Spacetimes have conventionally been described by a global Lorentzian metric on a differentiable four-manifold. Herein we explore the possibility of spacetimes defined by a connection, which is locally but not globally Levi-Civita. The general method of obtaining such connections is presented for the nondegenerate case followed by an example that modifies the Robertson-Walker spacetimes for flat...

Frobenius manifold structures on the spaces of abelian integrals were constructed by I. Krichever. We use D-modules, deformation theory, and homological algebra to give a coordinate-free description of these structures. It turns out that the tangent sheaf multiplication has a cohomological origin, while the Levi–Civita connection is related to 1-dimensional isomonodromic deformations.

Symmetric connections that are compatible with semi-Riemannian metrics can be characterized using an existence result for an integral leaf of a (possibly non integrable) distribution. In this paper we give necessary and sufficient conditions for a left-invariant connection on a Lie group to be the Levi–Civita connection of some semi-Riemannian metric on the group. As a special case, we will con...

We study submanifolds of a Riemannian manifold with a semi-symmetric non-metric connection. We prove that the induced connection is also a semi-symmetric non-metric connection. We consider the total geodesicness, total umbilicity and the minimality of a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection. We have obtained the Gauss, Codazzi and Ricci equations wit...

We construct tame differential calculi coming from toral actions on a class of $\mathrm{C}^*$-algebras. Relying the existence unique Levi-Civita connection such calculus, we prove version Bianchi identity. A Gauss-Bonnet theorem for canonical calculus rank two is studied.

Abstract A new derivation of the flow metrics in Type IIA is given. It better adapted to formulation as a variant Laplacian flow, and it uses projected Levi–Civita connection themselves instead their conformal rescalings.

We analyze the Dirac Laplacian of a one-parameter family of Dirac operators on a compact Lie group, which includes the Levi-Civita, cubic, and trivial Dirac operators. More specifically, we describe the Dirac Laplacian action on any Clifford module in terms of the action of the Lie algebra’s Casimir element on finitedimensional irreducible representations of the Lie group. Using this descriptio...

This is an in-progress set of notes describing the often-neglected geometrical, dynamical interpretation of a connection on a manifold. The essential idea is that, while the Levi-Civita connection is associated to the sliding action of parallel transport, it is perhaps more natural to think of a tangent space to a surface being rolled around as we move from point to point. The difference manife...

Using natural lifting operations, we give a coordinate-free proof of the fact that connection used by Bordemann, Neumaier and Waldmann to construct Fedosov standard ordered star product on cotangent bundle Riemannian manifold is obtained symplectification complete lift corresponding Levi-Civita connection, in sense Yano Patterson. In terms local coordinates, this has already been shown Plebansk...

A natural connection with torsion is defined, and it called the first on Riemannian Π-manifold. Relations between introduced Levi–Civita are obtained. Additionally, relations their respective curvature tensors, Ricci scalar curvatures in main classes of a classification Π-manifolds presented. An explicit example dimension five provided.