arXiv research

Estimates geodesics on surfaces without conjugate points.

New manifolds found without interior conjugate points.

Study Price inequalities and Betti number growth on specific manifolds.

In this short note, we prove that the usual $Θ$ function on a Riemannian manifold without conjugate points is uniformly bounded from below. This extends a result of Green in two dimensions. This elementary lemma implies that the Bérard remainder in the Weyl law is valid for a manifold without conjugate points, without …

We show that in the fundamental groups of closed manifolds without conjugate points centralizers of all elements virtually split.

The study examines continuous mean curvature functions on manifolds without conjugate points.

Condition for Anosov flows on non-compact manifolds without conjugate points.

Study geodesic flows on hyperbolic manifolds without conjugate points, proving unique measure of maximal entropy.

Locally extremal geodesic loops are closed geodesics in Riemannian manifolds.

The paper connects geodesic flows and limit sets on visibility manifolds.

Study equilibrium measures on manifolds without conjugate points with visibility covering.

Study ergodic properties of geodesic flows on specific manifolds without conjugate points.

Euclidean metrics without conjugate points are proven to be flat.

Geodesic flows on specific manifolds are structurally stable.

New examples of stable Lorentzian surfaces without conjugate points are found.

Study non-convex manifolds' rigidity properties without convexity assumption.

Geodesic flows on compact manifolds without conjugate points are shown to have a unique measure of maximal entropy.

The paper solves the Dirichlet problem at infinity and defines Poisson boundaries for certain manifolds.

We prove flatness of complete Riemannian planes and cylinders without conjugate points under optimal conditions on the area growth.

We show that given a point on a Finsler surface, one can always find a neighborhood of the point and isometrically embed this neighborhood into a Finsler torus without conjugate points.

Study light ray transform on Lorentzian manifolds without conjugate points.

Geodesic flows on certain surfaces are shown to be semi-conjugate to expansive flows.

Let $\mathcal{W}^{n}$ be the class of $C^{\infty }$ complete simply connected $n-$dimensional manifolds without conjugate points. The hyperbolic space as well as Euclidean space are good examples of such manifolds. Let $$ and let $A$ be a subset of $W$. This article aims at characterization and bu…

In this paper, we are interested in the location of conjugate points along a geodesic in the volumorphism group of a compact three-dimensional manifold without boundary (the configuration space of an ideal fluid). As shown in the author's previous work, these are typically pathological, i.e., they can occur in clusters…

The paper counts geodesic loops on surfaces without conjugate points.

Let $(M,g)$ be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that $M$ is a flat manifold, provided $M$ is asymptotically harmonic of constant $h = 0$.

Global rigidity theorem for metrics on Σ×S1 without conjugate points.

Unique entropy measure found for geodesic flows on certain surfaces.

Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature, provided M is asymptotically harmonic of constant h > 0.

We study the geodesic X-ray transform $I_Γ$ of tensor fields on a compact Riemannian manifold $M$ with non-necessarily convex boundary and with possible conjugate points. We assume that $I_Γ$ is known for geodesics belonging to an open set $Γ$ with endpoint on the boundary. We prove generic s-injectivity and a stabilit…

Abstract manifolds split with no conjugate points using special functions.

Study analyzes Lévy process structure on manifolds with conjugate points.

Classifies 3D manifold diffeos without heteroclinic curves.

The paper proves rigidity for Anosov geodesic flows on certain manifolds.

Paper proves existence of conjugate points on ellipsoids but not on spheres.

We consider an equivariant analogue of a conjecture of Borcherds. Let $Y$ be a real $K3$ surface without real points. Let $g$ be a Ricci-flat Kaehler metric on $Y$ invariant under the complex conjugation. We shall prove that the equivariant determinant of the Laplacian of $(Y,g)$ with respect to the complex conjugation…

Study on conjugate points in a $C^*$-algebra's Grassmann manifold.

Existence of a conjugate point proven on 3D ellipsoid.

We prove that the topology, smooth structure, and metric of a compact Lorentzian manifold with boundary is uniquely determined by data at the boundary. The data consists of the lengths and directions of future-directed once-broken geodesics connecting points on the boundary, which are first timelike and then lightlike.…

Study finds conjugate points in geodesics of Kolmogorov flows on torus.

From a spray space $S$ on a manifold $M$ we construct a new geometric space $P$ of larger dimension with the following properties: 1. Geodesics in $P$ are in one-to-one correspondence with parallel Jacobi fields of $M$. 2. $P$ is complete if and only if $S$ is complete. 3. If two geodesics in $P$ meet at one point, the…

We show a Hopf type rigidity for thermostats without conjugate points on a 2-torus

Extends E. Hopf's theorem to magnetic systems without conjugate points.

What are appropriate geometric conditions ensuring that a complete Riemannian 2-cylinder without conjugate points is flat? Examples with nonpositive curvature show that one has to assume that the ends of the cylinder open sublinearly. We show that sublinear growth of the ends is indeed sufficient if it is measured by t…

Study geodesic ray transform on 2D manifolds with conjugate points.

Support theorem proved for X-ray transform on certain non-compact manifolds.

Study on sub-Riemannian manifolds, focusing on conjugate points and caustic stability.

A complete Riemannian manifold without conjugate points is called asymptotically harmonic if the mean curvature of its horospheres is a universal constant. Examples of asymptotically harmonic manifolds include flat spaces and rank one locally symmetric spaces of noncompact type. In this paper we show that this list exh…