In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off–diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/...

In the present paper, we discuss singular minimal surfaces in Euclidean $3-$space $\mathbb{R}^{3}$ which are minimal. Such a surface is nothing but plane, trivial outcome. However, non-trivial outcome obtained when modify usual condition of minimality by using special semi-symmetric metric connection instead Levi-Civita on $\mathbb{R}^{3}$. With this new connection, prove that, besides planes, ...

We develop a theory in which there are couplings amongst Dirac spinor, dilaton and non-Riemannian gravity and explore the nature of connection-induced dilaton couplings to gravity and Dirac spinor when the theory is reformulated in terms of the Levi-Civita connection. After presenting some exact solutions without spinors, we investigate the minimal spinor couplings to the model and in conclusio...

We study the quantum sphere Cq [S] as a quantum Riemannian manifold in the quantum frame bundle approach. We exhibit its 2-dimensional cotangent bundle as a direct sum Ω ⊕ Ω in a double complex. We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operators. We show that the q-monopole as spin connection induces a natural Levi-Civita type connection and find its Ricci ...

We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes[Co], which extends the Lichnerowicz vanishing theorem [L] to foliated manifolds with ...

A noncommutative-geometric generalization of the classical formalism of frame bundles is developed, incorporating into the theory of quantum principal bundles the concept of the Levi-Civita connection. The construction of a natural differential calculus on quantum principal frame bundles is presented, including the construction of the associated differential calculus on the structure group. Gen...

We construct finite element approximations of the Levi-Civita connection and its curvature on triangulations oriented two-dimensional manifolds. Our construction relies Regge elements, which are piecewise polynomial symmetric (0,2)-tensor fields possessing single-valued tangential-tangential components along interfaces. When used to discretize Riemannian metric tensor, these tensor do not posse...

Since its introduction by Élie Cartan, the holonomy of a connection has played an important role in differential geometry. One of the best known results concerning holonomy is Berger’s classification of the possible holonomies of Levi-Civita connections of Riemannian metrics. Since the appearance of Berger [1955], much work has been done to refine his list of possible Riemannian holonomies. See...

If ∇ is a torsionless connection on the tangent bundle of a manifold M the Weyl curvature W is the part of the curvature in kernel of the Ricci contraction. We give a coordinate free proof of Weyl’s result that W vanishes if and only if (M,∇) is (locally) diffeomorphic to RP with ∇, when transported to RP, in the projective class of ∇LC , the Levi-Civita connection of the Fubini–Study metric on...