Journal of Noncommutative Geometry· 2026Q1
On two notions of torsion and metric compatibility of connections in noncommutative geometry
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- Q1SCImago
- 2026year
Short summary
Metric compatibility in Beggs–Majid noncommutative geometry corresponds to Hermitian connections in Mesland–Rennie, and torsion-free conditions are equivalent when the calculus is quasi-tame.
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Key points
- Metric compatibility (Beggs–Majid) equals Hermitian connections (Mesland–Rennie) for *-preserving connections.
- Torsion-free conditions are equivalent for both frameworks if the calculus is quasi-tame.
- Results prove existence/uniqueness of Mesland–Rennie Levi-Civita connections for unitary cocycle deformations of Riemannian manifolds and Heckenberger–Kolb calculi on quantized irreducible flag manifolds.
AI-generated from the title and abstract; the full text is not read.
Abstract
We compare the notions of metric compatibility and torsion of a connection in the frameworks of Beggs–Majid and Mesland–Rennie. It follows that for * -preserving connections, compatibility with a real metric in the sense of Beggs–Majid corresponds to Hermitian connections in the sense of Mesland–Rennie. If the calculus is quasi-tame, the torsion zero conditions are equivalent. A combination of these results proves the existence and uniqueness of Levi-Civita connections in the sense of Mesland–Rennie for unitary cocycle deformations of a large class of Riemannian manifolds, as well as the Heckenberger–Kolb calculi on all quantized irreducible flag manifolds.
The authors' abstract, as published at the source. Journal of Noncommutative Geometry, 2026 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics