We introduce $q$-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with $q$-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a general class of metrics. We also give metric connections on a class of projective modules over the quantum 2-sphere. Finally, we outline a generalization to any Hopf algebra with a (left) covariant calculus and associated quantum tangent space.
We prove that the $4D_\pm$ calculi on the quantum group $SU_q(2)$ satisfy a metric-independent sufficient condition for the existence of a unique bicovariant Levi-Civita connection corresponding to every bi-invariant pseudo-Riemannian metric.
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 als…
We explore the possibility of introducing q-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. For the module of 1-forms on the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a g…
We study connections on hermitian modules, and show that metric connections exist on regular hermitian modules; i.e finitely generated projective modules together with a non-singular hermitian form. In addition, we develop an index calculus for such modules, and provide a characterization in terms of the existence of a pseudo-inverse of the matrix representing the hermitian form with respect to a set of generators. As a first illustration of the above concepts, we find metric connections on the fuzzy sphere. Finally, the framework is applied to a class of noncommutative minimal surfaces, for w…
We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.
For bicovariant differential calculi on quantum groups various notions on connections and metrics (bicovariant connections, invariant metrics, the compatibility of a connection with a metric, Levi-Civita connections) are introduced and studied. It is proved that for the bicovariant differential calculi on $SL_{q}(N)$, $O_{q}(N)$ and $Sp_{q}(N)$ from the classification of Schmuedgen, K. and Schueler, A. there exist unique Levi-Civita connections.
We prove the existence and uniqueness of Levi-Civita connections for strongly sigma-compatible pseudo-Riemannian metrics on tame differential calculi. Such pseudo-Riemannian metrics properly contain the classes of bilinear metrics as well as their conformal deformations. This extends the previous results in references 9 and 10. Star-compatibility of Levi-Civita connections for bilinear pseudo-Riemannian metrics are also discussed.
We construct tame differential calculi coming from toral actions on a class of $\mathrm{C}^*$-algebras. Relying on the existence of a unique Levi-Civita connection on such a calculus, we prove a version of the Bianchi identity. A Gauss-Bonnet theorem for the canonical tame calculus of rank two is studied.
We give a new definition of Levi-Civita connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence-uniqueness result for a class of modules of one forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes-Landi deformations of spectral triples on the Connes-Dubois Violette-Rieffel-deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifo…
Given a tame differential calculus over a noncommutative algebra $\mathcal{A}$ and an $\mathcal{A}$-bilinear pseudo-Riemannian metric $g_0,$ consider the conformal deformation $ g = k. g_0, $ $k$ being an invertible element of $\mathcal{A}.$We prove that there exists a unique connection $\nabla$ on the bimodule of one-forms of the differential calculus which is torsionless and compatible with $g.$ We derive a concrete formula connecting $\nabla$ and the Levi-Civita connection for the pseudo-Riemannian metric $g_0.$ As an application, we compute the Ricci and scalar curvature for a general conf…
We construct a U_h(sp(4))-equivariant quantization of the four-dimensional complex sphere S^4 regarded as a conjugacy class, Sp(4)/Sp(2)x Sp(2), of a simple complex group with non-Levi isotropy subgroup, through an operator realization of the quantum polynomial algebra C_h[S^4] on a highest weight module of U_h(sp(4)).
Two toy models are considered within the framework of noncommutative differential geometry. In the first one, the Einstein action of the Levi-Civita connection is computed for the algebra of matrix valued functions on a torus. It is shown that, assuming some constraints on the metric, this action splits into a classical-like, a quantum-like and a mixed term. In the second model, an analogue of the Palatini method of variation is applied to obtain critical points of the Einstein action functional for $M\sb 4(R)$. It is pointed out that a solution to the Palatini variational problem is not neces…
