Geometry of $G$-Structures via the Intrinsic Torsion

We study the geometry of a $G$-structure $P$ inside the oriented orthonormal frame bundle ${\rm SO}(M)$ over an oriented Riemannian manifold $M$. We assume that $G$ is connected and closed, so the quotient ${\rm SO}(n)/G$, where $n=\dim M$, is a normal homogeneous space and we equip ${\rm SO}(M)$ with the natural Riemannian structure induced from the structure on $M$ and the Killing form of ${\rm SO}(n)$. We show, in particular, that minimality of $P$ is equivalent to harmonicity of an induced section of the homogeneous bundle ${\rm SO}(M)\times_{{\rm SO}(n)}{\rm SO}(n)/G$, with a Riemannian metric on $M$ obtained as the pull-back with respect to this section of the Riemannian metric on the considered associated bundle, and to the minimality of the image of this section. We apply obtained results to the case of almost product structures, i.e., structures induced by plane fields.


Introduction
Existence of a geometric structure on an oriented Riemannian manifold is equivalent to saying that the structure group SO(n) of the oriented orthonormal frame bundle reduces to a certain subgroup G. For example, for G equal SO(m) × SO(n − m), U(n/2), U(n/2) × 1, Sp(n/4)Sp(1), G 2 , Spin (7) we have almost product, almost Hermitian, almost contact, almost quaternion-Kähler, G 2 and spin structures, respectively. It is natural to ask if the holonomy group of the Levi-Civita connection ∇ is contained in given G. On the other hand the list of possible irreducible Riemannian holonomies is limited by the Berger list [17].
The defect of the Levi-Civita connection to be a G-connection measures the intrinsic torsion ξ, which is the difference of ∇ and a G-connection ∇ G (with torsion), The study of possible intrinsic torsions, i.e. the decomposition of the space of intrinsic torsions into irreducible modules, was initiated by Gray and Hervella [14] in the case of almost Hermitian manifolds. Later, many authors considered other possible cases (see, for example, [4,5,8,16,18,21]).
The other possible direction, initiated by Wood [22] and generalized to the general case by Gonzalez-Davila and Martin Cabrera [12], is to consider differential properties of intrinsic torsion induced by condition of harmonicity of the unique section of the associated homogeneous bundle. More precisely, a G- where π SO(M ) is the projection in the orthonormal frame bundle SO(M). Assume the decomposition so(n) = g ⊕ m, where m is the orthogonal complement to g with respect to the Killing form, on the level of Lie algebras is reductive. Equipp SO(M)/G with the natural Riemannian metric induced from the Killing form on m and Riemannian metric g on M. Then we say that a G-structure P is harmonic if the section σ is a harmonic. The correspondence of the notion of harmonicity with the intrinsic torsion follows from the fact that the intrinsic torsion ξ can be considered as a section of the bundle T * M ⊗ m P , where m P is the adjoint bundle P × adG m. This follows from the observation that the m-component of the connection form ω with respect to the reductive decomposition so(n) = g ⊕ m can be projected to the tangent bundle T M (see the next section for the details).
To seek for the 'best' possible non-integrable G-structures we consider the third possible approach. Namely, we focus on the minimality of a G-structure in the oriented orthonormal frame bundle SO(M). It is not surprising that minimality is related with the harmonicity of a G-structure. More generaly, we use the concept of intrinsic torsion to obtain some results on the geometry of Gstructures. The idea comes from the results obtained by the author [19] in the case of a single submanifol. We deal with the intrinsic and extrinsic geometry of a G-structure. To be more precise, we define the Riemannian metric on SO(M) by inducing it from the Riemannian metric g on M and Killing form B on the structure group SO(n). It is interesting that the Levi-Civita connection on P depends on the G-connection and Levi-Civita connection∇, which comes from the modification of the Riemannian metric g. This deformationg depends on the intrinsic torsion and equals the pull-back of the Riemannian metric on the associated homogeneous bundle SO(M)/G with respect to the section σ.
The main emphasis is put on the minimality of a G-structure P in the Riemannian structure SO(M). The main theorem can be stated as follows.
Theorem. The following conditions are equivalent In the fourth section we introduce a tensor, which transfers the Riemannian metric g to the mentioned aboveg. Its properties are crutial in the main considerations. With the Riemannian metricg and the horizontal distribution H ′ of the G-structure P induced by the minimal G-connection ∇ ′ , the projection π P : P → M becomes the Riemannian submersion.
The fifth section is the main section of the general outline and deals with the geometry of G-structures. Firstly, we consider intrinsic geometry focusing on the curvatures and, secondly, we consider extrinsic geometry, with the main result concerning minimality of a G-structure.
We end the article with some relevant examples. We consider almost product and almost hermitian structures.
Throughout the paper we will use the following notation and identification: For any associated bundle E = P × G S with the fiber S and induced by the principal bundle P (M, G) any element in E will be denoted by [[p, s]]. Moreover, we have

Intrinsic torsion
In this section we will review the basic facts concerning inrinsic torsion of a G-structure [17,12,23]. To make the article self contained we present all the proofs Let (M, g) be oriented Riemannian manifold, SO(M) its oriented orthonormal frame bundle. Denote by ω the connection form induced by the Levi-Civita connection ∇ on M. Let H and V be horizontal and vertical distributions on SO(M), respectively.
Assume the structure group SO(n) reduces to a subgroup G such that the quotient SO(n)/G is normal reductive, i.e. the subspace m = g ⊥ ⊂ so(n) defines ad(G) invariant decomposition so(n) = g⊕m, where g is the Lie algebra of G and the orthogonal part is taken with respect to the Killing form B. Denote by P ⊂ SO(M) the reduced subbundle. The g-component ω g of ω defines the connection form on P . Deonte by H ′ and V ′ the horizontal and vertical distributions on P with respcet to ω g , respectively.
Moreover, consider the following associated bundles -the adjoint bundles over M: so(n) P = P × adG so(n), g P = P × adG g and m P = P × adG m.
Notice that so(n) P can be realized as the bundle so(M) of skew-symmetric endomorphisms of the tangent bundle T M via the identification where we treat element p ∈ P x as a linear isomorphism p : R n → T x M. Analogously, we have the bundles g(M) and m(M). Moreover, so(n) P is isomorphic to the bundle V equiv over M of equivariant vertical vector fields on P . Namely, the map [[p, A]] → A * p , settles this isomorphism, where A * denotes the vertical vector induced by the matrix A ∈ so(n).
, it follows that ξ(X) ∈ m P . Section ξ ∈ Γ(T * M ⊗m P ) is called the intrinsic torsion of a G-structure P . By above ientifications ξ(X) = ξ X ∈ m(M).
The properties of the intrinsic torsion are gathered in the following proposition. Let us first introduce necessary notions. Let ∇ ′ be the connection on M induced by ω g on P . We call ∇ ′ the minimal connection of a G-structure.
Remark 2.2. One can show that δξ = −Θ, where Θ is the torsion of ω and δ : Γ(T * M ⊗ g P ) → Γ(T M) ⊗ Λ 2 (T * M) is the anty-symmetrization. This justifies the name for intrinsic torsion. On the level of M, by above Proposition, this is equivalent to the following equality Intrinsic torsion and the curvature tensor R obey the following formulas.
. Proof. First two relations follow by Proposition 2.1 (2) and from the fact that ∇ ′ respects the decomposition g(M) ⊕ m(M). They imply, immediately, the remaining ones.
Notice also that Let H N and V N be horizontal and vertical distributions on N, respectively, where H N is induced by ω. We have where SO(n) acts on T (SO(n)/G) by the differential of the natural action of The g-component ω g of ω defines the connection on ζ : N → M. Denote the horizontal lift of the vector U ∈ T N by U h,ζ ∈ T SO(N). Since the following diagram commutes and connection forms on π P : P → M and ζ : SO(M) → N are both equal to ω g it follows that Moreover, we will frequently identify sections of N with equivariant maps from SO(M) to SO(n)/G. Denote the equivariant function corresponding to σ by f . Then vσ * (X) = f * (X h p ), X ∈ T x M, π(p) = x, where vZ denotes the vertical component of Z ∈ T N. Indeed, if γ is a curve on M such thatγ(0) = X, denoting by γ h its horizontal lift to γ h (0) = p, by above isomorphisms, we have Since f is constant on P , by Proposition 2.1 where we consider ξ X as an element of m N . Moreover, denote by ϕ : T N → m N the following map i.e. ϕ settles the described above isomorphism of V N onto m N and is zero elsewhere. The Riemannian metric on N is induced by g and the Killing form on m, namely, Denote by ∇ N the Levi-Civita connection of the metric ·, · on N. To study vertical properties it is convinent to introduce two natural connections ∇ c and ∇ m on the adjoint bundle m N .
The connection form ω in SO(M) induces the connection ∇ ω in the bundle . This connection can be described as follows Indeed, if f β is the equivariant function corresponding to β, then, by the fact that β|P is equivariant for so(n) P we have (p ∈ P ) In m P we can consider the connection ∇ m induced from the connection form ω g on P . Thus, by above, In order to study the geometry of N it is necessary to consider sections of the bundle V N of vertical vector fields on N. This bundle is isomorphic to m N . Let ∇ c be the connection in m N associated with the connection form ω g of the bundle ζ : SO(M) → N. Let us compare ∇ c with the Levi-Civita connection ∇ N on N. For this purpose, define the homogeneous curvature form Φ, which is a 2-form on N with values in m N , by the formula Indeed, by the Koszul formula for the Levi-Civita connection ∇ N and the fact that ∇ c is metric for B we have . Moreover, taking the m-component of the structure equation Ω = dω + [ω, ω] and by the fact that equivariant function corresponding to ϕ(U) equals ω m (Ũ), we get

If one of U and V is horizontal and the other one vertical then
Finally, by the natural reductivity implied by the normal reductivity of SO(n)/G we get (2.5). Gathering all above results we have the following usefull relations.
Proposition 2.4. The following relations between ∇ ω , ∇ m and ∇ N , ∇ c hold Remark 2.5. Notice that π * N m P = m N and the pull-back connection π * N ∇ m equals ∇ c .
In this section we will derive the formula for the derivative ∇σ * and the tension field of σ.
Equip N with the Riemannian metric ·, · given by (2.2) and M with a Riemannian metricg on M, which may vary from g. We will consider σ as a map Let E(σ) denotes the energy functional where the norm · is taken with respect tog and ·, · , i.e., where (ẽ i ) is ag-orthonormal basis on M.
In order to study harmonic sections it is convenient to study variations of the functional (3.2) in the class of all sections of N. Ifg = g, then critical points of this functional are called harmonic G-structures [12]. We will consider the general caseg = g in the class of all maps from M to N. In other words, we will seek for σ to be a harmonic map. One can show that harmonicity of σ is equivalent to vanishing of the Euler-Lagrange equation where∇ is the Levi-Civita connection ofg and ∇ σ is the pull-back connection in the bundle We call τ (σ) the tension field of σ. Taking the decomposition of τ (σ) with repsect to above decomposition, harmonicity of σ is equivalent to vanishing of hτ (σ) and vτ (σ). Denote by Πg and Π g the differential ∇σ * with respect tog and g, respectively. Moreover, let S be the difference between∇ and ∇, i.e. S(X, Y ) =∇ X Y − ∇ X Y . Then τg(σ) = trgΠg = trg(Π g − σ * S).
We will show (see [23,12]) that For the proof of the first equality, by Proposition 2.4, Remark 2.5 and (3.1) we have 3) follows by Proposition 2.3 (see also (2.1)). For the proof of the second equality, computing the covariant derivative of the equality π N * σ * = id T M we get Since π N : N → M is a Riemannian submersion with totally geodesic fibers, by the following equations (see [1,20,23]) and by (3.5) Now, using (3.1) and the formula In order to describe explicitly the condition for the harmonicity of σ let us introduce certain curvature operator [19]. For any α ∈ so(M) let Then R α ∈ so(M) and the following formula holds Indeed, Now, we can state the formula for the harmonicity of σ.

Properties of certain transfer tensor
In this section we will introduce invertible tensor induced by the intrinsic torsion, the Riemannian metric defined by this tensor and state the properties of the Levi-Civita connection of this new metric. Results in this section are generalizations of the results obtained by the author in [19].
Adopt the notation from the previous section. For α ∈ m(M) put where we consider the intrinsic torsion as an element of m(M). It is easy to show that Let L be the endomorphism of the tangent bundle of the form In order to derive some properties of L recall the definition of the Riemannian metric on the bundle SO(M) induced by the metric g on M and by the Killing form B on the structure group SO(n). We define the Riemannian metric g SO(M ) on SO(M) as follows In particular, L is a symmetric automorphism of T M. Moreover, the covariant derivative of L is related with the intrinsic torsion ξ X by the formula Proof. By (4.1) and Proposition 2.1, Thus L is symmetric and positive definite. Moreover, by the fact that ∇ ω is metric for B and equals usual connection ∇, we have By Proposition 4.1 the symmetric and bilinear form g(X, Y ) = g(X, LY ), X, Y ∈ T M, defines a Riemannian metric on M. We call L the transfer tensor between g and g. Notice, that the projection π P : P → M is a Riemannian submersion with respect to g SO(M ) on P andg on M. Denote by∇ the Levi-Civita connection of g. One can show that [9] Thus by Proposition 4.1 we get

Geometry of G-structures
In this section we study the geometry of a G-structure P in SO(M). In this case we need to consider the Levi-Civita connection ∇ SO(M ) of the Riemannian metric g SO(M ) . One can show [19] that (

1) The orthogonal projection T SO(M) → T P equals
Proof. We will use frequently Proposition 2.1 and formula (4.1). Recall that T P = H ′ ⊕ m P . For any Y ∈ T M Proposition follows from the fact that and α * g − L −1 (ξ · α m ) h ′ is tangent to P . 5.1. Intrinsic geometry. For the intrinsic geometry we will compute the Levi-Civita connection ∇ P of (P, g SO(M ) ), the curvature tensor, Ricci tensor and sectional and scalar curvatures. We will compare obtained geometry with the geometry of the base manifold M.
Let us first introduce operator Q α and establish some relations. For X ∈ T M and α ∈ so(M) put Generalizing Proposition 2.3 we have the following lemma.

Theorem 5.3. The Levi-Civita connection ∇ P of P with the induced Riemannian metric G SO(M ) from SO(M) is of the following form
where X, Y ∈ T M, α, β ∈ g P .
Proof. First, by (4.1) and (3.7) we havẽ Thus, using (2.1) and (4.3) We showed that By the formula for the connection ∇ SO(M ) by Proposition 5.1, we obtain Thus, by the definition of Q α and formula (5.1) we get the desired formula for ∇ P X h ′ Y h ′ . For the proof of the second formula we will use Lemma 5.2 and Proposition 5.1. We have We prove the remaining relations analogously.
Proposition 5.4. The curvature tensor R P of ∇ P equals

Proof. Follows directly by Theorem 5.3, Jacobi identity and the relations
and (ẽ i ) is an orthonormal basis with respect tog on M and (α A ) is an orthonormal basis of g P with respect to B.
Proof. First, notice that which follows by the definition of Q α , Lemma 5.2 and Proposition 2.3. Hence Q α is skew-symmetric with respect tog. Now, it suffices to use the formulas for the curvature tensor R P .
Corollary 5.6. The sectional curvatures of ∇ P are given by the following formulas where X, Y are orthonormal with respect tog and α, β ∈ g P orthonormal with respect to B.
Proof. First and last relations follow immediately by the formulas for the curvature tensor R P and by (5.2). For the proof of the second one it sufices to use the skew-symmetry of Q β with respect tog (see the proof of Corollary 5.5).
Corollary 5.7. The scalar curvature of ∇ P equals By the above formulas for the curvatures we have the following relations between the geometry of (P, g SO(M ) ) and M.
Corollary 5.8. We have: (1) If dim M > 2, then P is never of constant sectional curvature.
(4) If R ′ = 0 and scalar curvature of (M,g) is positive, then the scalar curvature of P is positive.

Extrinsic geometry.
The properties of the extrinsic geometry are encoded in the second fundamental form, which we will derive explicitely. Moreover, we will compute the mean curvature vector of P in SO(M) and relate the minimality of P with the harmonicity of an induced section σ with appropriate Riemannian structure. Adopt the notation from the previous sections. For α ∈ m P let Then α + ∈ T ⊥ P .
Theorem 5.9. The second fundamental form Π P of P in SO(M) satisfies the following relations Proof. By Proposition 2.3 (see also proof of Theorem 5.3) we have which implies the first equality. Moreover, which proves the second equality. Since [g, g] ⊂ g, it follows that m P -component of ∇ SO(M ) α ast β * vanishes.
By above theorem we get the following impliation.
whereas, by Theorem 5.9, minimality of P is equivalent to the following condition Claerly, (H1) and (H2) imply (M). Conversely, assume (M) holds. It suffices to show that (H2) holds. By (4.3) and (M) we have By Proposition 2.3 and natural reductivity of SO(n)/G we get the desired equality.

Minimality of related structures
Let, as before, π : SO(M) → M be an oriented orthonormal frame bundle over a Riemannian manifold (M, g). Let G ⊂ SO(n), n = dim M, be the closed subgroup such that the quotient SO(n)/G is normal reductive. Let P be the reduced G-structure and denote by σ the induced section of the associated bundle N = SO(M) × SO(n)SO(n)/G over M. We equip SO(M) and N in natural Riemannian metrics as in previous sections. Namely, the Riemannian metric ·, · on N, by (2.2), equals where X, Y ∈ T M, α, β ∈ m(M). For simplicity denote the element ζ * α * by α † .
By (3.1) and (3.5) for any X, Y ∈ T M we have σ * (X), σ * (Y ) =g(X, Y ). Thus σ * ·, · =g and the map σ : (M,g) → (N, ·, · ) is an isometric imbedding. Therefore the harmonicity of σ is equivalent to its minimality [6]. For the sake of completeness, let us compute the minimality condition of an image σ(M) in N. Notice that the distribution σ * (T M) ⊥ in T N is spanned by the elements

Examples
In this section we ilustrate obtained results for the G-structures, where G = SO(m) × SO(n − m) or G = U( n 2 ). The case of other possible G-structures, for example coming from the Berger list of possible Riemannian holonomy groups, will be studied by the author independently. for some m = 1, . . . , n − 1. The quotient SO(n)/G is a symmetric space, which is the oriented Grassmannian G o m (R n ) of oriented m-dimensional subspaces in the Euclidean space R n . The reduction of the oriented orthonormal frame bundle SO(M) to the subbundle P with the structure group G is equivalent to the existence of m-dimensional distribution E on M and, hence, its orthogonal complement F = E ⊥ . We call M with the distinguished distribution E an almost product structure.
The connection ∇ ′ induced by the connection form ω g , where ω is the connection form of the Levi-Civita connection ∇, takes the form where the decomposition X = X ⊤ + X ⊥ is taken with respect to T M = E ⊕ F . In other words it is the sum of two connections induced by ∇ -connections in the vector bundles E and F over M. The intrinsic torsion of almost product structure equals Let us provide some examples by considering examples known in the literature satisfying the third condition of above corollary (see [10] and bibliography therein). To fit these examples to our context we will analyze each in more detail concentrating on the intrinsic torsion.
(1) Let (M, g, X 0 ) be a K-contact manifold with the Reeb vector field X 0 , i.e. X 0 is a unit Killing vector field and there exist one-form η and endomorphism ϕ such that for all X, Y ∈ Γ(T M). One can show that [2,13] ∇ X X 0 = −ϕX, The one dimensional distribution E tangent to X 0 defines the almost product structure on M. Since X 0 is geodecis vector field, the distribution E is totally geodesic. It can be shown, that E ⊂ Gr o n−1 (T M) is a minimal submanifold or, in other words, the immersion σ : (M,g) → (Gr o n−1 (T M), ·, · ), σ(x) = E x , x ∈ M, is minimal. Therefore, the SO(n− 1)×SO(1)-structure P induced by E is minimal in the orthonormal frame bundle SO(M). Let us provide more details.
Denote by X ⊤ and X ⊥ the tangent and orthogonal parts of the vector X with respect to the decomposition T M = E⊕E ⊥ . By the above properties, the intrinsic torsion ξ takes the form In particular, ξ X 0 = 0. Thus LX = X + 2X ⊥ and g(X, Y ⊤ ) = g(X, Y ⊤ ),g(X, Y ⊥ ) = 3g(X, Y ⊥ ).
7.2. Almost Hermitian structures. Let (M, g) be a 2n-dimensional manifold with the almost complex structure J, i.e. J is an endomorphism of the tangent bundle T M such that J 2 = −I. Assume J is hermitian with respect to g, g(JX, JY ) = g(X, Y ), X, Y ∈ T M.
Let P be a U(n)-structure with respect to J. The holonomy of M reduces to U(n) if (M, g, J) is integrable. The intrinsic torsion of U(n)-structure P is given by the formula [7] ξ X Y = 1 2 J(∇ X J)Y.
Denote by Z(M) the twistor bundle over M, i.e. the associated bundle to SO(M) with the fiber of the totallity SO(2n)/U(n) of almost complex structures in R 2n . Theorem 5.11 can be stated as follows.