arXiv research

The paper studies minimal surfaces in deformed hyperbolic spaces and their properties.

We show that any two non-conjugate points on a forward or backward complete connected Finsler manifold can be joined by infinitely many geodesics which are not covered by finitely many closed ones, provided that the Betti numbers of the based loop space grow unbounded.

We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points.

Estimates geodesics on surfaces without conjugate points.

The paper studies Jacobi fields and conjugate points in projective sprays.

Paper finds new criteria for conjugate points in fluid flows.

Identifies conjugate points in spherical harmonics solutions of quasi-geostrophic equations.

Study normal operators of double fibration transforms with conjugate points.

We study the geodesic X-ray transform $X$ on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of $X$ and relate it to the con…

Study analyzes Lévy process structure on manifolds with conjugate points.

The study shows conditions for thermostats to have no conjugate points.

The paper studies conjugate points on Lie groups with specific metrics.

Proves robust transitivity for geodesic flows from metrics with conjugate points.

Existence of a conjugate point proven on 3D ellipsoid.

The study identifies conjugate and cut points in ideal fluid motion configurations.

Study shows deterministic equivalent for neural network kernel convergence.

We revisit an example of a semi-Riemannian geodesic that was discussed by Musso, Pejsachowicz and Portaluri in 2007 to show that not every conjugate point is a bifurcation point. We point out a mistake in their argument, showing that on this geodesic actually every conjugate point is a bifurcation point. Finally, we pr…

Given two points of a Generalized Robertson-Walker spacetime, the existence, multiplicity and causal character of geodesic connecting them is characterized. Conjugate points of such geodesics are related to conjugate points of geodesics on the fiber, and Morse-type relations are obtained. Applications to bidimensional …

The paper counts geodesic loops on surfaces without conjugate points.

We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case we prove a strengthened Price inequality. We employ these inequalit…

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

We recall the notion of Jacobi fields, as it was extended to systems of second-order ordinary differential equations. Two points along a base integral curve are conjugate if there exists a non-trivial Jacobi field along that curve that vanishes on both points. Based on arguments that involve the eigendistributions of t…

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

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

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.

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

The conjugate locus of a point on a surface is the envelope of geodesics emanating radially from that point. In this paper we show that the conjugate loci of generic points on convex surfaces satisfy a simple relationship between the rotation index and the number of cusps. As a consequence we prove the `vierspitzensatz…

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 prove that any asymptotically Euclidean metric on $\mathbb{R}^n$ with no conjugate points must be isometric to the Euclidean metric.

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

The paper explores conjugate points in Lorentzian spaces, comparing different definitions and proving related theorems.

Existence of a conjugate point in the incompressible Euler flow on a sphere and an ellipsoid is considered. Misiolek (1996) formulated a differential-geometric criterion (we call M-criterion) for the existence of a conjugate point in a fluid flow. In this paper, it is shown that no zonal flow (stationary Euler flow) sa…

In this work, we present direction-of-arrival (DoA) estimation algorithms based on the Krylov subspace that effectively exploit prior knowledge of the signals that impinge on a sensor array. The proposed multi-step knowledge-aided iterative conjugate gradient (CG) (MS-KAI-CG) algorithms perform subtraction of the unwan…

Improved Gaussian process regression with tighter log marginal likelihood bounds.

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

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

Infinitely many new examples of compact Lorentzian surfaces without conjugate points are given. Further, we study the existence and the stability of this property among Lorentzian metrics with a Killing field. We obtain a new obstruction and prove that the Clifton- Pohl torus and some of our examples are as stable as p…

A conjugate Bayesian method detects change points in Hawkes processes efficiently.

We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems,…

The conjugate locus of a point $p$ in a surface $\mathcal{S}$ will have a certain number of cusps. As the point $p$ is moved in the surface the conjugate locus may spontaneously gain or lose cusps. In this paper we explain this `bifurcation' in terms of the vanishing of higher derivatives of the exponential map; we der…

Study shows almost all Arnold stable solutions have no conjugate points.

We introduce the notion of a conjugation-free geometric presentation for a fundamental group of a line arrangement's complement, and we show that the fundamental groups of the following family of arrangements have a conjugation-free geometric presentation: A real arrangement L, whose graph of multiple points is a union…

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…

We determine the complete conjugate locus along all geodesics parallel or perpendicular to the center (Theorem 2.3). When the center is 1-dimensional we obtain formulas in all cases (Theorem 2.5), and when a certain operator is also diagonalizable these formulas become completely explicit (Corollary 2.7). These yield s…

Helfer in [Pacific J. Math. 164/2 (1994), p. 321--350] was the first to produce an example of a spacelike Lorentzian geodesic with a continuum of conjugate points. In this paper we show the following result: given an interval $[a,b]$ of $IR$ and any closed subset $F$ of $IR$ contained in $]a,b]$, then there exists a Lo…

A well-known Lemma in Riemannian geometry by Klingenberg says that if $x_0$ is a minimum point of the distance function $d(p,\cdot)$ to $p$ in the cut locus $C_p$ of $p$, then either there is a minimal geodesic from $p$ to $x_0$ along which they are conjugate, or there is a geodesic loop at $p$ that smoothly goes throu…

We derive an interpolation version of constrained matrix Li-Yau-Hamilton estimate on Kähler manifolds. As a result, we first get a constrained matrix Li-Yau-Hamilton estimate for heat equation on a Kähler manifold with fixed Kähler metric. Secondly, we get a corresponding estimate for forward conjugate heat equation on…

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