arXiv research

The authors define a class of functions on Riemannian manifolds, which is called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming i…

As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic $\varphi$-convex function and deduce some basic properties of $\varphi$-convex function and geodesic $\varphi$-convex function. We also introduce the concept of geodesic $\varphi$-convex set and $\varphi$-epigraph and in…

The paper defines new types of geodesic functions on Riemannian manifolds and explores their properties.

Study on Mabuchi functional's convexity using ε-geodesics.

New characterization of geodesic currents via curve functionals.

New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applicat…

The paper connects geodesic nets to distance function critical points.

Constructs a function to count closed geodesics on Riemannian manifolds.

The paper studies geodesics and isoparametric functions on Finsler spheres.

We study singularities of geodesics flows in two-dimensional generalized Finsler spaces (pseudo-Finsler spaces). Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows to consider isotropic lines as (unparametrized…

New method to bound Laplacian eigenvalues of geodesic balls.

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…

Let $(M,g)$ be a simple Riemannian manifold. Under the assumption that the metric $g$ is real-analytic, it is shown that if the geodesic ray transform of a function $f\in L^{2}(M)$ vanishes on an appropriate open set of geodesics, then $f=0$ on the set of points lying on these geodesics. The approach is based on a micr…

Geodesics of contactomorphisms on a specific manifold are characterized by Hamiltonian functions.

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…

Geodesics found in a metric space of m-subharmonic functions.

Defines weak geodesics on specific subsets of manifolds.

New proof of energy functional monotonicity via geodesics in measure space.

The paper connects Schrödinger equations to geodesics on a 2-surface.

Strong geodesic convex function and strong monotone vector field of order $m$ on Riemannian manifolds have been established. A characterization of strong geodesic convex function of order $m$ for the continuously differentiable functions has been discussed. The relation between the solution of a new variational inequal…

Extends curve functions to geodesic currents with a simple criterion.

Geodesics in Kähler metrics connect metrics with constant scalar curvature.

Geodesically convex functions are continuous on Riemannian manifolds.

Geodesic concavity and hypersymplectic structures in $G2$-structures space.

We show that the existence of a function in $L^{1}$ with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions with constant geodesic X-ray transform always …

Given a hyperbolic surface $§$, a classic result of Birman and Series states that for each $K$, all complete geodesics with at most $K$ self-intersections can only pass through a certain nowhere dense, Hausdorff dimension 1 subset of $§$. We define a self-intersection function for each complete geodesic, which bounds t…

Paper finds critical metrics with pinched curvature are geodesic balls.

Convex optimization is a vibrant and successful area due to the existence of a variety of efficient algorithms that leverage the rich structure provided by convexity. Convexity of a smooth set or a function in a Euclidean space is defined by how it interacts with the standard differential structure in this space -- the…

Study geodesics on Grassmann manifold for functions vanishing on subsets of a set X.

In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length $l(γ)/k$. We employ energy methods to provide a relationship between the 1/k-geodesics and what we define as the balanced points of the uniform energy. We show that classes of balanced points of the uniform energy pe…

Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.

We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofa…

Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.

Given a compact orientable surface with finitely many punctures $Σ$, let $\Cal S(Σ)$ be the set of isotopy classes of essential unoriented simple closed curves in $Σ$. We determine a complete set of relations for a function from $\Cal S(Σ)$ to $\bold R$ to be the geodesic length function of a hyperbolic metric with geo…

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

New study shows acceleration in hyperbolic spaces is impossible for strongly geodesically convex functions.

The paper connects geodesic flows and higher-dimensional Reidemeister torsion for hyperbolic orbifolds.

Research extends geodesic length function study to three holed sphere.

Extends DCP framework to Hadamard manifolds for geodesically convex functions.

Geodesic descent optimizes likelihood in dually flat spaces.

Study of complex Hessian equations using subharmonic functions and geodesics.

Study geodesics on Grushin spaces, proving upper bounds on conjugate times.

The main goal of the paper is to prove the sandwich theorem for geodesic convex functions in a complete Riemannian manifold. Then by using this theorem we have proved an inequality in a manifold with bounded sectional curvature. Finally, we have shown that the gradient of a convex function is orthogonal to the tangent …

Study finds geodesic networks for surfaces with convex boundary.

Assigns compact set distance-like functions to non-compact geodesic spaces.

The paper characterizes strong Hamel functions using symmetries and proves their preservation properties.

Given a geodesic line $γ$ the hyperbolic space $\mathbb H^n$ we formulate a necessary and sufficient condition for a function along this geodesic which measure the mean curvature of totally umbilical leaves of a foliation orthogonal to $γ$. Then we extend the result to $γ$ being a hypercycle i.e. a geodesic on a hypers…

Proves existence of curves with constant curvature in a sphere.