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- W260064236 abstract "For a compact connected Lie group $G$ we prove that the bi-invariant affine connections which induce derivations on the corresponding Lie algebra $frak{g}$ coincide with the bi-invariant metric connections. In the sequel, we focus on the geometry of a naturally reductive space $(M=G/K, g)$ endowed with a family of $G$-invariant connections ${nabla^{alpha} : alphainmathbb{R}}$ whose torsion is a multiple of the torsion of the canonical connection $nabla^{c}$, i.e. $T^{alpha}=alphacdot T^{c}$. For the spheres $S^{6}$ and $S^{7}$ we prove that the space of $G_2$ (resp. $Spin(7)$)-invariant affine or metric connections consists of the family $nabla^{alpha}$. In the compact case we examine the flatness condition $R^{alpha}equiv 0$ and we state a refinement of the classical Cartan-Schouten theorem. The constancy of the induced Ricci tensor $Ric^{alpha}$ is also described. We prove that any compact isotropy irreducible naturally reductive Riemannian manifold, which is not a symmetric space of Type I, carries at least two $nabla^{alpha}$-Einstein structures with skew-torsion, namely these which occur for $alpha=pm 1$. A generalization of this result is given also for a class of compact normal homogeneous spaces $M=G/K$ with two isotropy summands. We introduce a new 2-parameter family of $G$-invariant connections on $M=G/K$, namely $nabla^{s, t}$ with $sinmathbb{R}$ and $tinmathbb{R}_{+}$; for the Killing metric $t=1/2$ skew-torsion appears and we examine the $nabla^{s, 1/2}$-Einstein condition. We show that $M$ is normal Einstein, if and only if, $M$ is a $nabla^{s, 1/2}$-Einstein manifold with skew-torsion for one of the values $s=0, 2$. In this way we provide a series of new examples of manifolds admitting these structures.admitting these structures." @default.
- W260064236 created "2016-06-24" @default.
- W260064236 creator A5048525568 @default.
- W260064236 date "2014-08-05" @default.
- W260064236 modified "2023-09-27" @default.
- W260064236 title "Invariant connections with skew-torsion and $nabla$-Einstein naturally reductive manifolds" @default.
- W260064236 hasPublicationYear "2014" @default.
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