An integral formula for Riemannian $G$-structures with applications to almost hermitian and almost contact structures

For a Riemannian $G$-structure, we compute the divergence of the vector field induced by the intrinsic torsion. Applying the Stokes theorem, we obtain the integral formula on a closed oriented Riemannian manifold, which we interpret in certain cases. We focus on almost harmitian and almost contact metric structures.


Introduction
Equipping a manifold M with a Riemannian metric g is equivalent to reduction of a frame bundle L(M) to orthogonal frame bundle O(M), i.e. to action of a structure group O(n). Assuming moreover that M is oriented we can consider the bundle SO(M) of oriented orthonormal frames. Existence of additional geometric structure can be considered as a reduction of a structure group SO(n) to a certain subgroup G. For example, almost hermitian structure gives U( n 2 )-structure, almost contact metric structure U( n−1 2 ) × 1-structure, etc. If ∇ is a Levi-Civita connection of (M, g) we may measure the defect of ∇ to be a G-connection. This leads to the notion of an intrinsic torsion. If this (1, 2)tensor vanishes (in such case we say that a G-structure is integrable) then ∇ is a G-connection, which implies that the holonomy group is contained in G. We may classify non-integrable geometries by finding the decomposition of the space of all possible intrinsic torsions into irreducible G-modules. This approach was initiated by Gray and Hervella for U( n 2 )-structures [10] and later considered for other structures by many authors [5,11,12,7,6]. Each so called Gray-Hervella class, gives some restrictions on the curvature.
One possible approach to curvature restrictions on compact G-structures can be achieved by obtaining integral formulas relating considered objects. This has been firstly done, in a general case, by Bor and Hernandez Lamoneda [3]. They uses Bochner-type formula for forms being stabilizers of each considered subgroup in SO(n). They obtained integral formulas for G = U( n 2 ), SU( n 2 ), G 2 and Spin 7 and continued this approach for Sp(n)Sp (1) in [4]. The case G = U( n−1 2 ) × 1 has been done later in [8].
In this article, we show how mentioned formulas can be obtained in a different way. The nice feature of our approach is that the main integral formula (3.11) is stated in a general case of any G-structure for G ⊂ SO(n). This is achieved by considering, so called, characteristic vector field induced by the intrinsic torsion and calculating its divergence. For some Gray-Hervella classes the characteristic vector field vanishes, and then we get point-wise formula relating an intrinsic torsion with a curvature. Moreover, our integral formula, which can be reformulated in such a way that it is equivalent to formulas by Bor and Hernandez Lamoneda [5], gives a priori different information that the ones in [5].
We concentrate on almost hermitian and almost contact metric structures. In the way described above we recover many well known relations. Let us state some of the consequences of the main itegral formula (the objects used in these statements will be defined in appropriate sections): (1) Let M be a G-structure such that the orthogonal complement g ⊥ of the Lie algebra g of Lie group G in so(n) satisfies [g ⊥ , g ⊥ ] ⊂ g. Assume that the characteristic vector field vanishes. Then we have the following pointwise formula 1 2 where s g ⊥ is a g ⊥ -component of the scalar curvature and ξ alt , ξ sym are skew-symmetric and symmetric parts of the intrinsic torsion ξ. In particular, if the intrinsic torsion is totally skew-symmetric then (2) Assume (M, g, J) is closed hermitian manifold of Gray-Hervella type W 4 . Then M s − s * > 0, where s is a scalar curvature and s * is * -scalar curvature. (3) On closed SU(n)-structure of type W 1 ⊕W 2 ⊕W 5 we have M s = 5 M s * . (4) If closed almost contact metric structure is of type C 5 ⊕ . . . ⊕ C 10 , then M s − s * = M Ric(ζ, ζ).

Intrinsic torsion
Let (M, g) be an oriented Riemannian manifold. Denote by SO(M) the bundle of oriented frames over M. Let ∇ be the Levi-Civita connection of g and let ω be the induced connection form. Let G ⊂ SO(n), where n = dim M, be a Lie subgroups such that on the level of Lie algebras we have the following decomposition where the orthogonal complement is taken with respect to the Killing form. Then ω decomposes as ω = ω g ⊕ ω g ⊥ . The component ω g is a connection form in the G-reduction P ⊂ SO(M), if such exists, and therefore defines a Riemannian connection ∇ G on M. The difference , defines a (1, 2)-tensor called the intrinsic torsion of a G-structure. ξ satisfies some skew-symmetry conditions by the fact that ξ X ∈ g ⊥ (T M) ⊂ so(T M) where g ⊥ (T M) is the associated bundle of the form P × ad(G) g ⊥ . In particular, By a definition, the intrinsic torsion measures defect of the Levi-Civita connection to be a G-connection. In particular, if ξ vanishes, then the holonomy of ∇ is contained in G. Study of the intrinsic torsion and its decomposition into irreducible summands was initiated by Gray and Hervella in the case of G = U( n 2 ) [10]. Since then, other possible cases, mainly coming from the Berger classification of non-symmetric irreducible holonomy groups, has been considered (see, for example, [5,11,12,7,6]).

An integral formula
Let (M, g) be an oriented Riemannian manifold with the Levi-Civita connection ∇. Assume M is a G-structure, with G ⊂ SO(n) such that (2.1) holds and let ξ be the associated intrinsic torsion. Define a vector field χ = χ G by where (e i ) is any orthonormal basis. We call χ the characteristic vector field of a G-structure M. Notice that if ξ is skew-symmetric with respect to X and Y then χ vanishes. This is the case, for example, for nearly Kähler manifolds (see the following sections). Additionally, Thus, vanishing of the characteristic vector field is equivalent to the fact that divergences with respect to ∇ and ∇ G coincide.
In this section we compute the divergence of χ with respect to ∇. First, let us recall well-known curvature identities involving the intrinsic torsion: where R and R G are the curvature tensors of ∇ and ∇ G , respectively. We use the following convention for the curvature Denoting the scalar curvatures of ∇ and ∇ G by s and s G , respectively, by the above identity we get Notice that (∇ X ξ) Y is skew-symmetric, since ξ X is skew-symmetric, Thus the first and second sum on the right hand side of (3.4) are opposite. Moreover, Let us compute the last term in (3.4), Then (3.8) i,j g(ξ e j e i , ξ e i e j ) = |ξ sym | 2 − |ξ alt | 2 .
Proposition 3.1. On an oriented G-structure M such that (2.1) holds we have We will improve above divergence formula a little bit, by getting rid of the component s G replacing it by g ⊥ -component of s and some additional term, which vanishes in many cases. Namely, by (3.3), (3.6) and (3.8) we have Denote the last component in the above formula by s alt g ⊥ , i.e.
Since s = s g + s g ⊥ , (3.9) can be rewritten in the form contained in the proposition below.
Proposition 3.2. On an oriented G-structure M such that (2.1) holds we have If M is, additionally, closed, then the following integral formula holds Remark 3.3. Notice that elements are quadratic invariants of the representation of SO(n) in the space of (1, 2)tensors with the symmetries of the intrinsic torsion, i.e. the space T * M ⊗so(T M). This implies that |ξ| 2 and |ξ alt | 2 − |ξ sym | 2 are also quadratic invariants. Thus, for an irreducible submodule U of the representation T * M ⊗so(T M), since the space of its quadratic invariants in one dimensional [2], then the number Here ξ U denotes the U-component of ξ with respect to decomposition into irreducible summands. This approach is also valid for any irreducible module G-module in the space of possible intrinsic torsions. This kind of approach, was used in [3] to get integral formulas for many G-structures.
We have an immediate consequence of the formula (3.10).
g and the characteristic vector field vanishes, then In particular, if the intrinsic torsion is totally skew-symmetric, then The consequences of the integral formula will be presented in the following section for certain choices of G.

Applications to certain Riemannian G-structures
In this section we rewrite formulae (3.10) and (3.11) for certain G-structures. We also give some applications of these relations. We will show that obtained formulas are consistent with the Bochner type formulae obtained, using representation theory, in [3].

4.1.
Almost product structures. We show that the divergence and integral formulae obtained in the previous section agree with the Walczak formulas [14]. Since this integral formula has found many applications, we will only concentrate on deriving it from (3.11) and state its one corollary, which will be needed later. Let Let ∇ be the Levi-Civita connection of g. Since the orthogonal projection is just a restriction to non-diagonal blocks, it follows that the intrinsic torsion equals where Y ⊤ and Y ⊥ denotes the components of Y in D and D ⊥ . Notice that ξ is made of shape operators and fundamental forms of distributions D and D ⊥ . Recall, that the second fundamental form, for example, of D is a (1, 2)-symmetric tensor B = B D of the form Additionally, we will use integrability tensor T = T D being just Notice that B(X, Y ) + T (X, Y ) = (∇ X Y ) ⊥ , hence B and T are symmetrization and alternation of (a minus of) a part of the intrinsic torsion reduced to D.
For an orthonormal basis (e i ) adapted to decomposition D ⊕ D ⊥ , denote by e A components of (e i ) in D and by e α components of (e i ) in D ⊥ . The characteristic vector field χ equals where H and H ⊥ are mean curvature vectors of D and D ⊥ respectively. Since and analogously interchanging e α with e A , then Denoting by s mix so called mixed scalar curvature, we get Putting all facts together (3.10) implies Walczak formula [14] −div(H Assuming M is closed, the following Walczak integral formula holds [14] (4.2) This formula has found many applications. Let us only state one of its consequences for D of codimension 1, since it will be used in one of forthcoming subsections. In this case, clearly, T ⊥ = 0 and B ⊥ = H ⊥ . Denoting the unit positively oriented vector field orthogonal to D by ζ, we have H = −(divζ)ζ. Moreover, Therefore, (4.2) becomes In particular, [u(n) ⊥ , u(n) ⊥ ] ⊂ u(n), thus s alg u(n) ⊥ = 0. The orthogonal projection from so(n) to u(n) ⊥ equals A → 1 2 (A + JAJ). Thus the u(n)-component of R is given by Moreover, the intrinsic torsion, being informally the projection of −∇ to u(n) ⊥ , is given by the formula Hence, the characteristic vector field χ is the following Let us describe the intrinsic torsion with the use of the Nijenhuis tensor N and the Kähler form Ω. Recall that and Ω(X, Y ) = g(X, JY ).
It is a famous theorem by Newlander and Nirenberg that vanishing of the Nijenhuis tensor is equivalent to integrability of J, i.e. existence of complex coordinates adapted to J. In can be shown [1] that Unfortunately, this shows that ξ has no particular symmetries and it is hard to give nice interpretations for the symmetrized and skew-symmetrized intrinsic torsion ξ sym and ξ alt , respectively. Therefore, it is convenient to consider some restrictions or decomposition of the intrinsic torsion. The space of all possible intrinsic torsions is, in this case, T * M ⊗u(n) ⊥ (T M). Decomposing this space into irreducible modules with respect to U(n)-action, we get so called Gray-Hervella classes [10] T where each class can be characterized as follows: W 3 : ξ X Y = ξ JX (JY ) and χ = 0, It can be shown [10] that W 1 ⊕ W 2 and W 3 ⊕ W 4 are described by relations (4.10) Now, we will discus the relations between elements in the divergence formula (3.10) in each pure class separately. We will proceed by studying quadratic invariants of the U(n)-representation on the space of intrinsic torsions T * M ⊗ u(n) ⊥ (T M) [10]: where α(X, Y, Z) = g(ξ X Y, Z). Notice that It is not hard to see by definitions of each pure class W i and conditions (4.8) the following relations contained in the Decompose ξ and χ with respect to the Gray-Hervella classes as follows Thus, by above considerations (see also Remark 3.3), we have Hence (4.11) divχ = |ξ 1 | 2 − 1 2 which implies the integral formula by Bor and Hernandez Lamoneda [3] (assuming M is closed) Let us list two applications: one which was not stated in [3] but, although not directly, can be found [13] and the second one being reformulation of Corollary 3.4.  Proof. Follows directly by (4.10) and by Corollary 3.4.
Example 4.3. Consider a six sphere S 6 with a natural Riemannian metric g induced from R 7 and an almost complex J structure induced from cross product on R 7 . It is well known that J is g-orthogonal and nearly-Kähler (W 1 ). It can be shown that s = 30, s * = 6 and |∇J| = 24 [9]. This justifies Proposition 4.2.

4.3.
Special almost hermitian structures. Assume (M, g, J) is an almost hermitian manifold equipped with a complex volume form Ψ = ψ + + iψ − such that Ψ, Ψ C = 1, where the inner product is a natural extension of an inner product for real valued forms. This structure defines reduction of a structure group to special unitary group SU(n), hence a SU(n)-structure. On the level of Lie algebras we have Then A ∈ su(n) if and only if A ∈ u(n) and trA 1 = 0. Notice that Thus, the orthogonal projection from so(2n) to su(n) ⊥ = u(n) ⊥ ⊕ R equals The intrinsic torsion ξ equals ξ = ξ U (n) + η, where ξ U (n) is the intrinsic torsion of related U(n)-structure and Denote the one form on the right-hand-side of the above formula evaluated on JX not X also by η, so η X Y = η(JX)JY . This convention will appear to be useful. Denote the class in the space of all possible intrinsic torsions induced by η by W 5 . Thus we have a decomposition Remark 4.4. As it was noticed in [3] module W 5 is not in general orthogonal to other modules W i , i = 1, 2, 3, 4. This follows from the fact that W 5 ≡ T * M ≡ W 4 , thus there are isomorphic modules or, in other words, the multiplicity of W 5 or W 4 is 2. This also implies that these two mentioned modules are not orthogonal but they are orthogonal to remaining ones. We will proceed in a different way than in [3].
Let us begin by describing all objects contained in the divergence formula (3.10). We have , where we split s su(n) ⊥ into s u(n) ⊥ and s R with respect to the decomposition su(n) ⊥ = u(n) ⊥ ⊕ R. Applying the Stokes theorem, assuming M is closed we get the following integral formula.  To compute s alt Hence, Denote by ξ U (n),12 and ξ U (n),34 the W 1 ⊕ W 2 and W 3 ⊕ W 4 components of ξ U (n) . By (4.8) we get i,j g(ξ e j e i , ξ Je j Je i ) = −|ξ U (n),12 | 2 + |ξ U (n),34 | 2 .
We have proved the following corollary. In particular, if ξ U (n) ∈ W 1 ⊕ W 2 ⊕ W 3 ⊕ W 5 , then Remark 4.7. The above integral formula, however formulated in a different way, can be found in [3]. Let us be more precise. In [3] authors state some consequences of their formula for almost hermitian structures with vanishing first Chern class c 1 (M). Let us derive the first Chern class in our setting. It is known [9] that the first Chern form γ is given by Thus, using the same arguments as before Corollary 4.6, we get where tr * γ = i γ(e i , Je i ). Notice that vanishing of the first Chern class, i.e., γ = dα for some 1-form, is equivalent to the fact that M tr * γ = 0. Thus by  It suffices to notice that by (4.10), (4.9), 1 4 (s − s * ) = |ξ U (n) | 2 . 4.4. Almost contact metric structures. Let (M, g) be a (2n+1)-dimensional manifold together with a 1-form and its dual unit vector field ζ and ϕ ∈ End(T M) such that (4.14) ϕ 2 X = −X + η(X)ζ, g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ).
Notice that ϕ defines almost complex structure which is g-orthogonal on the distribution kerη. Thus we get U(n) × 1-structure. On the level of Lie algebras we have so(2n + 1) = u(n) ⊕ u(n) ⊥ , where u(n) ⊥ is isomorphic to the space of block matrices of the form B a −a ⊤ t , B ∈ u(n) ⊥ , a ∈ R 2n , t ∈ R.
Since here ζ = e 2n+1 , ϕ is a natural complex structure on R 2n and zero on ζ, it is easy to see that the orthogonal projection from so(2n + 1) onto u(n) ⊥ equals Rewriting this formula with the use of the one-form η(≡ ζ ⊤ ), the intrinsic torsion satisfies the following relation This, moreover, implies the formula for the intrinsic torsion By (4.14) it follows that Thus, we may write the intrinsic torsion in an alternative way Hence, the characteristic vector field in this case equals The condition ξ ∈ T * M ⊗ u(n) ⊥ (T M) is equivalent to the relation (4.16). Decomposing the space T * M ⊗ u(n) ⊥ (T M) into irreducible U(n) × 1-modules, we get twelve classes C 1 , . . . , C 12 [5]. First four are isomorphic to Gray-Hervella classes W 1 , . . . , W 4 . Let us describe these spaces in more detail. Put Each of above spaces is characterized as follows [5]: Class D 1 : ξ ζ Y = ξ X ζ = 0. Applying formluas (4.17) and (4.18) we obtain ∇ζ = 0 and hence χ = − 1 2 ϕ(divϕ), as expected, since in this case, being not very precise, ξ is the intrinsic torsion on the almost hermitian structure kerη.
Concluding we have the following result.  If additionally, M is closed, then the following integral formula holds Proof. The only explanation is needed for the proof of the integral formula. It follows by (4.21) and integral formula (4.3).
Let us list some direct consequences of above fact. Proof. By classification of almost contact metric structures (see [5, Table III] and remark below), if ξ ∈ C 5 ⊕ . . . ⊕ C 10 , then ∇ ζ ϕ = 0. Now, it suffices to apply Proposition 4.9.
Note that we should be careful with studying irreducible modules C 1 , . . . , C 12 , since the correspondence α ↔ β interchanges some of the modules, which is underlined in the table below. Table 2. Module correspondence via α ↔ β α C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10 C 11 C 12 β C 1 C 2 C 3 C 4 C 6 C 5 C 8 C 7 C 9 C 10 C 11 C 12 In an analogous way as for U(n)-structures, it can be shown, with a little bit more effort, that the integral formula (3.11) in this case is equivalent with the integral formula obtained in [8].