We investigate harmonic forms of geometrically formal metrics, which are defined as those having the exterior product of any two harmonic forms still harmonic. We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal Kähler metric are parallel w.r.t. the Levi-Civita connection. In the general Riemannian case a formal metric with ma...

∇XV = λX for every vector field X. (1.2) Here ∇ denotes the Levi-Civita connection of g. We call vector fields satisfying (1.2) closed conformal vector fields. They appear in the work of Fialkow [3] about conformal geodesics, in the works of Yano [7–11] about concircular geometry in Riemannian manifolds, and in the works of Tashiro [6], Kerbrat [4], Kühnel and Rademacher [5], and many other aut...

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...

A new framework to perturbative quantum gravity is proposed following the geometry of nonholonomic distributions on (pseudo) Riemannian manifolds. There are considered such distributions and adapted connections, also completely defined by a metric structure, when gravitational models with infinite many couplings reduce to two–loop renormalizable effective actions. We use a key result from our p...

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter. In Part I, we compute the Levi-Civita connection for these metrics. The connection and curvature forms take values in pseudodifferential operators (ΨDOs), and we compute the top symbols of these forms. In Part II, we develop a theory of Chern-Simons classes CS ...

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter s. In Part I, we compute the Levi-Civita connection for these metrics for s ∈ Z. The connection and curvature forms take values in pseudodifferential operators (ΨDOs), and we compute the top symbols of these forms. In Part II, we develop a theory of Wodzicki-Che...

We discuss the propagation of fermions on generic, curved branes in Ishibashi-Kawai-Kitazawa-Tsuchiya-type matrix models. The Dirac operator can be understood either terms a Weitzenb\"ock connection, or Levi-Civita connection with an extra torsion term. detail coupling spin to background geometry using Jeffreys-Wentzel-Kramers-Brillouin approximation. Despite absence local Lorentz invariance un...

In this paper, a group model for real extension of the de Sitter plane is pro-posed. This contains special matrices, which subgroup general linear group. It established that there exists left-invariant contact metric structure on group, normal and, therefore, para-Sasakian. The basis vector fields Lie algebra infinitesimal automorphisms are found. has maximum dimension in addition to Levi-Civit...

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...