Levi-Civita connections and vector fields for noncommutative differential calculi
Jyotishman Bhowmick, Debashish Goswami, Giovanni Landi
Abstract
We study covariant derivatives on a class of centered bimodules [Formula: see text] over an algebra [Formula: see text] We begin by identifying a [Formula: see text]-submodule [Formula: see text] which can be viewed as the analogue of vector fields in this context; [Formula: see text] is proven to be a Lie algebra. Connections on [Formula: see text] are in one-to-one correspondence with covariant derivatives on [Formula: see text]. 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.
Topics & Concepts
MathematicsCovariant transformationCovariant derivativeNoncommutative geometryLie derivativeConnection (principal bundle)Pure mathematicsVector fieldLie algebraTorsion (gastropod)Algebra over a fieldAffine connectionGauge covariant derivativeDifferential geometryVector spaceDifferential formExterior algebraNoncommutative algebraic geometryMultiplicative functionVector bundleCompatibility (geochemistry)Differential calculusMetric (unit)Lie groupLie bracket of vector fieldsCharacteristic classMetric connectionLie algebroidHomotopy and Cohomology in Algebraic TopologyAdvanced Topics in AlgebraAdvanced Operator Algebra Research