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Abstract
We propose a general procedure to construct noncommutative deformations of an embedded submanifold M of {mathbb {R}}^n determined by a set of smooth equations f^a(x)=0. We use the framework of Drinfel’d twist deformation of differential geometry of Aschieri et al. (Class Quantum Gravity 23:1883, 2006); the commutative pointwise product is replaced by a (generally noncommutative) star -product determined by a Drinfel’d twist. The twists we employ are based on the Lie algebra Xi _t of vector fields that are tangent to all the submanifolds that are level sets of the f^a (tangent infinitesimal diffeomorphisms); the twisted Cartan calculus is automatically equivariant under twisted Xi _t. We can consistently project a connection from the twisted {mathbb {R}}^n to the twisted M if the twist is based on a suitable Lie subalgebra {mathfrak {e}}subset Xi _t. If we endow {mathbb {R}}^n with a metric, then twisting and projecting to the normal and tangent vector fields commute, and we can project the Levi–Civita connection consistently to the twisted M, provided the twist is based on the Lie subalgebra {mathfrak {k}}subset {mathfrak {e}} of the Killing vector fields of the metric; a twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can be characterized in terms of generators and star -polynomial relations. We present in some detail twisted cylinders embedded in twisted Euclidean {mathbb {R}}^3 and twisted hyperboloids embedded in twisted Minkowski {mathbb {R}}^3 [these are twisted (anti-)de Sitter spaces dS_2,AdS_2].
Highlights
The notion of a submanifold N of a manifold M is a fundamental concept in geometry, playing a crucial role in various branches of mathematics and physics.A metric, connection, etc., on M uniquely induces a metric, connection, etc., on N
In the last few decades, various deep physical and mathematical reasons have stimulated the generalization of differential geometry to so-called noncommutative geometry (NCG) [15,41,45,47,48]
H = U g equipped with ε, S is a Hopf algebra; this means that a number of properties are fulfilled, in particular ( ⊗ id) ◦ = ◦, ( ⊗ id) ◦ = id = ◦, μ ◦ (S ⊗ id) ◦ = η ◦ = μ ◦ ◦ [μ : H ⊗ H → H denotes the product in H, μ(a ⊗ b) = ab, and η : C → H is defined by η(α) = α1]
Summary
The notion of a submanifold N of a manifold M is a fundamental concept in (differential) geometry, playing a crucial role in various branches of mathematics and physics. Identifying vector fields with first-order differential operators, we denote as := {X = Xi ∂i | Xi ∈ X } the Lie algebra of smooth vector fields X on D f ; here and below we abbreviate ∂i ≡ ∂/∂ xi Those vector fields X ∈ such that X ( f a) belong to C for all a, or equivalently such that X (h) belongs to C if h does (i.e., vanishes when restricted to M) make up a Lie subalgebra C, which is a X -bimodule; those such that X (h) belongs to C for all h ∈ X make up a smaller Lie subalgebra CC, which is an ideal in C and itself a X -bimodule. X over C[[ν]] (the ring of formal power series in ν with coefficients in C): as a
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