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 Hermitian and almost contact metric structures.

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 Hernández Lamoneda [5]. They use 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 [6]. The case G = U ( n− 1 2 ) × 1 has been studied later in [12] by other authors. 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 is valid for any G-structure on closed M for compact G ⊂ SO(n). Let us roughly describe the approach and all used objects in this formula. We consider, so-called, characteristic vector field χ = i ξ e i e i induced by the intrinsic torsion ξ and calculate its divergence. ξ alt and ξ sym denote the skew-symmetric and symmetric components of ξ , ξ alt , whereas s g ⊥ and s alt g ⊥ are, in a sense, g ⊥ components of a scalar curvature (see the following sections for more details). For some Gray-Hervella classes, the characteristic vector field vanishes, and then we get point-wise formula relating an intrinsic torsion to a curvature.
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 integral formula (the objects used in these statements will be defined in appropriate sections): 1. Assume (M, g, J ) is closed Hermitian manifold of Gray-Hervella type W 4 such that s = s * , where s is a scalar curvature and s * is a * -scalar curvature. Then, M is Kähler (compare [22]). 2. On a closed SU (n)-structure of type W 1 ⊕ W 5 , we have M s = 5 M s * . 3. Let (M, g, ϕ, η, ζ ) be an almost contact metric structure with the intrinsic torsion ξ ∈ D 2 .
In the end, we consider some examples focusing on (reductive) homogeneous spaces. We show, which is an immediate consequence of the formula for the Levi-Civita connection, that in these examples the characteristic vector field vanishes. Hence, the main divergence formula is point-wise.
where the orthogonal complement is taken with respect to the Killing form. Hence, ω decomposes as where 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 skewsymmetry 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 the defect of the Levi-Civita connection to be a G-connection. In particular, if ξ vanishes, then the holonomy of ∇ is contained in G. The 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 ) [14]. Since then, other possible cases, mainly coming from the Berger classification of non-symmetric irreducible holonomy groups, have been considered (see, for example, [7,8,11,20,21]).

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), 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. Moreover, put In this section, we compute the divergence of χ with respect to ∇. First, let us recall well-known curvature identities involving the intrinsic torsion [12]: where R and R G are the curvature tensors of ∇ and ∇ G , respectively. We use the following convention for the curvature Denote by s and s G the scalar curvatures of R and R G , respectively.

Proposition 1 On an oriented G-structure M, we have
Proof By (5), we have Thus, the first sum and second sum on the right-hand side of (7) are opposite. Moreover, Let us compute the last term in (7), Substituting (8) and (9) into (7), we get (6).
We will improve the above divergence formula a little bit, by getting rid of the component s G and replacing it by g ⊥ -component of s and some additional term, which vanishes in some cases. Namely, denote by s alt g ⊥ the following quantity:

Proposition 2 On an oriented G-structure M, we have
If M is, additionally, closed, then the following integral formula holds Proof By (4) and (9), we have Since s = s g + s g ⊥ , (6) can be rewritten in form (10).

Remark 1 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) [14]. 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 is one-dimensional [4], then the number is a constant multiple of |ξ U | 2 . 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 [5] to get integral formulas for many G-structures.
We have an immediate consequence of the formula (10).

Corollary 1
Assume M is an oriented G-structure, where G = U ( n 2 ), n even, or G = SO(m) × SO(n − m). If the characteristic vector field vanishes, then In particular, if the intrinsic torsion is totally skew-symmetric, then with the equality if and only if the G-structure M is integrable (i.e., ξ = 0).
Proof For the listed choices of G, we have [g ⊥ , g ⊥ ] ⊂ g; thus, s alt g ⊥ vanishes.
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 formulas (10) and (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 formulas obtained, using representation theory, in [5].

Almost product structures
We show that the divergence and integral formulas obtained in the previous section agree with the Walczak formulas [23]. Since this integral formula has found many applications, we will only concentrate on deriving it from (11) and state its one corollary, which will be needed later. Let Let ∇ be the Levi-Civita connection of g. Since the orthogonal projection to m is just a restriction to non-diagonal blocks, it follows that the intrinsic torsion equals where Y and Y ⊥ denote 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 the decomposition D⊕D ⊥ , denote by e A elements of (e i ) in D and by e α elements of (e i ) in D ⊥ . The characteristic vector field χ equals where H and H ⊥ are mean curvature vectors of D and D ⊥ , respectively. We may now state and show that the Walczak formula [23] is an integral formula (11) for G = SO(m) × SO(n − m).

Proposition 3 ([23]) On a closed Riemannian manifold equipped with a pair of complementary orthogonal and oriented distributions, the following Walczak integral formula holds
where s mix is a mixed scalar curvature defined by and analogously interchanging e α with e A , then Moreover, Putting all these facts together (10) implies Walczak divergence formula [23] − Assuming M is closed, the Walczak integral formula holds.
Formula (13) 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 χ = (divζ )ζ − ∇ ζ ζ . Moreover, Therefore, (14) and (13) can be rewritten in the following well-known way.

Proposition 4 On a Riemannian manifold with an orientable codimension one distribution
D, we have the following divergence formula: and, assuming M is closed, the following integral formula

Almost Hermitian structures
Assume (M, g, J ) is an oriented Riemannian manifold with an almost complex structure J , i.e., J 2 = −id T M , which is Hermitian, i.e., g(J X, Then, (M, g, J ) is of even dimension 2n and induces an U (n)-structure. On the level of Lie algebras, we have 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 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 [2] that Unfortunately, this shows that ξ has no particular symmetries and using (19), 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 [14] T where each class can be characterized as follows: In the characterization of W 4 , θ is a one-form often called the Lee form.
The following proposition contains well-known and useful properties of almost Hermitian W 1 , . . . , W 4 classes.

Proposition 5 ([14])
We have the following characterization of Gray-Hervalla classes: Moreover, The left-hand side has a nice interpretation, which is valid for all Gray-Hervella classes. Define Ricci and * -Ricci tensors by These induce, taking traces, scalar curvatures Then, by (16) Now, we will discus relations between elements in the divergence formula (10) in each pure class W i 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) [14]: Thus, the divergence formula (10) may be rewritten, using these invariants and formula (22) as follows: It is not hard to see by definitions of each pure class W i and Proposition 5 that the following fact holds. Table 1. In a table, i (k) j denotes an invariant i j considered for a class W k .

Proposition 6 Quadratic invariants characterize pure Gray-Hervella classes W i as listed in
Using Proposition 6 and relation (23), we can derive some useful relations for each pure class. These relations are well known. (See the references listed in the proposition below.) Let us enlarge on this. In analogy to Gray [15], we consider the following curvature condition: Notice that the class SC contains Gray class G 1 , which by definition, denotes almost Hermitian structures for which the curvature tensor satisfies In particular, there is no locally conformally Kähler non-Kahler structure defined on a closed manifold which satisfies SC condition.
Proof It suffices to apply (10), Proposition 6 and use the fact that |ξ | 2 = 1 4 |∇ J | 2 . For a W 4 case, notice that Now, we show that the main integral formula (11) in an almost Hermitian case is equivalent to a Bor-Lamoneda formula [5]. Decompose ξ and χ with respect to the Gray-Hervella classes as follows: i.e., χ k = i ξ k e i e i , and let It can be shown that Thus, by above considerations (see also Remark 1), we have Hence, which implies the integral formula by Bor and Hernández Lamoneda [5] (assuming M is closed)

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 tr A 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 [5,8,20], where ξ U (n) is the intrinsic torsion of related U (n)-structure and Define a one-form η by the relation η X Y = η(J X)J Y . This convention will appear to be useful. Denote the class in the space of all possible intrinsic torsions induced by η by W 5 . Split s su(n) ⊥ into s u(n) ⊥ and s R with respect to the decomposition su(n) ⊥ = u(n) ⊥ ⊕ R.

Proposition 8
On a closed SU(n)-structure (M, g, J ) with the W 5 -component induced by the 1-form η, we have the following integral formula: In particular, if (M, g, J ) is of Gray-Hervella class W 1 ⊕W 2 ⊕W 3 treated as U (n)-structure, Thus, Assuming M is closed and applying the Stokes theorem, we get (26).
The values of s R and s alt su(n) ⊥ can be computed explicitly, which gives an alternative version of formula (26). Firstly, introduce two components of the intrinsic torsion, ξ U (n),12 ∈ W 1 ⊕ W 2 and ξ U (n),34 ∈ W 3 ⊕ W 4 . Now, it suffices to apply (26).

Remark 2
The above integral formula, however formulated in a different way, can be found in [5]. Let us be more precise. In [5], 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 [13] that the first Chern form γ is given by It is not hard to see that Thus, using the same arguments as before Corollary 2, we get It suffices to notice that by (22) and (21), we have 1 4 (s − s * ) = |ξ U (n) | 2 .

Almost contact metric structures
Let (M, g) be a (2n + 1)-dimensional manifold together with a 1-form η (and its dual unit vector field ζ ) and ϕ ∈ End(T M) such that 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 where u(n) ⊥ is isomorphic to the space of block matrices of the form 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 [12] By (28), it follows that Thus, we may write the intrinsic torsion in an alternative way [12] ξ Hence, the characteristic vector field in this case equals The condition ξ ∈ T * M ⊗ u(n) ⊥ (T M) is equivalent to relation (31). Decomposing the space T * M ⊗ u(n) ⊥ (T M) into irreducible U (n) × 1-modules, we get 12 classes C 1 , . . . , C 12 [7]. First four are isomorphic to Gray-Hervella classes W 1 , . . . , W 4 .

Remark 3
Note that in [7] types of almost contact metric structures, i.e., irreducible modules of T * M ⊗ u(n) ⊥ (T M), were classified with respect to α(X , ϕY ). It is well known that this is equivalent to considering the intrinsic torsion as a map β(X , Y , Z ) = g(ξ X Y , Z ). The correspondence follows from the fact that 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 Table 2.

Lemma 1
The following relations hold: Proof Denote by (B, a) an element in u(n) ⊥ of form (29). Note that a = (B, a)e 2n+1 ∈ R 2n ⊂ R 2n+1 . Then, where a ∧ã is an endomorphism of R 2n given by (a ∧ã)v = ã, v a − a, v ã. Thus where ϕ 0 is a restriction of ϕ to R 2n , thus defining an almost complex structure. By these considerations, we are ready to compute s alt u(n) ⊥ . We have (here i, j = 1, . . . , 2n) For s u(n) ⊥ by (30), we have By Lemma 1 and above considerations, we may rewrite formula (10) as Moreover, by a classification of each module C i by quadratic invariants [7, Table I], we have the following observation.

Corollary 4 If the intrinsic torsion of an almost contact metric structure belongs to D 2 class and satisfies Janssen-Vanhecke C(α) condition, then
Proof Follows immediately by Proposition 10(1).

Remark 4
In an analogous way as for U (n)-structures, it can be shown, with a little bit more effort, that the integral formula (11) in this case is equivalent to the integral formula obtained in [12].

Examples
In this section, we apply obtained results to certain almost Hermitian and almost contact metric structures. First examples are simple illustration of obtained results for almost contact metric structures. We deal with this structure only since it has not been investigated from this point of view elsewhere. Almost Hermitian case is, due to well-known facts contained in Proposition 7, well understood from this perspective. In the end, we focus on more involving examples concerning homogeneous spaces, where we treat both cases-almost Hermitian and almost contact metric structures.
It follows that the second divergence formula in Proposition 10 is justified.

Examples on reductive homogeneous spaces
We will show that for a certain choice of G-structures on reductive homogeneous spaces induced from one-parameter deformations of invariant Riemannian metrics, the characteristic vector field χ vanishes; hence, the divergence formula becomes point-wise formula. We justify this stating appropriate examples. We closely follow [3, p. 140] and [1]. Let K be a connected, compact Lie group and H its closed, connected Lie subgroup. The quotient K /H is a homogeneous space denoted by M. Assume additionally that on the level of Lie algebras k = h ⊕ m, where m is the orthogonal complement with respect to some ad(H )-invariant bilinear form B on k. Deform B in the following way: Assume m = m 0 ⊕m 1 , where For any t > 0, we put Form B t defines an invariant Riemannian metric g t on M. We will often write g instead of g t , if there is no confusion. The Levi-Civita connection of g may be described as a linear map Λ : m → so(m) defined as follows: [3] Λ where X , Y ∈ m 0 , A, B ∈ m 1 . Now, we consider a G-structure on M. Thus, we have a decomposition so(m) = g ⊕ g ⊥ , where we take orthogonal complement with respect to the Killing form on so(m). Then, Λ splits as Λ = Λ g + Λ g ⊥ . Λ g defines a G-connection ∇ G , whereas Λ g ⊥ corresponds to the intrinsic torsion ξ .
In the following two examples, we introduce a G-structure via the same procedure. Denote by x 0 the coset eH, and let ad : H → SO(m) be the isotropy representation. Let ϕ : m → m be a linear map, which intertwines the isotropy representation. Since all tensor bundles on M are associated with the bundle G → M with respect to the isotropy representation, it follows that ϕ induces (1, 1)-tensor field, in our case, almost Hermitian or almost contact metric structure. defines a one-parameter family of Riemannian metrics on F 1,2 . The following basis is orthonormal with respect to the given inner product on m: where e jk is a skew-symmetric matrix with the ( j, k) entry equal to 1 and s jk is a symmetric matrix with the ( j, k)-entry equal to i (and remaining elements except for (k, j)-entry equal to zero). We see that m 0 is spanned by where (t, r , s) denotes an element diag(e it , e is , e ir ) ∈ H and R θ is a rotation in R 2 through an angle θ . In order to define an almost Hermitian structure, it suffices to define isotropy The Levi-Civita connection of this almost Hermitian structure can be described by a map Λ : m → so(m), The curvature tensor R is given by where Ad : h → so(m) denotes the differential of the isotropy representation, Here, H k denotes the matrix 1 2 s kk . Now we are ready to compute s u(3) ⊥ . Deriving relations for the commutators in m and its components in h and then the curvature tensor R and projections of elements e i j to u(3) ⊥ , we get We could obtain the above relation by applying formula (22). It is easy to see that s = 2(−13 + 3t − 2 t ) and s * = 2(3 − 5t − 2 t ). Let us turn to computations of the intrinsic torsion and its components. We easily see that Λ : m → u(3) ⊥ . Hence, the minimal connection ∇ U (3) is induced by a zero map. Moreover, corresponds to |ξ sym | 2 − |ξ alt | 2 ; hence, the main divergence formula, which reduces to 1 2 s u(3) ⊥ = |ξ alt | 2 − |ξ sym | 2 = 4(2 − t), is justified.
Let us look at the Gray-Hervella classes induced by t for each choice of t > 0. Since χ = 0, the considered almost Hermitian structure is of type W 1 ⊕ W 2 ⊕ W 3 . Simple calculations show that Λ, hence ξ , satisfies ξ J X J Y = −ξ X Y . Thus, by Proposition 5, the considered structures are of type W 1 ⊕ W 2 . Moreover, it is nearly Kähler, i.e., in W 1 , if and only if t = 1 2 . By above considerations, we see that for t < 2, s u(3) ⊥ > 0 and for t > 2, s u(3) ⊥ < 0.

Example 5
We follow very closely the approach by Agricola [1]. Consider the fivedimensional Stiefel manifolds V 4,2 = SO(4)/SO (2). We embed S(2) as a lower diagonal block. We have the splitting so(4) = so(2) ⊕ m with respect to the Killing form B, where There is a one-parameter family g t of Riemannian metrics on V 4,2 constructed by Jensen [16], which are obtained from the invariant dot product on m The curvature tensor is then given by where Ad is the differential of the isotropy representation Ad : SO ( One can check that ϕ, in deed, defines an almost contact structure with the Reeb field ζ induced by E 5 and compatible with the metric g t . Notice that the fundamental form F(X , Y ) = g t (X , ϕ(Y )) induced by ϕ if F = e 13 + e 24 , which is proportional to d E 5 making the structure just Sasaki structure [1].