arXiv research

We assume a vector bundle $p: E\to M$ with a general linear connection $K$ and a classical linear connection $\Lam$ on $M$. We prove that all classical linear connections on the total space $E$ naturally given by $(\Lam, K)$ form a 15-parameter family. Further we prove that all connections on $J^1 E$ naturally given by…

The Chern-Simons forms for R-linear connections on Lie algebroids are considered. A generalized Chern-Simons formula for such R-linear connections is obtained. We it apply to define Chern character and secondary characteristic classes for R-linear connections of Lie algebroids.

The construction of a linear connection on a pullback bundle from a connection on a vector bundle is explained in terms of fiberwise linear approximation. This procedure clarifies the geometric meaning of the linearized connection as well as the associated parallel transport and curvature.

The paper characterizes compatible linear connections on 3D Finsler manifolds.

The parallel linear transports defined by flat linear connection are axiomatically described. On this basis a number of properties, some of which are new, of these transports and connections are derived.

We give coordinate formula and geometric description of the curvature of the tensor product connection of linear connections on vector bundles with the same base manifold. We define the covariant differential of geometric fields of certain types with respect to a pair of a linear connection on a vector bundle and a lin…

Novel ternary structures reveal new interpretations of linear connections.

New method for linear connections in ODEs with constraints.

Sprays on Frechet manifolds connect connections and tangent structures.

We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism o…

A linear connection on a Finsler manifold is called compatible to the metric if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a compatible linear connection. Since the compatibility to the Finslerian metric does not imply the…

A linear connection $D$ in a Lie algebroid is said to be metrizable if there exists a Riemannian metric $h$ in the Lie algebroid such that $Dh=0$. Conditions for the linear connection $D$ to be metrizable are investigated.

A linear connection is associated to a nonlinear connection on a vector bundle by a linearization procedure. Our definition is intrinsic in terms of vector fields on the bundle. For a connection on an affine bundle our procedure can be applied after homogenization and restriction. Several applications in Classical Mech…

We describe the ringed-space structure of moduli spaces of jets of linear connections (at a point) as orbit spaces of certain linear representations of the general linear group. Then, we use this fact to prove that the only (scalar) differential invariants associated to linear connections are constant functions, as wel…

New proof for unique semi-symmetric compatible linear connection on Finsler manifolds.

Study on plane curves with special connections and curvatures.

The paper explores conditions for Randers metrics to have compatible linear connections.

Extending Lévi-Civita's concept to non-quadratic spaces, this study finds extremal compatible linear connections.

Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.

Proves torsion and curvature are unique for smooth manifolds.

Spheres in curve graphs are connected, proving Gromov boundary linearity.

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibi\-li\-ty condition). By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Rie…

Families of linear connections are constructed on almost contact manifolds with Norden metric. An analogous connection to the symmetric Yano connection is obtained on a normal almost contact manifold with Norden metric and closed structural 1-form. The curvature properties of this connection are studied on two basic cl…

A generalized notion of a Lie algebroid is presented. Using this, the Lie algebroid generalized tangent bundle is obtained. A new point of view over (linear) connections theory on a fiber bundle is presented. These connections are characterized by o horizontal distribution of the Lie algebroid generalized tangent bundl…

Linear F-manifolds are studied with connections and dual spaces.

In this paper we describe the local Ricci and Bianchi identities for an h-normal N-linear connection DΓ(N) on the dual 1-jet space J^{1*}(T,M). To reach this aim, we firstly give the expressions of the local distinguished (d-) adapted components of torsion and curvature tensors produced by DΓ(N), and then we analyze th…

The aim of the present paper is to provide an \emph{intrinsic} investigation of the properties of the most important geometric objects associated with the fundamental linear connections in Finsler geometry. We investigate intrinsically the most general relations concerning the torsion tensor fields and the curvature te…

A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.

Let $π:E\to M$ be a vector bundle over a simply connected manifold and $\nabla$ a linear connection in $π$. Let $σ: U \rightarrow E$ be a $\nabla$-parallel section of $π$ defined on a connected open subset $U$ of $M$. We give sufficient conditions on $U$ in order to extend $σ$ to the whole $M$. We mainly concentrate to…

This work refines claims about neural network connectivity, showing that simultaneous linear connectivity is possible under certain conditions.

Due to the success of residual networks (resnets) and related architectures, shortcut connections have quickly become standard tools for building convolutional neural networks. The explanations in the literature for the apparent effectiveness of shortcuts are varied and often contradictory. We hypothesize that shortcut…

The reduction theorems for general linear and classical connections are generalized for operators with values in higher order gauge-natural bundles. We prove that natural operators depending on the $s_1$-jets of classical connections, on the $s_2$-jets of general linear connections and on the $r$-jets of tensor fields …

We define Dorfman connections, which are to Courant algebroids what connections are to Lie algebroids. Several examples illustrate this analogy. A linear connection $\nabla\colon \mathfrak{X}(M)\timesΓ(E)\toΓ(E)$ on a vector bundle $E$ over a smooth manifold $M$ is tantamount to a linear splitting $TE\simeq T^{q_E}E\op…

A geometric structure (FAP-structure), having both absolute parallelism and Finsler properties, is constructed. The building blocks of this structures are assumed to be functions of position and direction. A non-linear connection emerges naturally and is defined in terms of the building blocks of the structure. Two lin…

Diagonal linear networks converge to lasso regularization path during training.

In the present paper, the $(HM',S,T)$-Cartan connections on pseudo-Finsler manifolds, introduced by A. Bejancu and H.R. Farran, are obtained by the natural almost complex structure arising from the nonlinear connection $HM'$. We prove that the natural almost complex linear connection associated to a $(HM',S,T)$-Cartan …

We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.

The moduli space of jets of certain G-structures (basically those which admit a canonical linear connection) is shown to be isomorphic to the quotient of a natural G-module by G.

Calculates affine transformations for specific homogeneous spaces.

Let $F^{2n}=(M,M',F^{\ast})$ be an even-dimensional pseudo-Finsler manifold. We construct an almost hypercomplex structure on any chart domain of a certain atlas of $M'$ by using a considered non-linear connection. Then by using the almost hypercomplex structure we define two new families of Finsler connections. Also w…

We give a Finsler non-linear connection by a new simplified definition for not only regular case but also singular case. In regular case, it corresponds to non-linear connection part of Berwald's connection, but our connection is expressed not in line element space but in point-Finsler space. In this view we recognize …

Study of multiplicative connections in Lie groupoids.

The aim of this paper is to generalize the theory of nonlinear connections of Grifone ([3] and [4]). We adopt the point of view of Anona [1] and continue developing the approach established by the first author in [10]. The first part of the work is devoted to the problem of associating to each $L$-regular linear connec…

New approach connects Finsler geometry's metric and connections.

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integr…

Study Lie algebroid connections on principal bundles over complex projective varieties.

This work analyzes how bottleneck layers and skip connections affect linear denoising autoencoders' generalization.

The paper reveals surprising star-shaped connectivity in neural networks.