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arXiv · 2001.01545

Levi-Civita connections and vector fields for noncommutative differential calculi

Jyotishman Bhowmick·Debashish Goswami·Giovanni Landi
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Abstract

We study covariant derivatives on a class of centered bimodules $\mathcal{E}$ over an algebra A. We begin by identifying a $\mathbb{Z} ( A ) $-submodule $ \mathcal{X} ( A ) $ which can be viewed as the analogue of vector fields in this context; $ \mathcal{X} ( A ) $ is proven to be a Lie algebra. Connections on $\mathcal{E}$ are in one to one correspondence with covariant derivatives on $ \mathcal{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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Jyotishman Bhowmick, Debashish Goswami, Giovanni Landi. 2020-01-06. Levi-Civita connections and vector fields for noncommutative differential calculi. https://doi.org/10.1142/s0129167x20500652

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