arXiv research

Geodesic connectedness proved for statistical manifolds with divisible cubic forms.

Geodesics connect model modes in neural network loss landscapes.

Study geodesics of meromorphic connections on Riemann surfaces.

Proves geodesic connections on 2-torus without invariant tori.

Circle's metric is at least π/4 away from any simply connected geodesic space.

In this paper a functional definition of geodesics is introduced which allows to generalize the notion of a geodesic from smooth to topological manifolds. It is shown that in the smooth case the new definition coincides with the classical definition of geodesics of a linear connection. If the smoothness is not required…

New connections share geodesics with superintegrable systems.

Study on geodesics in Kropina metrics with applications.

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

We show that a conformal connection on a closed oriented surface $Σ$ of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it follows that the unparametrised geodesics of a Riemannian metric on $Σ$ determine th…

We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field assoc…

The Levi-Civita connection and geodesic equations for a stationary spacetime are studied in depth. General formulae which generalize those for warped products are obtained. These results are applicated to some regions of Kerr spacetime previously studied by using variational methods. We show that they are neither space…

Found a 6D Lie group with a closed geodesic.

We continue our study of the space of geodesics of a manifold with linear connection. We obtain sufficient conditions for a product to have a space of geodesics which is a manifold. We investigate the relationship of the space of geodesics of a covering manifold to that of the base space. We obtain sufficient condition…

The study constructs families of nilpotent Lie groups with geodesic orbit metrics.

Study confirms geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.

In the recent paper \cite{LoD1}, we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic can not be the only closed geodesic…

If we consider the moduli space of flat connections of a non trivial principal SO(3)-bundle over a surface, then we can define a map from the set of perturbed closed geodesics, below a given energy level, into families of perturbed Yang-Mills connections depending on a small parameter. In this paper we show that this m…

New data-driven Cartan connection tracks complex vascular structures.

In this paper, we study the relation between geodesic and harmonic mappings. Harmonic mappings are defined between Riemannian manifolds as critical points of the energy functional, on the other hand, geodesic mappings are defined in a more general setting (manifolds with affine connections). Using the well-established …

Geodesics in Kähler metrics connect metrics with constant scalar curvature.

Given a globally hyperbolic spacetime endowed with a complete lightlike Killing vector field and a complete Cauchy hypersurface, we characterize the points which can be connected by geodesics. A straightforward consequence is the geodesic connectedness of globally hyperbolic generalized plane waves with a complete Cauc…

A manifold with an arbitrary affine connection is considered and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dynamics associated with the given invariant affine connection. …

We classify pseudo-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a pseudo-Riemannian manifold. Also, we obtain the classification of the pseudo-Riemannian submersions with (para-)complex connected totally geodesic fibres from a (para-)complex pseudo-hyperbolic sp…

The present note deals with the dynamics of metric connections with vectorial torsion, as already described by E. Cartan in 1925. We show that the geodesics of metric connections with vectorial torsion defined by gradient vector fields coincide with the Levi-Civita geodesics of a conformally equivalent metric. By pullb…

Projective connections arise from equivalence classes of affine connections under the reparametrization of geodesics. They may also be viewed as quotient systems of the classical geodesic equation. After studying the link between integrals of the (classical) geodesic flow and its associated projective connection, we tu…

In this paper we show that two Lagrangian graphs over the torus in $\mathbb{C}^n$ with large Lagrangian phase can be connected via Lipschitz continuous geodesic with respect to the $L^2$ metric on the space of Lagrangian submanifolds. In particular, the geodesic for Lagrangian graphs over the torus in $\mathbb{C}^n$ ca…

Simply connected 4-manifolds with specific Weyl tensor are geodesic balls in space forms.

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two $C^\infty$--differentiable connected, complete Riemannian manifolds, $k:M_1\to\mathbb R$ a $C^\infty$--differentiable function, having $0<k_0<k(x)\leq K_0$, for any $x\in M_1$ and $g:=g_1-kg_2$ the semi Riemannian metric on the product manifold $M:=M_1\times M_2$. We associate to …

Paper proves geodesics and focal points unchanged by conformal changes in pseudo-Finsler manifolds.

We consider equitorsion second type almost geodesic mappings of a non-symmetric affine connection space in this article. Using different computational methods, we obtained some invariants of these mappings. Last generalized Thomas projective parameter and Weyl projective tensor as invariants of a second type almost geo…

Study precise rates of horizontal gap shrinkage on generic translation surfaces.

We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x_{1},...,x_{n}) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d > n) of hypersurfa…

Study of time-dependent metrics and connections in geometry.

We create a smooth manifold of triangular meshes with a geodesically complete metric.

Theorem shows generic metrics yield non-degenerate geodesic nets.

Convexity properties of Weil-Petersson geodesics on the Teichmüller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considera…

In this paper we show that on a complete Riemannian manifold of negative curvature and dimension $n>1$ every two points which realize a local maximum for the distance function are connected by at least $2n+1$ geometrically distinct geodesic segments (i.e. length minimizing). Using a similar method, we obtain that in th…

New results on geodesic flows using curve shortening flow.

Connections between Lie derivatives and the deviation equation has been investigated in spaces with affine connection. The deviation equations of the geodesics as well as deviation equations of non-geodesics trajectories have been obtained on this base. This is done via imposing certain conditions on the Lie derivative…

This paper explores the relation between convex functions and the geometry of space-times and semi-Riemannian manifolds (an investigation initiated by Gibbons-Ishibashi). Specifically, we study geodesic connectedness. We give geometric-topological proofs of geodesic connectedness for classes of space-times to which kno…

Introduces new Finsler metrics and connects them to information geometry.

Horizon saddle connections imply dense hyperbolic geodesics on dilation surfaces.

A pair of points in a riemannian manifold makes a secure configuration if the totality of geodesics connecting them can be blocked by a finite set. The manifold is secure if every configuration is secure. We investigate the security of compact, locally symmetric spaces.

In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions…

Study random walks on sub-Riemannian manifolds using retractions.

We investigate certain natural connections between subriemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a subriemannian geodesic shared between its projections onto the integrable …

Study spin chains and sigma models on flag manifolds, calculating spectra and geodesics.