On a problem of A. M. Vasil'ev (Q1901213)

Let \(G\) be a connected Lie group and \(H\) a closed subgroup of \(G\). We assume that the quotient space \(M = G/H\) is reductive. Reductive spaces belong to the class of homogeneous spaces having canonical invariant affine connections. These connections have a characteristic property: geodesics of these connections are trajectories of one-parameter subgroups of motion groups of the homogeneous space. Invariant affine connections of the reductive homogeneous space \(M = G/H\) having the above mentioned property will be called \(G\)-connections. The problem of A. M. Vasil'ev is to find invariant affine connections of the reductive homogeneous space \(M = G/H\), the geodesics of which are trajectories of one-parameter subgroups of normal extensions \({\mathcal G}\) of the Lie group \(G\) of their motions. Such connections will be called \(\Gamma\)- connections. Every \(G\)-connection is a \(\Gamma\)-connection. This is a simplified solution to the Vasil'ev problem. It may turn out that the Vasil'ev problem can be solved using its simplified variant. For this, it is necessary and sufficient that each \(\Gamma\)-connection of \(M = G/H\) be a \({\mathcal G}\)-connection of \(M = {\mathcal G}/{\mathcal H}\). This allows us to formulate a new problem: Is it true that each Vasil'ev problem is solved with the help of its simplified variant? In this paper, the author proves that the above problem is nontrivial and gives a partial solution to it. In particular, he proves that semisimple manifolds admit only a trivial solution to the Vasil'ev problem.