Journal of Lie Theory

\def\g{{\frak g}} For a compact connected Lie group $G$ we study the class of bi-invariant affine connections whose geodesics through $e\in G$ are the 1-parameter subgroups. We show that the bi-invariant affine connections which induce derivations on the corresponding Lie algebra $\g$ coincide with the bi-invariant metric connections. Next we describe the geometry of a naturally reductive 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 $\nabla^{c}$. For the spheres S$^6$ and S$^7$ we prove that the space of G$_2$ (respectively, Spin(7))-invariant affine or metric connections consists of the family $\nabla^{\alpha}$. We examine the ``constancy'' of the induced Ricci tensor Ric$^{\alpha}$ and prove 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 $\alpha\in\mathbb{R}$. We also provide examples of $\nabla^{\pm 1}$-Einstein structures for a class of compact homogeneous spaces $M=G/K$ with two isotropy summands.