Given the Finsler structure (M, F) on a manifold M, a Riemannian structure (M, h) and a linear connection on M are defined. They are obtained as the " average " of the Finsler structure and the Chern connection. This linear connection is the Levi-Civita connection of the Riemannian metric h. The relation between parallel transport of the Chern connection and the Levi-Civita connection of h are ...

The object of this paper is to study invariant submanifoldsM of Sasakian manifolds ̃ M admitting a semisymmetric nonmetric connection, and it is shown that M admits semisymmetric nonmetric connection. Further it is proved that the second fundamental forms σ and σ with respect to Levi-Civita connection and semi-symmetric nonmetric connection coincide. It is shown that if the second fundamental fo...

As is well known, a metric on a manifold determines a unique symmetric connection for which the metric is parallel: the Levi-Civita connection. In this paper we investigate the inverse problem: to what extent is the metric of a Riemannian manifold determined by its LeviCivita connection? It is shown that for a generic Levi-Civita connection of a metric h there exists a set of positive semi-defi...

The starting point of the famous structure theorems on Berwald spaces due to Z.I. Szabó [4] is an observation on the Riemann-metrizability of positive definite Berwald manifolds. It states that there always exists a Riemannian metric on the underlying manifold such that its Levi-Civita connection is just the canonical connection of the Berwald manifold. In this paper we present a new elementary...

This paper discusses the extent to which one can determine the space-time metric from a knowledge of a certain subset of the (unparametrised) geodesics of its Levi-Civita connection, that is, from the experimental evidence of the equivalence principle. It is shown that, if the space-time concerned is known to be vacuum, then the Levi-Civita connection is uniquely determined and its associated m...

LetM = (M,OM) be a smooth supermanifold with connection ∇ and Batchelor model OM ∼= ΓΛE∗ . From (M,∇) we construct a connection on the total space of the vector bundle E →M . This reduction of∇ is well-defined independently of the isomorphism OM ∼= ΓΛE∗ . It erases information, but however it turns out that the natural identification of supercurves inM (as maps from R1|1 toM) with curves in E r...

For any flag manifold G/T we obtain an explicit expression of its Levi-Civita connection with respect to any invariant Riemannian metric.

This paper investigates the relationship between two fundamental types of objects associated with a connection on a manifold: the existence of parallel semi-Riemannian metrics and the associated holonomy group. Typically in Riemannian geometry, a metric is specified which determines a Levi-Civita connection. Here we consider the connection as more fundamental and allow for the possibility of se...

We start from the pure Einstein-Hilbert action S = λ 2 R ⋆ 1 in Metric-Affine-Gravity, with the orthonormal metric g ab = η ab. We get an effective Levi-Civita Dilaton gravity theory in which the Dilaton field is related to the scaling of the gravitational coupling. When the Weyl symmetry is broken the resulting Einstein-Hilbert term is equivalent to the Levi-Civita one, using the projective in...

We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with Riemannian holonomy G2 can admit non-integrable G2 structures of type R⊕S 2 0 (R )⊕R in the sense of Fernández and Gray. This relies on the construction of some G2 solvmanifolds, whose Levi-Civita connection is known to give a parallel spinor, admitting a 2-parameter family of metric connections with non-zero skew-symm...