Diffeological Levi-Civita connections

A diffeological connection on a diffeological vector pseudo-bundle is defined just the usual one on a smooth vector bundle; this is possible to do, because there is a standard diffeological counterpart of the cotangent bundle. On the other hand, there is not yet a standard theory of tangent bundles, although there are many suggested and promising versions, such as that of the internal tangent bundle, so the abstract notion of a connection on a diffeological vector pseudo-bundle does not automatically provide a counterpart notion for Levi-Civita connections. In this paper we consider the dual of the just-mentioned counterpart of the cotangent bundle in place of the tangent bundle (without making any claim about its geometrical meaning). To it, the notions of compatibility with a pseudo-metric and symmetricity can be easily extended, and therefore the notion of a Levi-Civita connection makes sense as well. In the case when $\Lambda^1(X)$, the counterpart of the cotangent bundle, is finite-dimensional, there is an equivalent Levi-Civita connection on it as well

Kaluza-Klein Reduction of Low-Energy Effective Actions: Geometrical Approach

Equations of motion of low-energy string effective actions can be conveniently described in terms of generalized geometry and Levi-Civita connections on Courant algebroids. This approach is used to propose and prove a suitable version of the Kaluza-Klein-like reduction. Necessary geometrical tools are recalled.Comment: Some comments added based on the journal review. A few of minor typos correcte

Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and application

In this article we extend the Gallot-Tanno theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over such a manifold admits a parallel symmetric 2-tensor then it is incomplete and has non zero constant curvature. An application of this result to the existence of metrics with distinct Levi-Civita connections but having the same unparametrized geodesics is given

Null-projectability of Levi-Civita connections

We study the natural property of projectability of a torsion-free connection along a foliation on the underlying manifold, which leads to a projected torsion-free connection on a local leaf space, focusing on projectability of Levi-Civita connections of pseudo-Riemannian metric along foliations tangent to null parallel distributions. For the neutral metric signature and mid-dimensional distributions, Afifi showed in 1954 that projectability of the Levi-Civita connection characterizes, locally, the case of Patterson and Walker's Riemann extension metrics. We extend this correspondence to null parallel distributions of any dimension, introducing a suitable generalization of Riemann extensions.Comment: 15 page

Pair of associated Schouten-van Kampen connections adapted to an almost contact B-metric structure

There are introduced and studied a pair of associated Schouten-van Kampen affine connections adapted to the contact distribution and an almost contact B-metric structure generated by the pair of associated B-metrics and their Levi-Civita connections. By means of the constructed non-symmetric connections, the basic classes of almost contact B-metric manifolds are characterized. Curvature properties of the considered connections are obtained.Comment: 12 page

Levi-Civita connections from toral actions

We construct tame differential calculi coming from toral actions on a class of C*-algebras. Relying on the existence of a unique Levi-Civita connection on such a calculi, we prove a version of the Bianchi identity. A Gauss-Bonnet theorem for the canonical rank 2-calculus is studied.Comment: This is a replacement for the first versio