Examples of Minimal G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{G}$$\end{document}-structures

Let M be an oriented Riemannian manifold and SO(M) its oriented orthonormal frame bundle. Assume there exists a reduction P⊂SO(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\subset SO(M)$$\end{document} of the structure group SO(dimM)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO(\dim M)$$\end{document} to a subgroup G. We say that a G-structure M is minimal if P is a minimal submanifold of SO(M), where we equip SO(M) in the natural Riemannian metric. We give non-trivial examples of minimal G-structures for G=U(dimM/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=U(\dim M/2)$$\end{document} and G=U((dimM-1)/2)×1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=U((\dim M-1)/2)\times 1$$\end{document} having some special features—locally conformally Kähler and α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-Kenmotsu manifolds, respectively.


Introduction
Existence of a geometric structure compatible with a Riemannian metric on an oriented manifold is equivalent to reduction of the structure group of oriented orthonormal frame bundle SO(M ) to certain subgroup G ⊂ SO(n), n = dim M . For example, almost Hermitian structure is defined by the unitary group U (m) ⊂ SO(2m), almost contact structure by U (m)×1 ⊂ SO(2m+1) or an almost quaternion-Hermitian structure by Sp(m)Sp(1) ⊂ SO(4m). Considering additionally the Levi-Civita connection ∇ we may ask if this connection is compatible with the given reduction. The failure is measured by the intrinsic torsion. In particular, if the intrinsic torsion vanishes, the holonomy algebra is contained in the Lie algebra g of the structure group.
We may classify intrinsic torsion with respect to the action of G obtaining irreducible components, often called Grey-Hervella classes. Another approach was initiated by Wood [11,12] and, in general case, by Martin-Cabrera and

Minimal G-structures via the Intrinsic Torsion
All the information in this section can be found in [3,8]. Let (M, g) be an oriented Riemannian manifold. Consider an oriented orthonormal frame bundle SO(M ). Let ∇ denote the Levi-Civita connection of g. It induces the horizontal distribution H ⊂ T SO(M ). Any vector X ∈ T M has the unique lift X h p to H p , p ∈ SO(M ). Vertical distribution V = kerπ * , where π : SO(M ) → M is a natural projection, is pointwise, isomorphic to the Lie algebra so(n) of the structure group SO(n). Denote by A * the fundamental vertical vector field induced by an element A ∈ so(n). The Riemannian metric on SO(M ) is given as follows: where X ∈ T M, A ∈ so(n). Define a structure on M by restricting the structure group SO(n) to a subgroup G such that on the level of Lie algebras, the following decomposition is ad(G)-invariant (g ⊥ denotes the orthogonal complement with respect to the Killing form). We say that a G-structure M is minimal if the induced subbundle P with the structure group G is minimal as a submanifold of SO(M ). Let us now define the intrinsic torsion and formulate the condition of harmonicity and minimality using this notion. Let ω be the connection form of the horizontal distribution H (induced by ∇). By the invariance of the splitting (1) the decomposition ω = ω g + ω g ⊥ defines a connection ω g on P . Denote the horizontal distribution induced by ω g by H and the associated horizontal lift of X ∈ T M by X h . Define the intrinsic torsion by the formula By ad(G)-invariance of ω g ⊥ and the horizontal lift, it follows that ξ X is defined up to the adjoint action, thus is an element of the adjoint bundle g ⊥ P = P × ad(G) g ⊥ . Thus we may treat ξ X as an endomorphism ξ X : T M → T M. It follows by above considerations that The reduction P ⊂ SO(M ) defines the unique section σ P of the associated bundle N = SO(M ) × SO(n) (SO(n)/G), where e ∈ SO(n) is the identity element. We define a Riemannian metric on N by inducing from the Riemannian metric g on M and the Killing form restricted to g ⊥ . We say that a G-structure M is harmonic if a section σ P : M → N is a harmonic section. Denote by vW the vertical component in T N of a vector W ∈ T N. Then [3] vσ P * (X) = − ξ X , thus harmonicity is coded in the intrinsic torsion. Moreover, we say that a G-structure is a harmonic map, if the unique section σ P is a harmonic map. Notice that notions of harmonicity and harmonicity as a map of a G-structure are different. Harmonic section do not need to be a harmonic map. In the former case we consider variations of the energy functional of the norm of the differential σ * among sections, whereas in the later case among all maps from M to N (compare Proposition 1 below).
Let us state results obtained in [3,8] concerning harmonicity and minimality of G-structures. For any endomorphism T : (2)

Proposition 1 ([3]). A G-structure M is harmonic if and only if the following condition holds
where (e j ) is a g-orthonormal basis. Moreover, a G-structure M is a harmonic map if it is a harmonic G-structure and Consider a Riemannian metricg on M defined bỹ where (e j ) is any g-orthonormal basis.

Proposition 2 ([8]). A G-structure M is minimal if and only if the following condition holds
where (ẽ j ) is anyg-orthonormal basis. Alternatively, if and only if the section σ P : M → N is a harmonic map, where we consider the Riemannian metricg instead of g on M .
Remark 1. Recall that condition for harmonicity of a map σ P : (M,g) → N is of the following form where S is the difference of the Levi-Civita connection∇ of the metricg and the Levi-Civita connection ∇ of the metric g [8].
Notice that in [8] the author considered intrinsic torsion differing by the sign form the intrinsic torsion considered in this article and by other authors.

Examples of Minimal G-structures
In this section we will compute the condition (5) for G = U (n) and G = U (n)× 1 assuming that considered structures satisfy some additional properties. Let us begin by explaining our choices. In general, the space of all possible intrinsic torsions T * M ⊗ g ⊥ P (with the notation from the previous section) is large, so we restrict our approach to certain submodules. In some cases, the intrinsic torsion is given by concrete formula depending on a vector field or a function. This happens for locally conformally Kähler structures in the case G = U (m) and α-Kenmotsu manifolds in the case G = U (n) × 1. Then the intrinsic torsion, hence the condition of minimality, depends on a (closed) one form θ, called the Lee form, or a single function α, respectively. In these cases it is possible to find appropriate examples of manifolds satisfying condition of minimality.

Locally Conformally Kähler Structures
Let (M, g, J) be a Hermitian manifold, i.e., J 2 = − id T M , J is integrable and g-invariant, Assume that M is locally conformally Kähler (LcK, for short) [7,10]. Then, there exists closed one-form θ, called the Lee form, such that where Ω is the Kähler form, Ω(X, Y ) = g(X, JY ), X, Y ∈ T M. Moreover [7], where J is considered here as a block matrix ( 0 −I I 0 ). The projection pr u(n) ⊥ : so(2n) → u(n) ⊥ respecting above decomposition is given by Thus the intrinsic torsion ξ X is given by the formula [2] which, by (6) implies In further considerations, we will use the notion of * -Ricci tensor, which is defined as follows where (e j ) is any orthonormal basis.
Remark 2. Our definition of * -Ricci tensor differs slightly from the one considered, for example, in [3]. We have Ric * (X, Y ) = Ric * (Y, X), where Ric * is the * -Ricci tensor in [3] and here we consider these tensors as (0, 2)-tensors. Notice that * -Ricci is not, in general, symmetric.
We will compute the condition of minimality of a G-structure induced by LcK manifold. First of all, let us derive the formula for the Riemannian metric g. Denote by (e j ) any g-orthonormal basis. Theñ Denote by D the J-invariant distribution spanned by the vector fields θ , Jθ . Let D ⊥ be the orthogonal complement of D in T M with respect to g. Notice that X ∈ D ⊥ if and only if θ(X) = θ(JX) = 0, which implies that for X ∈ D ⊥ we haveg(X, Y ) = 1 + 1 4 |θ | 2 g(X, Y ). Thus by dimensional reasons, orthogonal complement to D with respect tog is just D ⊥ . Hence, there should be no confusion in writing D ⊥ . Moreover, if X ∈ D, theng(X, Y ) = g(X, Y ). For a g-orthonormal basis (e j ) such that e 2n−1 = 1 |θ | θ and e 2n = Before computing minimality condition, let us introduce one useful notation. For a vector X ∈ T M put Let us collect properties of the assignment X → X in the Proposition below.

g(X , Y ) = g(X, Y ) for any X, Y ∈ T M.
After lengthy computations we get where the divergence div J equals div J = j (∇ẽ j J)ẽ j . By (6) and Proposition 3 the assignment is symmetric with respect to X and Y . Moreover, Jθ . Thus the bilinear map (X, Y ) → θ(JX)g(div J, Y ) is also symmetric.
For any X ∈ T M, Hence j R ξẽ j (ẽ j ) = − 1 2 By (7) we get j g(ξ R ξẽ j (ẽj ) Y, Z) = 1 8 where, to simplify notation, we put Concluding, by Proposition 2, a U (n)-structure on locally conformally Kähler manifold (M, g, J) with the Lee form θ is minimal U (n) if and only if the To check validity of the condition (8), we may restrict to certain vectors Y, Z. Indeed, since the right hand side is skew-symmetric with respect to Y and Z, by linearity, we have the following four possibilities: Now we will use the fact that dθ = Alt(∇θ) = 0. In the cases (i) and (iv), (8) is trivially satisfied. Finally, the cases (ii) and (iii) lead to the same condition Thus we have proved the following result. . , x n , y n ) with J ∂ ∂xi = ∂ ∂yi . Let f be arbitrary smooth function depending only on x 1 , y 1 and consider the conformal deformation g = e −2f g 0 . We will compute the condition of minimality of (M, g, J). The Lee form equals θ = df . Therefore Thus ∇ 0 X θ = ∇ 0 JX Jθ = 0 for X ∈ D ⊥ . Hence, by the formula for the Levi-Civita connections of conformally related metrics we get Recall that the curvature tensor R is given by the formula where L(X, Y ) = (∇ 0 X df )Y + df (X)df (Y ) and hessian is computed with respect to the Levi-Civita connection ∇ 0 of the Euclidean metric g 0 [9]. Simple calculations lead to the equality Since f depends only on x 1 and y 1 , it follows that both L(θ , Y ) and L(Jθ , JY ) vanish. Thus (9) holds. (M, g, J) is with parallel Lee form θ. If the U (n)-structure on locally conformally Kähler manifold (M, g, J) is a harmonic map, then it is a minimal. In particular, the U (n)-structure on any Hopf manifold is minimal.

Theorem 2. Assume
Before we will prove the above theorem let us recall the notion of Hopf manifold [10]. Consider the complex space C n \{0} without the origin and denote by Δ λ , where λ is nonzero complex number such that |λ| = 1, the cyclic group generated by the transformation z → λz, z ∈ C n . The Hopf manifold is a quotient (C n \{0})/Δ λ equipped with the Hermitian metric induced from the Hermitian metric h = 1 j z jzj j dz j ⊗d z j on C n \{0}. It can be shown that Hopf manifold is diffeomorphic to the product S 1 × S 2n−1 .
Proof (of Theorem 2). By (6) and the fact that θ is parallel we have Moreover, by (7), we have Since, by assumption, a U (n)-structure on M is a harmonic map, then (see Proposition 1) j (∇ ej ξ) ej = 0 and j R ξe j (e j ) = 0. Thus, by above considerations, j (∇ẽ j ξ)ẽ j = 0 and j R ξẽ j (ẽ j ) = 0. In particular, by Proposition 2, a U (n)-structure on M is minimal. Thus we have proved the first part of the theorem. The second part follows by the fact that Hopf manifolds are examples of LcK manifolds, which define U (n)-structures being harmonic maps [3].