arXiv research

New cutoff phenomenon found for geodesic paths on hyperbolic manifolds.

Let $M$ be a Riemannian manifold and ${\mathcal P}M$ be the space of all smooth paths on $M$. We describe geodesics on path space ${\mathcal P}M$. Normal neighbourhood structure on ${\mathcal P}M$ has been discussed. We identify paths on $M$ under "back-track" equivalence. Under this identification we show that if $M$ …

3D hyperbolic spaces have endless simple paths.

We study the behaviour of quasi-geodesics in Out(F_n). Given an element f in Out(F_n) there are several natural paths connecting the origin to f in Out(F_n); for example, paths associated to sequences of Stallings folds and paths induced by the shadow of greedy folding paths in Outer Space. We show that none of these p…

Geodesics connect model modes in neural network loss landscapes.

In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on Bär and Pfäffle's use of piecewise geodesic paths, the kernel of the time evolution operator is the heat kernel for the Laplacian on forms. The path integral is approximated by the integral of a form on …

Bicycle paths form geodesics in 3D subspaces, related to Kirchhoff rods.

Proposes Geodesic Integrated Gradients (GIG) for more accurate feature attributions in deep networks.

Approximate symmetries of geodesic equations on 2-spheres are studied. These are the symmetries of the perturbed geodesic equations which represent approximate path of a particle rather than exact path. After giving the exact symmetries of the geodesic equations, two different approaches to study the approximate symmet…

Given two points on a soup can or conical cup with lid, we find and classify all paths of minimal length connecting them. When the number of minimal paths is finite, there are at most four on a can and three on a cup. At worst, minimal paths are piece-wise smooth with three components, each of which is a classical geod…

Study geodesic paths on flat surfaces, comparing length and singularity counts.

EntroPath learns manifold geometry from diffusion paths.

We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into th…

Sub-Riemannian geometry connects bike paths to mathematical curves.

Consider the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold $(M,\om)$ with the Hofer $L^{\infty}$-norm. A path in $\Ham^c(M)$ will be called a geodesic if all sufficiently short pieces of it are local minima for the Hofer length functional $\Ll$. In this paper, we giv…

Characterizes paths minimizing anisotropic lengths in Euclidean space.

The paper studies metrics that match prescribed geodesics and introduces a variational problem.

Study connects contact structures to cone geodesics and contactomorphisms.

Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimi…

We construct all Finsler metrics on the two-sphere for which geodesics are circles and show that any (reversible) path geometry on a two-dimensional manifold is locally the system of geodesics of a Finsler metric.

Homogeneous magnetic paths found in Heisenberg space.

Novel geodesic results on affine and Lorentzian manifolds.

Average signature measures geodesics in Lie groups.

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

Study on geodesics in spacetime, proving properties of multiple maximizing paths.

We develop a Malliavin calculus on the horizontal path space of a totally geodesic Riemannian foliation. As a first application, under suitable assumptions, we prove a log-Sobolev inequality for a natural one-parameter family of infinite-dimensional Ornstein-Uhlenbeck type operators. As a second application, we obtain …

Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite …

The displacement and deviation vectors in spaces (manifolds), the tangent bundle of which is endowed with a transport along paths, are introduced. In case these spaces are equipped with a linear connection, the deviation equations (between arbitrary, geodesic or not, paths) in such spaces are investigated.

Paper finds shortest geodesic paths on hyperbolic surfaces.

The study explores ends in coarse homotopy of proper geodesic spaces.

The paper sparsifies networks by finding efficient paths in their functional space.

Study efficient geodesics in curve complex using dot graphs.

Study path spaces and their homology, extending loop products and coproducts.

We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space. We apply this to characterize hyperbolically embedded subgroups in terms of nice actions on (relatively) hyperbolic spaces. We also study the d…

In this survey article, we review the relation between heat kernels and path integrals. In particular, we review recent results on the approximation of the Wiener measure on compact manifold by measures on (finite-dimensional) spaces of piece-wise geodesics.

Geodesic envelopes stay uniformly bounded in specific Teichmüller spaces.

In this paper we study the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop $α$, the length of $α$ along a balanced folding path is not larger than the maximum of its lengths at th…

The main theorem of this paper classifies the quasi-geodesics in a Coxeter group that are tracked by geodesics. As corollaries, we show that if a Coxeter group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are tracked by Cayley graph geodesics, all special subgroups of the Coxeter group are quasi-…

The linear transports along paths in vector bundles introduced in Ref. [1] are applied to the special case of tensor bundles over a given differentiable manifold. Links with the transports along paths generated by derivations of tensor algebras are investigated. A possible generalization of the theory of geodesics is p…

The paper computes special values of combinatorial zeta functions to reveal topological properties of manifolds.

Geodesics on Kähler manifold potentials are paths of least action.

We extend the notion of unicorn paths between two arcs introduced by Hensel, Przytycki and Webb to the case where we replace one arc with a geodesic asymptotic to a lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the ar…

In 3D, conformal geodesics are variational.

Proves CLT for Brownian paths on pinched negative curvature manifolds.

Paper develops a new algorithm to find shortest paths on surfaces.

We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be interpreted as mos…

This paper describes a novel framework for computing geodesic paths in shape spaces of spherical surfaces under an elastic Riemannian metric. The novelty lies in defining this Riemannian metric directly on the quotient (shape) space, rather than inheriting it from pre-shape space, and using it to formulate a path energ…

Study geodesics on graphs with random lengths, proving bi-infinite paths exist.