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Invariant connections with skew-torsion and $\nabla$-Einstein manifolds
Differential Geometry · Submitted 5 Aug 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Then we examine the "constancy" of the induced Ricci tensor ${\rm Ric}^{\alpha}$ and prove that any compact simply-connected isotropy irreducible standard homogeneous Riemannian manifold, which is not a symmetric space of Type I, is a $\nabla^{\alpha}$-Einstein manifold for any $\alpha\in\mathbb{R}$.
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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 $\frak{g}$ 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 ${\rm S}^{6}$ and ${\rm S}^{7}$ we prove that the space of ${\rm G}_2$ (resp. ${\rm Spin}(7)$)-invariant affine or metric connections consists of the family $\nabla^{\alpha}$. Then we examine the "constancy" of the induced Ricci tensor ${\rm Ric}^{\alpha}$ and prove that any compact simply-connected 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.79 | 0.79 | 0.03 | asserted 0.77 |
| s1 | We show that the bi-invariant affine connections which induce derivations on the corresponding Lie algebra $\frak{g}$ coincide with the bi-invariant metric connections. | result 0.99 | 0.93 | 0.11 | asserted 1.00 |
| s2 | 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 $\nabla^{c}$. | contribution 0.96 | 0.85 | 0.06 | asserted 0.98 |
| s3 | For the spheres ${\rm S}^{6}$ and ${\rm S}^{7}$ we prove that the space of ${\rm G}_2$ (resp. ${\rm Spin}(7)$)-invariant affine or metric connections consists of the family $\nabla^{\alpha}$. | result 0.98 | 0.93 | 0.09 | asserted 1.00 |
| s4 | Then we examine the "constancy" of the induced Ricci tensor ${\rm Ric}^{\alpha}$ and prove that any compact simply-connected isotropy irreducible standard homogeneous Riemannian manifold, which is not a symmetric space of Type I, is a $\nabla^{\alpha}$-Einstein manifold for any $\alpha\in\mathbb{R}$. | result 0.99 | 0.94 | 0.56 | asserted 1.00 |
| s5 | We also provide examples of $\nabla^{\pm 1}$-Einstein structures for a class of compact homogeneous spaces $M=G/K$ with two isotropy summands. | result 0.89 | 0.94 | 0.09 | asserted 1.00 |
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Original abstract
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 $\frak{g}$ 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 ${\rm S}^{6}$ and ${\rm S}^{7}$ we prove that the space of ${\rm G}_2$ (resp. ${\rm Spin}(7)$)-invariant affine or metric connections consists of the family $\nabla^{\alpha}$. Then we examine the "constancy" of the induced Ricci tensor ${\rm Ric}^{\alpha}$ and prove that any compact simply-connected 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.
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For an alternative study of homogeneous $∇$-Einstein structures on odd-dimensional spheres we refer to [ref] .
Invariant affine connections on odd-dimensional spheres · The $∇^$-Einstein condition on naturally reductive spaces · 0.83
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