}-structure decomposes, according to the above irreducible decomposition of W, into 4 summands, ∇ω = (∇ω)_{1} ⊕ … ⊕ (∇ω)_{4}.

Remarks:

1. Note that W = 0 for n = 1 and W_{1} = W_{3} = 0 for n = 2 (i.e. for manifolds of real dimension 2 and 4 resp.).

2. For n ≥ 3 the four summands are non-zero and distinct (as U_{n}-representations), hence mutually orthogonal.

In [13] one can find the geometric meaning of the components of ∇ω and the terminology associated with the vanishing of certain subgroups of the 4 components. We list in the following table some of the possibilities:

Vanishing components of ∇ω Name

Geometric meaning

all  Kähler  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

Other types of U_{n}-structures have also received attention (for example the locally conformally Kähler manifolds); we refer the reader to [13] for the full list.

5.3 Summary of calculations for U_{n}

summand w_{i} ∈ W_{i} ⊗ ℂ  |w_{i}|_{T}^{2}  |alt(w_{i})|^{2} |int(w_{i})|^{2}
W_{1}  z_{1} ∧ z_{2} ∧ z_{3}  1/6  1  0
W_{2}  (not important)  (not important)  0  0
W_{3}  z_{1} ⊗ (z̄_{2} ∧ z̄_{3})  1/2  1  0
W_{4}  ∑_{j=2}^{n} z_{j} ⊗ (z̄_{j} ∧ z̄_{1})  (n − 1)/2  n − 1  (n − 1)^{2}

Explanation:

• To explain the zeros in the table, one verifies that the U_{n} representation Λ^{1} is not isomorphic to W_{1}, W_{2} or W_{3}, hence int(W_{1}) = int(W_{2}) = int(W_{3}) = 0, and that Λ^{3} does not contain a subspace isomorphic to W_{2} hence alt(W_{2}) = 0.

• For the remaining entries, we pick an element w_{i} in each irreducible subspace W_{i} ⊂ W, 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}.

• Let w_{1} = z_{1} ∧ z_{2} ∧ z_{3}. Then alt(w_{1}) = w_{1}, hence |w_{1}|_{T}^{2} = 1/6, |alt(w_{1})|^{2} = 1.

• Let w_{3} = z_{1} ⊗ (z̄_{2} ∧ z̄_{3}). Then alt(w_{3}) = z_{1} ∧ z̄_{2} ∧ z̄_{3}, hence |w_{3}|_{T}^{2} = 1/2, |alt(w_{3})|^{2} = 1.

• Let w_{4} = Σ_{j=2}^{n}z_{j} ⊗ (z̄_{j} ∧ z̄_{1}) (this corresponds to ω ∧ z̄_{1}, up to a constant, under the isomorphism Λ^{1,2} ≅ Λ^{1,0} ⊗ Λ^{0,2}.) Then

alt(w_{4}) = Σ_{j=2}^{n}z_{j} ∧ z̄_{j} ∧ z̄_{1}, int(w_{4}) = (n − 1)z̄_{1},

hence

|w_{4}|_{T}^{2} = ∑_{j=2}^{n} |z̄_{j} ∧ z̄_{1}|_{T}^{2} = (n − 1)/2, |alt(w_{4})|^{2} = (n − 1), |int(w_{4})|^{2} = (n − 1)^{2}.

5.4 The U_{n} formula

According to Proposition 2, the curvature term in Formula (4) can be written as

⟨R̃ω, ω⟩ = d tr (R, 𝔲_{n}^{⊥}),

for some constant d > 0, given by d = |α · ω|^{2} for a unitary α ∈ 𝔲_{n}^{⊥} ⊗ ℂ = Λ^{2,0} ⊕ Λ^{0,2}. Now for any α ∈ Λ^{2,0} we have α · ω = −ω · α = 2iα, so d = 4.

We would like now to express this ⟨R̃ω, ω⟩ term using the more standard U_{n}-invariants s (the scalar curvature) and s^{∗} (the ∗−scalar curvature), defined by

s = ∑_{i≠j}⟨R_{ij}(e_{j}), e_{i}⟩ = 2 tr (R),
s^{∗} = ∑_{i≠j}⟨R_{ij}(Je_{j}), Je_{i}⟩ = 2 tr (JR).

Now 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⟩,

Denote by ∇ω = (∇ω)_{1} + (∇ω)_{2} + (∇ω)_{3} + (∇ω)_{4} the decomposition of ∇ω, where (∇ω)_{i} ∈ W_{i}. Then dω = alt(∇ω) = ∑_{i} alt(∇ω)_{i}. Now since the alt(∇ω)_{i} lie in distinct U_{n}-representations (see remark above) they are mutually orthogonal, hence we get from the table

|dω|^{2} = |alt(∇ω)|^{2} = ∑_{i=1}^{4} |alt(∇ω)_{i}|^{2} = 6|(∇ω)_{1}|^{2} + 2|(∇ω)_{3}|^{2} + 2|(∇ω)_{4}|^{2}.

Similiarly,

|d^{∗}ω|^{2} = |int(∇ω)|^{2} = 2(n − 1)|(∇ω)_{4}|^{2}.

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 (4) all our previous calculation, our formula for the U_{n} case reduces to

2E_{1} − E_{2} + (n − 1)E_{4} = \frac{1}{2} ∫ s − s^{∗}  (6)

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

5.5 Examples

5.5.1. The 6-sphere. Let M = S^{6} with the standard (round) metric. If we view ℝ^{7} as the space of imaginary Cayley numbers then there is an almost-complex structure on S^{6} defined as follows: for an x ∈ S^{6} and v ∈ T_{x}S^{6} = x^{⊥} we define J(v) as the cross-product of x and v. This cross product in ℝ^{7} is defined in terms of the multiplication of Cayley numbers in the same way as the usual cross-product in ℝ^{3}(=imaginary quaternions) is defined via the multiplication of quaternions: the cross product of two imaginary Cayley numbers is the imaginary part of their Cayley product (the real part is just the usual euclidean dot product). One can verify [11] that this gives a U_{3}−structure of type W_{1} on S^{6}, a so-called “nearly-Kahler” structure, i.e. ∇ω = dω. Now for the round metric we have R = id_{Λ^{2}}, hence

s = 2 tr (R) = 2 dim(Λ^{2}) = 30,
s^{∗} = 2 tr (JR) = 2[dim(u_{3}) − dim(u_{3}^{⊥})] = 2(9 − 6) = 6.

Our formula (4) for this example thus reads E_{1} = 6 V ol(S^{6}). Since this an homogeneous structure (G_{2}- invariant), dω must have constant norm, hence we get |dω|^{2} = 6, which one should be able to verify directly.

5.5.2. Product of odd spheres. Let M = S^{2m+1} × S^{2n+1} ⊂ ℂ^{m+1} × ℂ^{n+1}, with the product of standard metrics. Then M admits a compatible (orthogonal) complex structure due to Hopf, Calabi-Eckmann, defined as follows: for (v, w) ∈ T_{(x,y)}(S^{2m+1} ×S^{2n+1}), where v ∈ T_{x}(S^{2m+1}) and w ∈ T_{y}(S^{2n+1}), decompose orthogonally v = v_{0} + iax, w = w_{0} + iby, where v_{0} ∈ (ℂx)^{⊥}, w_{0} ∈ (ℂy)^{⊥} and a, b ∈ ℝ. Then define

J(v, w) = (iv_{0} − ibx, iw_{0} + iay).

One can verify that this gives a homogeneous (U_{m+1} × U_{n+1}-invariant) U_{m+n+1}−structure of type W_{3} ⊕ W_{4} on M and of type W_{4} if n = 0, [13].

We have

R_{S^{2m+1}×S^{2n+1}} = R_{S^{2m+1}} ⊕ R_{S^{2n+1}} = Id_{Λ^{2}(T S^{2m+1})} ⊕ Id_{Λ^{2}(T S^{2n+1})},

hence

s = 2 tr (R_{M}) = 2 [\binom{2m + 1}{2} + \binom{2n + 1}{2}] = 2[m(2m + 1) + n(2n + 1)].

To calculate s^{∗} = 2 tr (RJ) we pick a basis of Λ_{(x,y)}^{2}M and sum the diagonal matrix elements of RJ. To this end, note that by definition of the almost complex structure J we can find an orthonormal basis of T_{(x,y)}(S^{2m+1} × S^{2n+1}) of the form

{e_{1}, Je_{1}, …, e_{m}, Je_{m}, e_{m+1}, Je_{m+1}, f_{1}, Jf_{1}, …, f_{n}, Jf_{n}}

with {e_{1}, Je_{1}, …, e_{m}, Je_{m}, e_{m+1}} a basis for T_{x}(S^{2m+1}) and {Je_{m+1}, f_{1}, Jf_{1}, …, , f_{n}, Jf_{n}} a basis for T_{y}(S^{2n+1}). By wedging pairs of elements of this basis we get an orthonormal basis of Λ_{(x,y)}^{2}M and one checks that the only elements of this induced basis whose contribution to tr (RJ) is non-vanishing are e_{1} ∧ Je_{1}, …, e_{m} ∧ Je_{m}, f_{1} ∧ Jf_{1}, …, f_{n} ∧ Jf_{n}, and that the contribution of each of these is 1. We thus get

s^{∗} = 2(m + n),

and so

s − s^{∗} = 4(m^{2} + n^{2}).

Our U_{m+n+1}−formula for this case thus reads

(m + n)E_{4} = 2(n^{2} + m^{2})V ol(M),

therefore by homogeneity, |(∇ω)_{4}|^{2} = 2(n^{2} + m^{2})/(m + n) identically .

5.6 Applications

Some of the following applications appeared in [15] and [10], 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.

5.6.1. Complex structures on hyperbolic manifolds. In dimension 2 it is well-known that all oriented riemannian surfaces admit a compatible complex structure (i.e. an orthogonal almost complex structure which is integrable): take J to be a rotation by 90 degrees in the “positive” direction and observe that with this J the resulting U(1)-structure is automatically Kähler (W ≡ 0 for n = 1, in the terminology of this section); in particular, the almost-complex structure J is integrable. In higher dimensions the situation is considerably more delicate, with large classes of riemannian metrics which do not admit orthogonal complex structures. The simplest such class to study, from our point of view, is that of hyperbolic manifolds.

Recall that a hyperbolic manifold is a complete riemannian manifold whose curvature operator is R = −Id_{Λ^{2}}. An immediate consequence of our Bochner formula for U_{n}, n ≥ 2, is that

• a compact hyperbolic manifold of dimension ≥ 4 does not admit an orthogonal complex structure, [15].

For this we only need to observe that we have different signs on the two sides of the equation (6): integrability of the almost complex structure is equivalent to E_{1} = E_{2} = 0, hence the left hand side is non negative, whereas the integrand on the right hand side is (s − s^{∗})/2 = 2 tr (R, 𝔲_{n}^{⊥}) = −2 dim(𝔲_{n}^{⊥}) < 0.

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 4manifold does not admit a complex structure (orthogonal or not), see [18]. 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.

Finally, we notice that our argument above in fact yields the somewhat stronger result:

• R < 0 ⟹ E_{2} ≠ 0 (on a compact riemannian manifold with negative definite curvature operator the component (∇ω)_{2} cannot vanish identically).

5.6.2. Conformally flat manifolds. A conformally flat manifold (i.e. locally conformal to euclidean space) is characterized by the vanishing of its Weyl tensor (see [2] p. 60). For example, a hyperbolic manifold is conformally flat. In the presence of an orthogonal complex structure on such a manifold the right hand side of formula (6) simplifies due to the following lemma:

LEMMA 1 On a conformally flat almost-hermitian 2n-dimensional manifold one has

s = (2n − 1)s^{∗}.

Proof. The space of curvature-like tensors on ℝ^{2n} with vanishing Weyl tensor is U_{n}-equivariantly (even O_{2n}- equivariantly) isomorphic to the space of quadratic forms on ℝ^{2n} (essentialy 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 our formula (6) reduces to

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

For an hermitian structure (E_{1} = E_{2} = 0) only the E_{4} term survives, hence • on a conformally flat, hermitian compact manifold ∫_{M} s ≥ 0, with equality iff the structure is Kahler.

This was proved by Gauduchon in [10]. Observe that this includes our previous result, that on a compact real hyperbolic manifold there can be no orthogonal complex structure, since a hyperbolic manifold is conformally flat with s < 0.

A similar application is to an almost-Kähler manifold (E_{1} = E_{3} = E_{4} = 0), where only the E_{2} term survives, hence

• On a conformally flat, almost-Kähler compact manifold ∫_{M} s ≤ 0, with equality iff the structure is Kähler.

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. (Such a manifold is sometimes called also “half conformally flat”). This class of manifolds received considerable attention because of the so-called “twistor construction”, relating them to a certain class of complex 3-folds (see for example [2]). We thus get,

• In dimension 4, the last two results remain valid for the larger class of compact ASD almost hermitian manifolds.

Further information on the relation between hermitian geometry and the Weyl tensor may be found in [10] and the references therein.

5.6.3. Locally symmetric spaces. A locally symmetric space is by definition a riemannian manifold covered by a symmetric space G/K. Some symmetric spaces G/K admit a G-invariant orthogonal complex structure, unique up to a sign if G is simple. These are called hermitian symmetric spaces (though they are in fact Kähler). If G is a product of k simple groups then we get 2^{k} G-invariant orthogonal complex structures on G/K. We will now show that, for a compact locally symmetric space of non-compact type (the covering space G/K is a product of irreducible non-compact non-flat symmetric spaces), the only way to get an orthogonal complex structure is by descending one of these G-invariant complex structures; namely,

• a compact locally symmetric space M of non-compact type with an orthogonal complex structure is necessarily the quotient of an hermitian symmetric space G/K of non-compact type, and the orthogonal complex structure on M (which is in fact Kähler) descends from one of the 2^{k} G-invariant orthogonal complex structures on G/K , where G is a product of k simple groups.

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

This result was proved by P. Gauduchon [10] in case G/K is irreducible and in [15] 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 (6), 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^{⊥} (w.r.t. the Killing form of 𝔤).

LEMMA 2 Suppose J ∈ End(𝔭) is an orthogonal almost 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 an 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 [7] .

LEMMA 3 Let R be the curvature operator of a symmetric space G/K of non-compact type. Then for any X, Y, X′, Y′ ∈ 𝔭 ≅ T_{[K]}G/K, ⟨R(X ∧ Y), X′ ∧ Y′⟩ = −⟨[X, Y], [X′, Y′]⟩.

This follows from [14], theorem 4.2, p. 215.

Now let us use these lemmas to prove our result. Let J be an orthogonal complex structure on M, fix a point x ∈ M and identify T_{x}M ≅ 𝔭. Next pick a unitary basis {u_{i}} for T_{x}^{1,0}M ≅ 𝔭^{1,0} and use lemma 3 above to calculate the integrand on the right hand of formula (6) at x:

(s − s^{∗})/2 = 2 tr (R, 𝔲_{n}^{⊥}) = 2 ∑_{i<j} ⟨R(u_{i} ∧ u_{j}), \overline{u_{i} ∧ u_{j}}⟩ = −2 ∑_{i<j}⟨[u_{i}, u_{j}], \overline{[u_{i}, u_{j}]}⟩ ≤ 0.

But on the left hand side of formula (6) we have a non-negative expression (E_{1} = E_{2} = 0 for an orthogonal complex structure) so we conclude that 𝔭^{1,0} is an abelian subalgebra of 𝔤 ⊗ ℂ. Lemma 2 above then implies that G/K is hermitian and the value of J at x comes from one of the (finitely many) G-invariant complex structures on G/K. By continuity, J corresponds to one of the G-invariant complex structures on G/K. QED

Note that for a locally symmetric space of compact type this result does not hold, as the example of the Calabi-Eckmann hermitian structure on the product of odd-sphere shows. However, in the compact-type case we can modify the above argument as follows: the curvature expression in lemma 3 changes sign, hence the right-hand side of formula (6) is non-negative and the left hand side for an almost-Kahler structure is non-positive. Thus

• a compact locally symmetric space M of compact type with a compatible symplectic structure is necessarily the quotient of an hermitian symmetric space G/K of compact type, and the symplectic structure on M (which is in fact Kähler) is one of the 2^{k} G-invariant symplectic structures on G/K, where G is a product of k simple groups.

For example, this determines the symplectic structures on ℂP^{n} (or products of them), compatible with the standard (Fubini-Study) metric.

6 SU_{n}, n ≥ 3.

6.1 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}. In fact, for n ≥ 3, SU_{n} is the stabilizer of ψ in O_{2n}.

Let us prove the last statement. We check it first on the level of Lie algebras. We decompose

Λ^{2} ≅ 𝔰𝔬_{2n} = 𝔰𝔲_{n} ⊕ ℝω ⊕ 𝔲_{n}^{⊥},

and argue that since these subspaces are SU_{n}-irreducible, the map ·ψ : α ↦ α · ψ is either 0 or injective when restricted to each of these, and since they are non-isomorphic the kernel of ·ψ is the direct sum of those for which it is 0 (this is where the argument fails for n = 2, since 𝔰𝔲_{2}^{⊥} = ℝω ⊕ 𝔲_{2}^{⊥} is a trivial 3-dimensional SU_{2}-representation). We check each of the three summands:

• 𝔰𝔲_{n}: SU_{n} stabilizes ψ (by definition) so 𝔰𝔲_{n} · ψ = 0.

• ℝω: by formula (5), ω · ψ = −inψ, hence ℝω · ψ ≠ 0.

• 𝔲_{n}^{⊥}: pick α = z_{1} ∧ z_{2} + z̄_{1} ∧ z̄_{2}, then

α · ψ = (z̄_{1} ∧ z̄_{2}) · ψ = [(z̄_{1} ∧ z̄_{2}) · (z_{1} ∧ z_{2})] ∧ z_{3} ∧ ⋯ ∧ z_{n} =
= −(z_{1} ∧ z̄_{1} + z_{2} ∧ z̄_{2}) ∧ z_{3} ∧ ⋯ ∧ z_{n} ≠ 0,

hence 𝔲_{n}^{⊥} · ψ ≠ 0 as well.

We conclude that 𝔰𝔲_{n} is the stabilizer of ψ in 𝔰𝔬_{2n}. It follows that the Lie algebra of the stabilizer of ψ in SO_{2n} is 𝔰𝔲_{n}.

To complete the argument we need to show that if an element g ∈ O_{2n} satisfies g^{∗}ψ = ψ then this implies g^{∗}ω = ω. Now, if g^{∗}ψ = ψ then we deduce from the equation

ω · ψ = −inψ

(see Equation (5) in the section on U_{n}) that

(g^{∗}ω) · ψ = (g^{∗}ω) · (g^{∗}ψ) = g^{∗}(ω · ψ) = −in(g^{∗}ψ) = −inψ = ω · ψ,

hence (g^{∗}ω − ω) · ψ = 0. But we know, from the first part of the proof, the 𝔰𝔲_{n} is the kernel of the map ·ψ : α ↦ α · ψ, hence (g^{∗}ω − ω) ∈ 𝔰𝔲_{n}. On the other hand, since g^{∗} commutes with α ↦ α · ψ it preserves its kernel 𝔰𝔲_{n}, and since it acts orthogonaly on Λ^{2} it also preserves the orthogonal complement 𝔰𝔲_{n}^{⊥}. Since ω ∈ 𝔰𝔲_{n}^{⊥} it follows that (g^{∗}ω − ω) ∈ 𝔰𝔲_{n}^{⊥}. Together with (g^{∗}ω − ω) ∈ 𝔰𝔲_{n} we get g^{∗}ω − ω = 0. QED

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

6.2 Decomposition of ∇η

The Lie algebra 𝔲_{n} ⊂ Λ^{2} of U_{n} decomposes under SU_{n} as

𝔲_{n} = 𝔰𝔲_{n} ⊕ ℝω,

hence

𝔰𝔲_{n}^{⊥} = ℝω ⊕ 𝔲_{n}^{⊥}.

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.

Next, to decompose Λ^{1}⊗[(𝔲_{n})^{⊥}·η] into irreducibles we use the decomposition of Λ^{1}⊗(𝔲_{n})^{⊥} into 4 irreducibles W_{1}, …, W_{4} from our calculation for the U_{n} case, and give their corresponding images the same names here. We thus obtain the decomposition

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

In fact, it will be useful for what follows to establish the precise relation between the (∇η)_{i} components and the (∇ω)_{i} components for i = 1, …, 4.

LEMMA 4 For an SU_{n} structure with Kähler form ω and volume form η as above, we have the orthogonal decomposition

∇η = \frac{1}{2}⟨∇η, η^{⊥}⟩η^{⊥} − \frac{1}{2}(∇ω) · η^{⊥},

with ·η^{⊥} : Λ^{1} ⊗ Λ^{2} → Λ^{1} ⊗ Λ^{n} given by θ ⊗ α ↦ θ ⊗ (α · η^{⊥}).

In other words,

(∇η)_{0} = \frac{1}{2}⟨∇η, η^{⊥}⟩η^{⊥} ∈ W_{0}, (∇η)_{i} = −\frac{1}{2}(∇ω)_{i} · η^{⊥} ∈ W_{i}, i = 1, …, 4.

Proof. Begin with

ω · ψ = −inψ,

so

(∇ω) · ψ + ω · (∇ψ) = −in(∇ψ).

Now decompose ∇ψ into types,

∇ψ = (∇ψ)^{n,0} + (∇ψ)^{n−1,1},

and so

ω · (∇ψ) = ω · (∇ψ)^{n,0} + ω · (∇ψ)^{n−1,1} =
= −in(∇ψ)^{n,0} − i(n − 2)(∇ψ)^{n−1,1},

and so

(∇ω) · ψ − in(∇ψ)^{n,0} − i(n − 2)(∇ψ)^{n−1,1} = −in[(∇ψ)^{n,0} + (∇ψ)^{n−1,1}],

thus

(∇ω) · ψ = −2i(∇ψ)^{n−1,1}.

Conjugating, we get

(∇ω) · \overline{ψ} = 2i(∇\overline{ψ})^{1,n−1}.

And so,

(∇ω) · η^{⊥} = (∇ω) · (\frac{ψ − \overline{ψ}}{i} ) = −2[(∇ψ)^{n−1,1} + conjugate] = −2 ∑_{i=1}^{4}(∇η)_{i}.

It follows, by SU_{n} equivariance, that

(∇ω)_{i} · η^{⊥} = −2(∇η)_{i}, i = 1, …, 4.  (7)

Remark: Note that in this case we do have two isomorphic summands: W_{0} ≅ W_{4} ≅ V. This complicates the situation, as we shall see later (appearance of “mixed terms” …).

6.3 Summary of calculations for SU_{n}, n ≥ 3.

summand element ∈ W_{i} ⊗ ℂ  |w_{i}|_{T}^{2}  |alt(w_{i})|^{2} = |int(w_{i})|^{2}
W_{0}  z_{1} η  2/n!  1
W_{1}  {z_{1}(z_{23} · η) + z_{2}(z_{31} · η) + \atop + z_{3}(z_{12} · η)}  6/n!  12
W_{2}  z_{1}(z_{23} · η) + z_{2}(z_{13} · η)  4/n!  2
W_{3}  z_{1}(z¯2¯3 · η)  (not important)  0
W_{4}  ∑_{k=2}^{n} z_{k}(zk̄¯1 · η)  2(n − 1)/n!  (n − 1)^{2}

Explanation:

• We use throughout a unitary basis z_{1}, …, z_{n} for Λ^{1,0}, and the notation ψ = z_{1} ∧ … ∧ z_{n}, η = ψ + \overline{ψ}, z_{ij} = z_{i} ∧ z_{j}, z_{ij̄} = z_{i} ∧ z̄_{j}, … etc.. We also drop, especially in long expressions, the use of the tensor product symbol ⊗. Thus, for example, z_{1}(z_{23} · η) stands for z_{1} ⊗ [(z_{2} ∧ z_{3}) · (z_{1} ∧ … ∧ z_{n} + z̄_{1} ∧ … ∧ z̄_{n})].

• We claim that for any w ∈ W, |alt(w)| = |int(w)|. Proof: first note that the pull-back action of SO_{2n} on the exterior algebra on V^{∗} commutes with the Hodge star operator ∗, hence the same holds for the Λ^{2} action on the exterior algebra on V^{∗}. Since ∗η = ±η, it follows that ∗(α · η) = ±α · η for any α ∈ Λ^{2}. Next, using the identity alt(θ ⊗ ϕ) = ± ∗ int(θ ⊗ ∗ϕ), holding for any θ ⊗ ϕ ∈ Λ^{1} ⊗ Λ^{p} and the fact that the Hodge star is an isometry of the exterior algebra, we obtain

|alt(θ ⊗ (α · η))| = |int(θ ⊗ ∗(α · η))| = |int(θ ⊗ (α · η))|.

Since W is spanned by elements of the form θ ⊗ (α · η), θ ∈ Λ^{1}, α ∈ Λ^{2}, the claim is proved.

• Let w_{0} = z_{1} ⊗ η. Then

alt(w_{0}) = z_{1} ∧ \overline{ψ},

hence

|w_{0}|_{T}^{2} = 2/n!, |alt(w_{0})|^{2} = 1.

• To obtain elements w_{i} ∈ W_{i} ⊗ ℂ, i = 1, …, 4, we take the elements we found for U_{n} in Λ^{1} ⊗ 𝔲_{n}^{⊥}, and map them under the map given by θ ⊗ α ↦ θ ⊗ (α · η). Now the map 𝔲_{n}^{⊥} → Λ^{n} given by α ↦ α · η is an homothetic embedding (the domain is irreducible), so it suffices to check its effect on a single element α ∈ 𝔲_{n}^{⊥} ⊗ ℂ, say α = z_{12}. Then

|α · η|^{2} = |(z_{1¯1} + z_{2¯2}) ∧ z̄_{3} ∧ ⋯ ∧ z̄_{n}|^{2} = 2 = 2|α|^{2},

hence for any θ ⊗ α ∈ Λ^{1} ⊗ 𝔲_{n}^{⊥}

|θ ⊗ (α · η)|_{T}^{2} = |θ|^{2}|α · η|_{T}^{2} = |θ|^{2} \frac{|α · η|^{2}}{n!} = |θ|^{2} \frac{2|α|^{2}}{n!} = |θ|^{2} \frac{4|α|_{T}^{2}}{n!}
= \frac{4}{n!}|θ ⊗ α|_{T}^{2} .

• w_{1} is the image of z_{1} ∧ z_{2} ∧ z_{3}, which is z_{1} ⊗ z_{23} + z_{2} ⊗ z_{31} + z_{3} ⊗ z_{12}, up to a constant, i.e.

w_{1} = z_{1} ⊗ (z_{23} · η) + … etc.,

where “… etc.” denotes 2 more similar terms obtained by cyclic permutation of 1, 2, 3. Now

alt(w_{1}) = alt (z_{1} ⊗ (z_{23} · η)) + … etc. = z_{1} ∧ (z_{23} · η) + … etc.
= 2 (z_{1¯12¯2} + z_{2¯23¯3} + z_{3¯31¯1}) ∧ z̄_{4} ∧ … ∧ z̄_{n},

and so

|w_{1}|_{T}^{2} = \frac{4}{n!}|z_{1} ⊗ z_{23} + … etc.|_{T}^{2} = \frac{4}{n!} · \frac{3}{2} = \frac{6}{n!}
|alt(w_{1})|^{2} = 4 · 3 = 12.

• w_{2} is the image of z_{1} ⊗ z_{23} + z_{2} ⊗ z_{13}, i.e.

w_{2} = z_{1} ⊗ (z_{23} · η) + z_{2} ⊗ (z_{13} · η),

so

alt(w_{2}) = (z_{1¯13¯3} − z_{2¯23¯3}) ∧ z̄_{4} ∧ … ∧ z̄_{n},

hence

|w_{2}|_{T}^{2} = \frac{4}{n!}(2 · \frac{1}{2}) = \frac{4}{n!},
|alt(w_{2})|^{2} = 2.

• w_{3} is the image of z_{1} ⊗ z¯2¯3, i.e. w_{3} = z_{1} ⊗ (z¯2¯3 · η), hence

alt(w_{3}) = ±alt (z_{1} ⊗ [z_{1} ∧ (z_{2¯2} + z_{3¯3}) ∧ z̄_{4} ∧ … ∧ z̄_{n}]) = 0.

• w_{4} is the image of Σ_{k=2}^{n}z_{k} ⊗ zk̄¯1, i.e.

w_{4} = Σ_{k=2}^{n}z_{k} ⊗ (zk̄¯1 · η)

so

alt(w_{4}) = (n − 1)z̄_{1} ∧ ψ,

hence

|w_{4}|_{T}^{2} = 2(n − 1)/n!,
|alt(w_{4})|^{2} = (n − 1)^{2}.

Denote by ∇η = (∇η)_{0} + (∇η)_{1} + (∇η)_{2} + (∇η)_{3} + (∇η)_{4} the decomposition of ∇η according to the above decomposition of W, and by F_{0}, …, F_{4} the corresponding L_{2}-norms of these components. We thus have (here we meet the fact that (∇η)_{0} and (∇η)_{4} are not neccesarily orthogonal)

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

= \frac{n!}{2}|(∇η)_{0}|_{T}^{2} + 2n!|(∇η)_{1}|_{T}^{2} + \frac{n!}{2}|(∇η)_{2}|_{T}^{2} + \frac{(n − 1)n!}{2}|(∇η)_{4}|_{T}^{2} +
+2⟨int(∇η)_{0}, int(∇η)_{4}⟩,

so then formula (4) gives

3F_{1} − F_{3} + (n − 2)F_{4} + \frac{4}{n!}F_{04} = \frac{1}{n!} ∫ ⟨R̃η, η⟩,  (8)

where

F_{04} = ∫_{M}⟨int(∇η)_{0}, int(∇η)_{4}⟩.

6.4 The curvature term

We proceed to interpret the various terms in equation (8). We begin with the curvature term.

LEMMA 5

⟨R̃η, η⟩ = \frac{s + s^{∗}}{2} .

Proof. According to proposition 2

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

with d_{1} = |ω · η|^{2}/|ω|^{2} and d_{2} = |α · η|^{2} for a unitary α ∈ Λ^{2,0}.

Let us calculate now each of the terms appearing on the right hand side of the last equation.

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

• d_{2}: for α = z_{12}, |α|^{2} = 1 and

z_{12} · η = z_{12} · \overline{ψ} = (z_{12} · z̄_{12}) ∧ z̄_{3} ∧ … ∧ z̄_{n} = (z_{1¯1} + z_{2¯2}) ∧ z̄_{3} ∧ … ∧ z̄_{n},

so |α · η|^{2} = 2 hence d_{2} = 2.

• tr (R, 𝔲_{n}^{⊥}): this was done in the U_{n} calculation, where we found

tr (R, 𝔲_{n}^{⊥}) = (s − s^{∗})/4.

• tr (R, ℝω): Let b be the Bianchi operator, then bR = 0. Now,

0 = ⟨(bR)ω, ω⟩ = ⟨Rω, ω⟩ − ∑_{j,k}⟨Rz_{jk̄}, z_{j̄k}⟩ − ∑_{j,k}⟨Rz_{jk}, zk̄j̄⟩,

thus

⟨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) = \frac{⟨Rω, ω⟩}{|ω|^{2}} = \frac{s^{∗}}{2n}.

Combining all the above we get the desired result.

Next, we have

LEMMA 6

|(∇η)_{i}|^{2} = \frac{1}{n!}|(∇ω)_{i}|^{2}, i = 1, 2, 3, 4.

Proof. This follows immediately by taking norms on both sides of the identity

2(∇η)_{i} = −(∇ω)_{i} · η^{⊥},  i = 1, 2, 3, 4  (9)

of Lemma 4. Since 𝔲_{n}^{⊥} ⊂ Λ^{2} is irreducible, the map 𝔲_{n}^{⊥} → Λ^{n} given by α ↦ α · η^{⊥} is a homothety onto its image. Pick say α = z_{12} ∈ 𝔲_{n}^{⊥} ⊗ ℂ, then

α · η^{⊥} = i(z_{1¯1} + z_{2¯2}) ∧ z̄_{3} ∧ … ∧ z̄_{n},

hence

|α · η^{⊥}|_{T}^{2} = \frac{1}{n!}|α · η^{⊥}|^{2} = \frac{2}{n!} = \frac{4}{n!}|α|_{T}^{2} ,

for all α ∈ Λ^{2}. It follows, by taking norms in equation (9), that

4|(∇η)_{i}|_{T}^{2} = |(∇ω)_{i} · η^{⊥}|_{T}^{2} = \frac{4}{n!}|(∇ω)_{i}|_{T}^{2} , i = 1, …, 4.

LEMMA 7

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

where iθ = ⟨∇ψ, \overline{ψ}⟩ = ⟨∇η, η^{⊥}⟩/2i.

Remark: By its definition, iθ is the connection form, relative to the section ψ, of the connection on the canonical bundle Λ^{n,0}, induced by the Levi-Civita connection (by projection).

Proof. Start with ⟨ψ, ψ̄⟩ = 1. Then

0 = d⟨ψ, ψ̄⟩ = ⟨∇ψ, ψ̄⟩ + ⟨ψ, ∇ψ̄⟩ = ⟨∇ψ, ψ̄⟩ + \overline{⟨∇ψ, ψ̄⟩},

hence ⟨∇ψ, ψ̄⟩ = iθ is imaginary (θ is a real 1-form.)

Next, the component (∇η)_{0} = (∇ψ + ∇ψ̄)_{0} is the component of ∇η in Λ^{1} ⊗ (Λ^{n,0} ⊕ Λ^{0,n}) , hence is given by

(∇η)_{0} = ⟨∇ψ, ψ̄⟩ ⊗ ψ + conjugate = −θ ⊗ η^{⊥}.  (10)

Next, according to formula (7),

(∇η)_{4} = −\frac{1}{2}(∇ω)_{4} · η^{⊥}.

Now by the SO_{2n}-equivariance of int : Λ^{1} ⊗ Λ^{n} → Λ^{n−1} we get that for any ϕ ⊗ α ∈ Λ^{1} ⊗ Λ^{2},

α · int(ϕ ⊗ η^{⊥}) = int(α · ϕ ⊗ η^{⊥}) + int(ϕ ⊗ α · η^{⊥}).

We claim that in fact the map defined on Λ^{1} ⊗ Λ^{2} by the left hand side of the last equation vanishes when restricted to W_{4}. By SU_{n}-equivariance, it is enough to check this on a single element in W_{4}, 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 trivialy on forms of type (0, p).

Furthermore, for ϕ ⊗ α ∈ Λ^{1} ⊗ Λ^{2} we have that α · ϕ = int(ϕ ⊗ α), and int(∇ω_{4}) = int(∇ω) = −d^{∗}ω (recall from our previous U_{n}-calculations that int(W_{i}) = 0, for i = 1, 2, 3), and so we conclude that

int(∇η)_{4} = \frac{1}{2}int [(int(∇ω)_{4}) ⊗ η^{⊥}] = −\frac{1}{2}int(d^{∗}ω ⊗ η^{⊥}).  (11)

Combining eqns. (10) and (11) we get

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

Finally, the map Λ^{1} → Λ^{n−1}, θ ↦ int(θ ⊗ 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} ⊗ ψ̄)| = |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 8

F_{04} = \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 the previous lemma.

And so our formula for the SU_{n} case now reduces to

3E_{1} − E_{3} + (n − 2)E_{4} + 2 ∫ ⟨κ, ω⟩ = \frac{1}{2} ∫ s + s^{∗}.  (12)

Observe that the formula, 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.

PROPOSITION 3 Formula (12) holds for any U_{n}-structure.

Proof. This follows from a version of formula (4) for the case of a p−form “twisted” by a vector bundle, ϕ ∈ Γ(Λ^{p}(M) ⊗ E), where E → M is a vector bundle with connection. All the operators in formula (4) extend naturally to this case and one checks that the formula still holds (see [21], p. 430). If we pick E to be the real part of Λ^{n,0} ⊕ Λ^{0,n}, equipped with the connection induced by the Levi-Civita connection (by projection), and take the section σ = η ⊗ η + η^{⊥} ⊗ η^{⊥} (basicaly the inclusion of E in Λ^{n}), then this section is defined globally, although η is defined only locally, and one checks easily that formula (4) applied to σ yields Formula (12). Details of this calculation are routine and are left to the reader.

6.5 The U_{n} examples revisited

The S^{6} example. Since it is of type W_{1} the (∇η)_{4} term vanishes hence also the F_{04} 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^{∗}.

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⟩.

Equivalently,

∫_{M} |∇ω|^{2} = \frac{1}{2m − 1} ∫_{M} [s − 2⟨κ, ω⟩]
∫_{M} ⟨κ, ω⟩ = \frac{1}{4m} ∫_{M} s + (2m − 1)s^{∗}.

6.6 Applications

Let us collect both formulas for easy reference:

PROPOSITION 4 Let (M^{2n}, g, J) be a compact, almost hermitian, 2n-mainifold, 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 may now specialize to certain types of almost hermitian manifolds; we give one example:

PROPOSITION 5 Let (M^{2n}, g, J), n ≥ 3, be an almost hermitian compact manifold 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,

• (Gray, [12]) if (M, g, J) is Nearly Kähler and c_{1} = 0, then ∫_{M} s = ∫_{M} 5s^{∗};

• (Sekigawa-Vanhecke, [20]) if (M, g, J) is almost Kähler and c_{1} = 0 then ∫_{M} |∇ω|^{2} = − ∫_{M} s and ∫_{M} s + s^{∗} = 0;

• (Sekigawa-Vanhecke, [20]) if (M, g, J) 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.

7 G_{2}

7.1 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..

Now the group G_{2} ⊂ SO_{7} is the stabilizer of the 3-form

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

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}. (In fact, according to Bryant [3], G_{2} is the stabilizer of ϕ in GL_{7}, and ϕ is generic in the sense that its GL_{7} orbit is open in Λ^{3}.)

It follows that a G_{2} structure on a 7-manifold is given by a 3-form ϕ of the type indicated above.

7.2 Decomposition of ∇ϕ

In the following decompositions (which are obtained by a combination of skill and luck, but it also helps alot to look up Bryant’s paper [3]) the subscripts on irreducible G_{2}-representations denote their dimensions. We omit 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 proposition is taken almost verbatim from [3] (see also [8] where the decomposition was first obtained).

PROPOSITION 6 Under G_{2}, we have the following decompositions into irreducible subspaces:

• Λ^{2} = Λ_{(14)}^{2} ⊕ Λ_{(7)}^{2}, corresponding to 𝔰𝔬_{7} = 𝔤 ⊕ 𝔤^{⊥}, where

– Λ_{(14)}^{2} = 𝔤 is the kernel of ·ϕ : Λ^{2} → Λ^{3}, α ↦ α · ϕ, and

– Λ_{(7)}^{2} = 𝔤^{⊥} ≅ Λ^{1} is the image of int_{·}ϕ : V → Λ^{2}, v ↦ int_{v}ϕ.

• Λ^{3} = Λ_{(1)}^{3} ⊕ Λ_{(7)}^{3} ⊕ Λ_{(27)}^{3}, where

– Λ_{(1)}^{3} = {multiples of ϕ}, and

– Λ_{(7)}^{3} ≅ Λ^{1} is the image of int_{·} ∗ ϕ : V → Λ^{2}, v ↦ int_{v} ∗ ϕ.

It follows from this proposition that

W := Λ^{1} ⊗ (𝔤^{⊥} · ϕ) = Λ^{1} ⊗ Λ_{(7)}^{3},

so 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

PROPOSITION 7 We have the following G_{2} decomposition:

W = Λ^{1} ⊗ Λ_{(7)}^{3} = W_{1} ⊕ W_{7} ⊕ W_{14} ⊕ W_{27}.

7.3 Summary of calculations for G_{2}

summand element w_{i} ∈ W_{i}  |w_{i}|_{T}^{2} |alt(w_{i})|^{2} |int(w_{i})|^{2}
W_{1}  ∗ϕ  7/24  7  0

W_{7}  θ_{2}(θ_{723} − θ_{135} + θ_{256} − θ_{671}) ; − θ_{4}(θ_{571} − θ_{613} + θ_{734} − θ_{456}) ; + θ_{5}(θ_{245} − θ_{357} + θ_{471} − θ_{123}) ; − θ_{6}(θ_{134} − θ_{246} + θ_{367} − θ_{712}) ; + θ_{3}(θ_{356} − θ_{461} + θ_{512} − θ_{234}) ; − θ_{7}(θ_{612} − θ_{724} + θ_{145} − θ_{567})  4  36  48

W_{14}  θ_{1}(θ_{571} − θ_{613} + θ_{734} − θ_{456}) ; − θ_{2}(θ_{467} − θ_{572} + θ_{623} − θ_{345}) ; + θ_{3}(θ_{245} − θ_{357} + θ_{471} − θ_{123}) ; − θ_{6}(θ_{612} − θ_{724} + θ_{145} − θ_{567})  8/3  0  8

W_{27}  {θ_{1}(θ_{467} − θ_{572} + θ_{623} − θ_{345}) \atop − θ_{2}(θ_{571} − θ_{613} + θ_{734} − θ_{456}}  4/3  4  0

Explanation:

• As we have seen above, W_{1} consists of multiples of ∗ϕ, so we take w_{1} = ∗ϕ. Then alt(w_{1}) = w_{1}, and int(w_{1}) = 0 (there are no G_{2}−invariant forms in Λ^{2}), hence

|w_{1}|_{T}^{2} = | ∗ ϕ|^{2}/4! = |ϕ|^{2}/4! = 7/24,  |alt(w_{1})|^{2} = 7,  |int(w_{1})|^{2} = 0.

• To get the next elements, we use repeatedly the G_{2}−isomorphism Λ^{1} ≅ V → Λ_{(7)}^{3}, v ↦ int_{v} ∗ ϕ, to move from Λ^{1} ⊗ Λ^{1} to W = Λ^{1} ⊗ Λ_{(7)}^{3}.

So, to obtain an element w_{7} ∈ W_{7}, we take an element in Λ_{(7)}^{2} , say int_{e_{1}}(ϕ) = θ_{24} + θ_{56} + θ_{37}, then write it as a 2-tensor, θ_{2} ⊗ θ_{4} − θ_{4} ⊗ θ_{2} + … etc., and apply the map v ↦ int_{v}(∗ϕ) to the second factor. The result is the following mess (we ommit the ⊗ symbol):

w_{7} = θ_{2}(θ_{723} − θ_{135} + θ_{256} − θ_{671}) − θ_{4}(θ_{571} − θ_{613} + θ_{734} − θ_{456})
+θ_{5}(θ_{245} − θ_{357} + θ_{471} − θ_{123}) − θ_{6}(θ_{134} − θ_{246} + θ_{367} − θ_{712})
+θ_{3}(θ_{356} − θ_{461} + θ_{512} − θ_{234}) − θ_{7}(θ_{612} − θ_{724} + θ_{145} − θ_{567}),

so that

alt(w_{7}) = 3(θ_{1235} + θ_{1267} + θ_{1457} + θ_{1346})

and

int(w_{7}) = 4(θ_{37} + θ_{56} + θ_{24}).

We thus get

|w_{7}|_{T}^{2} = 6 · 4/3! = 4,  |alt(w_{7})|^{2} = 9 · 4 = 36,  |int(w_{7})|^{2} = 16 · 3 = 48.

• To obtain an element w_{14} ∈ W_{14} we follow a similar route: pick an element in Λ_{(14)}^{2}, write it as a tensor, and apply the map v ↦ int_{v}(∗ϕ) to the second factor.

So starting with θ_{12} + θ_{36} ∈ Λ_{(14)}^{2}, we obtain

w_{14} = θ_{1}(θ_{571} − θ_{613} + θ_{734} − θ_{456}) − θ_{2}(θ_{467} − θ_{572} + θ_{623} − θ_{345})
+θ_{3}(θ_{245} − θ_{357} + θ_{471} − θ_{123}) − θ_{6}(θ_{612} − θ_{724} + θ_{145} − θ_{567}),

so that

int(w_{14}) = 2(θ_{12} + θ_{36}),

and

alt(w_{14}) = 0,

since there is no 14-dimensional summand in Λ^{4} ≅ Λ^{3}. Thus

|w_{14}|_{T}^{2} = 4 · 4/3! = 8/3,  |alt(w_{14})|^{2} = 0,  |int(w_{14})|^{2} = 4 · 2 = 8.

• To obtain an element w_{27} ∈ W_{27} we begin with a traceless quadratic form, say (θ_{1})^{2} − (θ_{2})^{2}, and apply the map v ↦ int_{v}(∗ϕ) to the second factor, obtaining:

w_{27} = θ_{1}(int_{e_{1}} ∗ ϕ) − θ_{2}(int_{e_{2}} ∗ ϕ)
= θ_{1}(θ_{467} − θ_{572} + θ_{623} − θ_{345}) − θ_{2}(θ_{571} − θ_{613} + θ_{734} − θ_{456})

hence

alt(w_{27}) = θ_{1467} − θ_{1345} + θ_{2347} + θ_{2456},

and

int(w_{27}) = 0,

since there is no 27-dimensional summand in Λ^{2}. Thus

|w_{27}|_{T}^{2} = 2 · 4/3! = 4/3,  |alt(w_{27})|^{2} = 4,  |int(w_{27})|^{2} = 0.

7.4 The G_{2} formula

Now denote by

∇ϕ = (∇ϕ)_{1} + (∇ϕ)_{7} + (∇ϕ)_{14} + (∇ϕ)_{27}

the decomposition of ∇ϕ corresponding to the decomposition of W. We thus have

|∇ϕ|^{2} = |(∇ϕ)_{1}|^{2} + |(∇ϕ)_{7}|^{2} + |(∇ϕ)_{14}|^{2} + |(∇ϕ)_{27}|^{2},

and our calculations give

|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. From these it follows that dϕ = d^{∗}ϕ = 0 implies ϕ is parallel.

Denote by E_{1}, …, E_{27} the L_{2} norms of (∇ϕ)_{1}, …, (∇ϕ)_{27} (resp.), then substituting into Equation (4), we get

6E_{1} + 5E_{7} − E_{14} − E_{27} = \frac{1}{3} ∫_{M}⟨R̃ϕ, ϕ⟩.  (13)

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

PROPOSITION 8 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. First we see how to derive the claimed identity, up to a constant, from a Schur lemma type argument. For that, we observe that both sides of the identity are G_{2}−invariant linear funcionals on the space of curvature type tensors, where the latter can be defined as the orthogonal complement of Λ^{4} in S^{2}(Λ^{2}). 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}) is also 1-dimensional. We conclude that the space of G_{2}−invariant linear funcionals on the space of curvature type tensors is 1-dimensional, hence s and ⟨R̃ϕ, ϕ⟩ are proportional.

It is thus sufficient to evaluate both sides of the identity on a particular curvature-type tensor (for which the outcome is non-zero). Let R = (θ_{12})^{2}. Clearly, R ∈ S^{2}(Λ^{2}) and alt(R) = θ_{12} ∧ θ_{12} = 0, hence R is of curvature type. Now

s := 2 tr (R) = −2⟨θ_{12}, θ_{12}⟩ = −2,
R̃ϕ := θ_{12} · (θ_{12} · ϕ) = θ_{12} · (−θ_{135} + θ_{562} − θ_{671} + θ_{723}) = −θ_{235} − θ_{561} − θ_{672} − θ_{713},

hence ⟨R̃ϕ, ϕ⟩ = −4 and the claim follows.

Hence we can write the formula for a G_{2}-manifold as follows

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

7.5 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 [8], 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 [6] 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|_{T}^{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 [9] 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 [8].

8 Spin_{7}

8.1 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}.

A Spin_{7} structure on an 8−manifold M is thus given by a 4−form Φ which is of the indicated type at each point of M. More precisely, at each point x ∈ M there should exist a linear isometry V → T_{x}M, pulling-back Φ(x) to the 4-form on V described above.

8.2 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

PROPOSITION 9 We have the following Spin_{7} decompositions:

• Λ^{2} = Λ_{(21)}^{2} ⊕ Λ_{(7)}^{2}, corresponding to so_{8} = 𝔤 ⊕ 𝔤^{⊥}, where

Λ_{(21)}^{2} = 𝔤 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. [3]. It follows from this proposition that

W := Λ^{1} ⊗ (𝔤^{⊥} · Φ) = Λ^{1} ⊗ Λ_{(7)}^{4},

so we decompose

W ≅ Λ^{1} ⊗ Λ_{(7)}^{2} = W_{1} ⊕ 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}.

8.3 Summary of calculations for Spin_{7}

summand element w_{i} ∈ W_{i}  |w_{i}|_{T}^{2} |alt(w_{i})|^{2} = |int(w_{i})|^{2}
W_{8}  ∑_{i=1}^{7} θ_{i} (θ_{0i} · Φ)  7/3  16 · 7
W_{48}  {θ_{0}[(θ_{12} + θ_{36}) · Φ] + θ_{1}[θ_{20} · Φ] \atop + θ_{2}[θ_{01} · Φ] + θ_{3}[θ_{60} · Φ] + θ_{6}[θ_{03} · Φ]}  4/3  8

Explanation:

• The 4-form Φ is ±self-dual. It follows, like in the previous case of SU_{n}, that |int(w)| = |alt(w)| for any w ∈ Λ^{1} ⊗ [𝔤^{⊥} · Φ].

• We pick w_{8} ∈ W_{8} as follows: we consider the composition of maps

Λ^{1} → Λ^{1} ⊗ Λ^{2} → Λ^{1} ⊗ Λ^{4},

where the 1st map is int^{∗} (the adjoint of interior product) and the 2nd is given by the map Λ^{2} → Λ^{4}, α ↦ α · Φ. Since these maps are Spin_{7}-equivariant, one easily sees, essentially by Schur’s lemma, that the composition is an isomorphism Λ^{1} ≅ W_{8}.

Starting with θ_{0}, we have that int^{∗}(θ_{0}) is, up to a constant, ∑_{i=1}^{7} θ_{i}θ_{0i} (remember that we are supressing the symbol ⊗), and so

w_{8} = ∑_{i=1}^{7} θ_{i}(θ_{0i} · Φ).

Now

int_{e_{0}}(Φ) = ϕ,  int_{e_{i}}(Φ) = int_{e_{i}}(∗ϕ) − θ_{0} ∧ int_{e_{i}}(ϕ),  i = 1, …, 7,

so that

θ_{0i} · Φ = θ_{i} ∧ ϕ − θ_{0} ∧ int_{e_{i}}(∗ϕ),

and so

w_{8} = ∑_{i=1}^{7} θ_{i} (θ_{i} ∧ ϕ − θ_{0} ∧ int_{e_{i}}(∗ϕ)),

int(w_{8}) = ∑_{i=1}^{7} int_{e_{i}}(θ_{i} ∧ ϕ − θ_{0} ∧ int_{e_{i}}(∗ϕ)) = ∑_{i=1}^{7} int_{e_{i}}(θ_{i} ∧ ϕ) = 4ϕ,

thus

|w_{8}|_{T}^{2} = ∑_{i=1}^{7} |θ_{i} ∧ ϕ − θ_{0} ∧ (int_{e_{i}}(∗ϕ))|^{2}/4!,

but now

|θ_{i} ∧ ϕ − θ_{0} ∧ int_{e_{i}} ∗ ϕ|^{2} = |θ_{i} ∧ ϕ|^{2} + |int_{e_{i}} ∗ ϕ|^{2} = 4 + 4 = 8,

so that

|w_{8}|_{T}^{2} = 7 · 8/4! = 7/3,
int(w_{8})|^{2} = 16|ϕ|^{2} = 16 · 7.

• We pick w_{48} ∈ W_{48} using a similar trick: consider the composition

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

where the 1st map is inclusion, the 2nd is alt^{∗} (the adjoint of alternation, or exterior product), and the 3rd is given as before by the Λ^{2}-action on Φ. Again, by Spin_{7}-equivariance of the maps involved, the composition is an isomorphism of the domain onto W_{48}.

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}, ie 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_{e_{i}}(∗ϕ)],

so that

w_{48} = θ_{2} (θ_{1} ∧ ϕ − θ_{0} ∧ int_{e_{1}} ∗ ϕ) − θ_{1} (θ_{2} ∧ ϕ − θ_{0} ∧ int_{e_{2}} ∗ ϕ) +
+θ_{6} (θ_{3} ∧ ϕ − θ_{0} ∧ int_{e_{3}} ∗ ϕ) − θ_{3} (θ_{6} ∧ ϕ − θ_{0} ∧ int_{e_{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}) = int_{e_{2}}(θ_{1} ∧ ϕ − θ_{0} ∧ int_{e_{1}} ∗ ϕ) − int_{e_{1}}(θ_{2} ∧ ϕ − θ_{0} ∧ int_{e_{2}} ∗ ϕ) +
+int_{e_{6}}(θ_{3} ∧ ϕ − θ_{0} ∧ int_{e_{3}} ∗ ϕ) − int_{e_{3}}(θ_{6} ∧ ϕ − θ_{0} ∧ int_{e_{6}} ∗ ϕ)
= … = −2α,

where “… ” denotes a moderately tedious, yet straight-forward, caculation.

Thus

|w_{48}|_{T}^{2} = 4|θ_{2} (θ_{1} ∧ ϕ − θ_{0} ∧ int_{e_{1}}(∗ϕ))|^{2}/4!
= (|θ_{1} ∧ ϕ|^{2} + |int_{e_{1}}(∗ϕ)|^{2})/3! = (4 + 4)/3! = 4/3,
|int(w_{48})|^{2} = 4|α|^{2} = 4 · 2 = 8.

8.4 The Spin_{7} formula

Denote by ∇Φ = (∇Φ)_{8} + (∇Φ)_{48} the decomposition of ∇Φ according to the decomposition of W. We thus have

|dΦ|^{2} = |d^{∗}Φ|^{2} = |int(∇Φ)|^{2} = |int((∇Φ)_{8})|^{2} + |int((∇Φ)_{48})|^{2} = 48|(∇Φ)_{8}|^{2} + 6|(∇Φ)_{48}|^{2};

in particular, dΦ = 0 implies ∇Φ = 0.

Denote by E_{8}, E_{48} the corresponding L_{2}-norms of these components, so then Formula (4) gives:

6E_{8} − E_{48} = \frac{1}{12} ∫ ⟨R̃Φ, Φ⟩.

The proof of the following lemma is very similar to the G_{2}-case and is left to the reader: LEMMA 9 ⟨R̃ϕ, ϕ⟩ = 2s

The formula for Spin_{7} is then

6E_{8} − E_{48} = \frac{1}{6} ∫ s.

8.5 Examples

S^{7} × S^{1}. In [5] 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.

References

[1] M. Berger, Sur les groupes d’holonomie des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279-330.

[2] A. Besse, Einstein manifolds, Springer Verlag (1987).

[3] R. Bryant, Metrics with exceptional holonomy, Ann. Math. 126 (1987), 525-567.

[4] R. Bryant, Recent advances in the theory of holonomy, Preprint math.DG/9910059 v2 October 1999.

[5] F.M. Cabrera, On Riemannian manifolds with Spin_{7}-structure, Publ. Math. Debrecen 46 (1995), 271-283.

[6] F.M. Cabrera, M.D. Monar, A.F. Swann, Classification of G_{2}-structures, J. London Math. Soc. 53 Part 2 (1996), 407.

[7] J. Carlson, D. Toledo, Harmonic Mappings of Kähler Manifolds to Locally Symmetric Spaces, Publ. Math. IHES 69 (1989), 173-201.

[8] M. Fernández, A. Gray, Riemannian manifolds with structure group G_{2}, Ann. Mat. Pura Appl. (IV) 32 (1982), 19-45.

[9] M. Fernández, L. Ugarte, Dolbeaut cohomology for G_{2}-manifolds, Geom. Dedicata 70 (1998), 57-86.

[10] P. Gauduchon, Complex Structures on Compact Conformal Manifolds of negative Type, in Complex Analysis and Geometry, Lect. Notes Pure Appl. Math. 173 (1995), 201-212.

[11] A. Gray, Riemannian manifolds with geodesic symmetries of order 3, J. Diff. Geom. 7 (1972), 343–369.

[12] A. Gray, The structure of nearly Kähler manifolds, Math. Ann. 223 (1976), 233-248.

[13] A. Gray, L.M. Hervella, The Sixteen Classes of Almost Hermitian Manifolds and Their Linear Invariants, Ann. Math. Pura Appl 103 (1978), 35–58.

[14] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press (1978).

[15] L. Hernández-Lamoneda, Curvature vs almost-hermitian structures, to appear Geom. Dedicata.

[16] D. Joyce, Compact riemannian 7-manifolds with holonomy G_{2}:I & II, J. Diff. Geom. 43 (1996), 291-328, 329-375.

[17] D. Joyce, Compact riemannian 8-manifolds with holonomy Spin_{7}, Inventiones Math. 123 (1996), 507-552.

[18] D. Kotschick, Remarks on geometric structures on compact complex surfaces, Topology 31 (1992), 317-321.

[19] S. Salamon, Riemannian geometry and holonomy groups, Pitman Research Notes Math. 201, Longman (1989).

[20] K. Sekigawa, L. Vanhecke, Almost hermitian manifolds with vanishing first Chern classes or Chern numbers, Rend. Sem. Mat. Univ. Pol. Torino 50 (1992), 195-208.

[21] H.-H. Wu, The Bochner Technique in Differential Geometry, Mathematical Reports 3, Harwood Academic Publishers.