The Dirac monopoles in 3-space and their generalization by C. N. Yang to 5-space are observed to be just the Levi-Civita spin connection of the cylindrical Riemannian metric on the 3and 5dimensional punctured spaces. Their straightforward generalization to higher dimensions is also investigated.

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.

We prove the Focal Index Lemma and the Rauch and Berger comparison Theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop space Ω(M) with the L2 (weak) Riemannian structure.

The maximal complexifications adapted to the Levi Civita connection for a distinguished one-parameter family of left-invariant metrics on a real, non-compact, semisimple Lie group G are determined. For G = SL2(R) their realization as invariant Riemann domains over G = SL2(C) is carried out and their complex-geometric properties are investigated. One obtains new non-univalent, non-Stein examples.

The purpose of this paper is to define a diagonal lift Dg of a Riemannian metric g of a manifold Mn to the cotangent bundle T (Mn) of Mn, to associate with Dg an Levi-Civita connection of T(Mn) in a natural way and to investigate applications of the diagonal lifts.

In this paper we prove a new geometric integrability theorem for almost complex manifolds. Let M be a connected, smooth 2m real dimensional manifold with an U(m) structure on its tangent bundle and ∇ be a compatible connection on it. Assume the complexified curvature of the connection has vanishing (0, 2) component. Then we claim that M can be given the structure of an m complex dimensional Käh...

We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may need refinement in theories of gravity with additional scalar fields. This argument is illustrated with a reformulation of the Brans-Dicke theory in terms of a spacetime connection with torsion determined dynamically in terms of the gradient of the Brans-Dicke scalar field. The perihelion shift in th...

In this paper we prove a new geometric integrability theorem for almost complex manifolds. Let M be a connected, oriented, smooth 2m real dimensional manifold with an U(m) structure on its tangent bundle. Moreover let ∇ be a compatible connection on it. Assume the complexified curvature of the connection has vanishing (0, 2) component. Then we claim that M can be given the structure of an m com...

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...