International Journal of Mathematics ( IF 0.604 ) Pub Date : 2020-06-23 , DOI: 10.1142/s0129167x20500652
Jyotishman Bhowmick; Debashish Goswami; Giovanni Landi

We study covariant derivatives on a class of centered bimodules $ℰ$ over an algebra $𝒜.$ We begin by identifying a $𝒵(𝒜)$-submodule $𝒳(𝒜)$ which can be viewed as the analogue of vector fields in this context; $𝒳(𝒜)$ is proven to be a Lie algebra. Connections on $ℰ$ are in one-to-one correspondence with covariant derivatives on $𝒳(𝒜)$. We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is also derived.

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