The geometry of special affinor structures of homogeneous \(\Phi\)-spaces of odd orders.

*(English. Russian original)*Zbl 0835.53057
Russ. Math. 38, No. 2, 82-84 (1994); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1994, No. 2 (381), 84-86 (1994).

Let \(M = G/H\) be a homogeneous \(\phi\)-space of order \(n = 2k + 1\); that means, for each point \(x \in M\) there is a diffeomorphism \(\phi_x\) of \(M\) having the point \(x\) as an isolated fixed point. Let \(g\) and \(h\) be the Lie algebras of \(G\) and \(H\), respectively. We assume that \(M = G/H\) is with canonical reductive decomposition, that means \(g = h \oplus m\), where \(m\) is the tangent space of \(M\) at its origin \(0 = H \in M\). It is known that \(m\) can be represented in the form \(m = m_1 \oplus m_2 \oplus \cdots \oplus m_k\) and on each \(m_i \neq \{0\}\) there exists a complex strucutre \(J_i\) defined by the relation \(\theta_i = a_i E + b_i j_i\), where \(a_i = \cos (2\pi/n)\), \(b_i = \sin (2\pi/n)\).

In this case, the complex structure \(J_0 = J_1 \oplus J_2 \oplus \cdots \oplus J_k\) on \(m\) defines an almost complex structure \(J\) on \(G/H\), invariant under \(G\) which is called a special almost complex structure. One can define by means of \(J\) \(k\) \(f\)-structures \((f^3 + f = 0)\) of a homogeneous \(\phi\)-space of order \(n = 2k + 1\), called special \(f\)-structures. One can also define \(p\) structures on \(M = G/H\), called special almost product structures.

The present article is devoted to the study of properties of the structures indicated above. He establishes relations between these structures and the Lie bracket on \(m\); that gives the possibility to obtain conditions for their integrability. He also studies relations between the special structures and invariant affine connections on \(G/H\) and examines how the subspace \(m^\varphi\) influences the geometry of these structures.

In this case, the complex structure \(J_0 = J_1 \oplus J_2 \oplus \cdots \oplus J_k\) on \(m\) defines an almost complex structure \(J\) on \(G/H\), invariant under \(G\) which is called a special almost complex structure. One can define by means of \(J\) \(k\) \(f\)-structures \((f^3 + f = 0)\) of a homogeneous \(\phi\)-space of order \(n = 2k + 1\), called special \(f\)-structures. One can also define \(p\) structures on \(M = G/H\), called special almost product structures.

The present article is devoted to the study of properties of the structures indicated above. He establishes relations between these structures and the Lie bracket on \(m\); that gives the possibility to obtain conditions for their integrability. He also studies relations between the special structures and invariant affine connections on \(G/H\) and examines how the subspace \(m^\varphi\) influences the geometry of these structures.

Reviewer: G.Tsagas (Thessaloniki)

##### MSC:

53C30 | Differential geometry of homogeneous manifolds |