arXiv research

The paper explores conditions for lifting maps between graphs to embeddings.

We study some properties of the tangent bundles with metrics of general natural lifted type. We consider a Riemannian manifold $(M,g)$ and we find the conditions under which the Riemannian manifold $(TM,G)$, where $TM$ is the tangent bundle of $M$ and $G$ is the general natural lifted metric of $g$, has constant sectio…

Let (M,g) be a pseudo-Riemannian manifold and $T^2M$ be its the second-order tangent bundle equipped with the deformed 2-nd lift metric g which obtained from the 2-nd lift metric by deforming the horizontal part with a symmetric (0,2)-tensor field c. In the present paper, we first compute the Levi-Civita connection and…

We study the conditions under which the tangent bundle $(TM,G)$ of an $n$-dimensional Riemannian manifold $(M,g)$ is conformally flat, where $G$ is a general natural lifted metric of $g$. We prove that the base manifold must have constant sectional curvature and we find some expressions for the natural lifted metric $G…

We show how to lift positive Ricci and almost non-negative curvatures from an orbit space $M/G$ to the corresponding $G$-manifold, $M$. We apply the results to get new examples of Riemannian manifolds that satisfy both curvature conditions simultaneously.

In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and differential topology of manifolds, liftings or desingulizations of maps of appropriate …

Starting from a bundle E over R, the dual of the first jet bundle, which is a co-dimension 1 sub-bundle of the cotangent bundle of E, is the appropriate manifold for the geometric description of time-dependent Hamiltonian systems. Based on previous work, we recall properties of the complete lifts of a type (1,1) tensor…

The paper discusses Gauss maps for Möbius surfaces in spheres and their applications to Willmore surfaces.

In the first part of this paper, we let $G$ be a finitely-generated amenable group such that $G/[G, G]$ is torsion-free. We suppose that $G$ acts by homeomorphisms homotopic to the identity on a manifold $M$, and give conditions on $M$ which imply that such an action must lift to an action on the universal cover $\tild…

A new method to rescale ReLU neural networks based on path-lifting.

Unified complexity bound for sampling logconcave distributions

Develops a lifting theory for exponential maps in semi-Riemannian geometry.

A variety of lifted inference algorithms, which exploit model symmetry to reduce computational cost, have been proposed to render inference tractable in probabilistic relational models. Most existing lifted inference algorithms operate only over discrete domains or continuous domains with restricted potential functions…

In this paper we study lifted left invariant $(α,β)$-metrics of Douglas type on tangent Lie groups. Let $G$ be a Lie group equipped with a left invariant $(α,β)$-metric of Douglas type $F$, induced by a left invariant Riemannian metric $g$. Using vertical and complete lifts, we construct the vertical and complete lifte…

The paper proves every closed curve on a bouquet of circles lifts to finite covers.

Study connects symmetries in dynamical systems to phase plane representations.

Invariance principle proved for lifted geodesic walks on Riemannian submersions.

Unified framework for lifted training and inversion of neural networks.

We provide new conditions for the Strong Atiyah conjecture to lift to finite group extensions. In particular, we show cocompact special groups satisfy these conditions, so the Strong Atiyah conjecture holds for virtually cocompact special groups.

Considering the class G of g-natural metrics on the tangent bundle of a Riemannian manifold (M, g), it is shown that the flatnees for g is a necessary and sufficient condition of weakly symmetry (recurrent or pseudo-symmetry) of G. In particular, the cases of weakly symmetric Sasakian lift metric studied by Bejan and C…

Let $(M,ω)$ be a symplectic manifold and $F$ be a Finsler structure on $M$. In the present paper we define a lift of the symplectic two-form $ω$ on the manifold $TM\backslash 0$, and find the conditions that the Chern connection of the Finsler structure $F$ preserves this lift of $ω$. In this situation if $M$ admits a …

Defines Jacobi fields in nonholonomic mechanics.

Given two univalent harmonic mappings $f_1$ and $f_2$ on $\mathbb{D}$, which lift to minimal surfaces via the Weierstrass-Enneper representation theorem, we give necessary and sufficient conditions for $f_3=(1-s)f_1+sf_2$ to lift to a minimal surface for $s\in[0,1]$. We then construct such mappings from Enneper's surfa…

New theory for local parameterization of deep ReLU networks.

We prove that under some purely algebraic conditions every locally homogeneous structure modelled on some homogeneous space is induced by a locally homogeneous structure modelled on a different homogeneous space.

We make a study of Poisson structures of T*M which are graded structures when restricted to the fiberwise polynomial algebra, and give examples. A class of more general graded bivector fields which induce a given Poisson structure w on the base manifold M is constructed. In particular, the horizontal lifting of a Poiss…

Existence and rigidity results for lifts in Carnot groups.

A coordinate-free proof of the Maximum Principle is provided in the specific case of an optimal control problem with fixed time. Our treatment heavily relies on a special notion of variation of curves that consist of a concatenation of integral curves of time-dependent vector fields with unit time component, and on the…

Study lift metrics and connections on tangent bundles of Riemannian manifolds.

In markets with transaction costs, consistent price systems play the same role as martingale measures in frictionless markets. We prove that if a continuous price process has conditional full support, then it admits consistent price systems for arbitrarily small transaction costs. This result applies to a large class o…

The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.

We show that, for an affine submersion $π: \mathbf{M}\longrightarrow \mathbf{B}$ with horizontal distribution, $\mathbf{B}$ is a statistical manifold with the metric and connection induced from the statistical manifold $\mathbf{M}$. The concept of conformal submersion with horizontal distribution is introduced, which i…

In this paper, we define a complete lift for semisprays. If $S$ is a semispray on a manifold $M$, its complete lift is a new semispray $S^c$ on $TM$. The motivation for this lift is two-fold: First, geodesics for $S^c$ correspond to the Jacobi fields for $S$, and second, this complete lift generalizes and unifies previ…

Unified approach to constructing integrable systems using Stäckel lifts.

Geometric structures are lifted to higher tangent bundles preserving statistical properties.

The article classifies liftings of connections on differential manifolds for geodesic modeling.

Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.

In the present paper, we study complete and vertical lifts of tensor fields from a smooth manifold $M$ to its Weil bundle $T^A M$ defined by a Frobenius Weil algebra $A$. For a Poisson manifold $(M,w)$, we show that the complete lift $w^C$ and the vertical lift $w^V$ of the Poisson tensor $w$ are Poisson tensors on $T^…

We study the conditions under which the cotangent bundle $T^*M$ of a Riemaannian manifold $(M,g)$, endowed with a Kählerian structure $(G,J)$ of general natural lift type (see \cite{Druta1}), is Einstein. We first obtain a general natural Kähler-Einstein structure on the cotangent bundle $T^*M$. In this case, a certain…

In this paper we continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role that the geometry of the extended manifold plays. Firstly we consider basic examples. We give a projective line of diff($M$)-equivariant pencil liftings for first order operators,…

We describe how the dynamical system of rolling two $n$-dimensional connected, oriented Riemannian manifolds $M$ and $\hat M$ without twisting or slipping, can be lifted to a nonholonomic system of elements in the product of the oriented orthonormal frame bundles belonging to the manifolds. By considering the lifted pr…

Study covers of sphere with homeomorphisms lifting property.

The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral $ζ$-invariants using lifted …

The paper constructs Sasakian lifts from Kähler manifolds and studies their properties.

New quantum code lacks sparse lift.

Criteria for lifting manifold diffeomorphisms to vector bundle automorphisms.

Simple lifts of non-simple curves on surfaces.

This paper extends braid lifting to coloured braid groupoids for all simple disc covers.