Invariant connections with skew-torsion and \(\nabla\)-Einstein manifolds (Q2810919)

The author studies three, still related, problems. Firstly, for a compact connected Lie group \(G\), he studies a class of bi-invariant affine connections whose geodesics through the identity are the \(1\)-parameter subgroups. He shows that the bi-invariant affine connections which induce derivations on the corresponding Lie algebra \(\mathfrak{g}\) coincide with the bi-invariant metric connections. Secondly, he describes the geometry of a naturally reductive homogeneous space \((M=G/K,g)\) endowed with a family of \(G\)-invariant connections \(\nabla^{\alpha}\) whose torsion is a multiple of the torsion of the canonical connection \(\nabla^c\). It is shown that for the spheres \(S^6\) and \(S^7\) the space of \(G_2\)-invariant (resp. \(\mathrm{Spin}(7)\)-invariant) affine or metric connections consists of the family \(\nabla^{\alpha}\). Finally, he examines the constancy of the induced Ricci tensor \(\mathrm{Ric}^{\alpha}\) and proves that any compact isotropy irreducible standard homogeneous Riemannian manifold, which is not a symmetric space of Type I, is a \(\nabla^{\alpha}\)-Einstein manifold for any real \(\alpha\). He also provides examples of \(\nabla ^{\pm 1}\)-Einstein structures for a class of compact homogeneous spaces \(M=G/K\) with two isotropy summands.