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 tha
t there exists a unique connection $ abla$ on the bimodule of one-forms of the differential calculus which is torsionless and compatible with $g.$ We derive a concrete formula connecting $ abla$ 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 conformal perturbation of the canonical metric on the noncommutative $2$-torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalar curvature turns out to be a negative constant.
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; $ mathca
l{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.
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 bimo
dule 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.
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 manifold, the definition of metric compatibility given in the paper by Frolich et al failed to ensure the existence of a unique Levi-Civita connection. In the case of the matrix geometry, the Levi-Civita connection that we get coincides with the unique real torsion-less unitary connection obtained by Frolich et al.
Suppose that a compact quantum group ${mathcal Q}$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^{ast}$ (co)-action $alpha$ on $C(M)$, such that $alpha(C^infty(M)) subseteq C^infty(M, {mathcal Q})$ and the linear span of
$alpha(C^infty(M))(1 otimes {mathcal Q})$ is dense in $C^infty(M, {mathcal Q})$ with respect to the Frechet topology. It was conjectured by the author quite a few years ago that ${mathcal Q}$ must be commutative as a $C^{ast}$ algebra i.e. ${mathcal Q} cong C(G)$ for some compact group $G$ acting smoothly on $M$. The goal of this paper is to prove the truth of this conjecture. A remarkable aspect of the proof is the use of probabilistic techniques involving Brownian stopping time.
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 co
nformal 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 prove that, given any smooth action of a compact quantum group (in the sense of cite{rigidity}) on a compact smooth manifold satisfying some more natural conditions, one can get a Riemannian structure on the manifold for which the corresponding $C
^infty(M)$-valued inner product on the space of one-forms is preserved by the action.
Suppose that a compact quantum group $clq$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^*$ (co)-action $alpha$ on $C(M)$, such that the action $alpha$ is isometric in the sense of cite{Goswami} for some Riemannian struc
ture on $M$. We prove that $clq$ must be commutative as a $C^{ast}$ algebra i.e. $clqcong C(G)$ for some compact group $G$ acting smoothly on $M$. In particular, the quantum isometry group of $M$ (in the sense of cite{Goswami}) coincides with $C(ISO(M))$.
If a compact quantum group acts faithfully and smoothly (in the sense of Goswami 2009) on a smooth, compact, oriented, connected Riemannian manifold such that the action induces a natural bimodule morphism on the module of sections of the co-tangent
bundle, then it is proved that the quantum group is necessarily commutative as a $C^{*}$ algebra i.e. isomorphic with $ C(G)$ for some compact group $G$. From this, we deduce that the quantum isometry group of such a manifold M coincides with $C(ISO(M))$ where $ISO(M) $ is the group of (classical) isometries, i.e. there is no genuine quantum isometry of such a manifold.
Suppose that a compact quantum group Q acts faithfully and isomet- rically (in the sense of [10]) on a smooth compact, oriented, connected Riemannian manifold M . If the manifold is stably parallelizable then it is shown that the compact quantum grou
p is necessarily commutative as a C ast algebra i.e. Q = C(G) for some compact group G. Using this, it is also proved that the quantum isometry group of Rieffel deformation of such manifold M must be a Rieffel-Wang deformation of C(ISO(M))
