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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.
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
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
Given a spectral triple of compact type with a real structure in the sense of [Dabrowski L., J. Geom. Phys. 56 (2006), 86-107] (which is a modification of Connes original definition to accommodate examples coming from quantum group theory) and refere
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
Applying the Pomeransky inverse scattering method to the four-dimensional vacuum Einstein equation and using the Levi-Civita solution for a seed, we construct a cylindrically symmetric single-soliton solution. Although the Levi-Civita spacetime gener