Noncommutative differential geometry over the Moyal algebra is developed following an algebraic approach. It is then applied to investigate embedded noncommutative spaces. We explicitly construct the projective modules corresponding to the tangent bundles of the noncommutative spaces, and recover from this algebraic formulation the metric, Levi-Civita connection and related curvature introduced...

Let G be a Poisson–Lie group equipped with left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on tangent bundle TG of G. In this paper, we induce metric TG, and express in different cases Levi-Civita connection curvature terms We prove that space differential forms Ω*(G) is graded algebra if, only Ω*(TG) algebra. Moreover, show Sanchez de Al...

We develop tools for studying the control of underactuated mechanical systems that evolve on a configuration space with a principal fiber bundle structure. Taking the viewpoint of affine connection control systems, we derive reduced formulations of the Levi-Civita and the nonholonomic affine connections, along with the symmetric product, in the presence of symmetries and nonholonomic constraint...

Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing the discrete analogue of the Atiyah sequence, with a connection corresponding to the choice of a splitting of the short exact sequence. Equivalent represent...

We show how an affine connection on a Riemannian manifold occurs naturally as cochain in the complex for Leibniz cohomology of vector fields with coefficients adjoint representation. The coboundary Levi-Civita can be expressed sum two terms, one Laplace-Beltrami operator and other Ricci curvature term. vanishing this has interpretation terms eigenfunctions Laplacian. Separately, we compute cert...

Abstract A real hypersurface M in a complex projective space inherits an almost contact metric structure from the Kählerian of ambient space. This allows us to define, for any nonzero number k , so-called -th generalized Tanaka–Webster connection. With this connection and Levi-Civita one we can associate two tensors type (1,2) Jacobi operator $$R_{\xi }$$ <mml:math xmlns:mml="http://www.w3.org/...