English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The well-known fact that $S^1$, $S^3$ and $S^7$ are parallelizable manifoldsadmitting flat connections is revisited. The role of torsion in theconstruction of those flat connections is made explicit, and the possibilitiesallowed by different metric signatures are examined. A necessary condition forparallelizability in an open region is that the torsion tensor must becovariantly constant. This property can be used to obtain a relation between atorsion-free and flat connections. Our treatment covers Riemannian andpseudo-Riemannian (non-Euclidean signature) hyperbolic manifolds of dimensionsthree and seven. Apart from the spherical cases mentioned above, the explicitflat $\mathfrak{so}(p,q)$ connections with $p+q=3,7$ are constructed for thecoset manifolds $SO(p,q+1)/SO(p,q)$ or $SO(p+1,q)/SO(p,q)$.

Flat $\mathfrak{so}(p,q)$-Connections for Manifolds of Non-Euclidean Signature