Study geodesic flow on graph-related nilmanifolds, finding integrable and non-integrable cases.
problem Understanding geodesic flow on specific geometric structures.
method Construction of first integrals to show complete integrability.
result Examples of integrable and non-integrable geodesic flows.
Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
problem Geodesic completeness and flow properties of compact Brinkmann spacetimes.
method Proof of geodesic completeness and flow properties of isotropic parallel vector fields in compact Brinkmann spaces.
result Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
Anosov geodesic flow proven in non-compact manifolds with negative curvature.
problem Proving Anosov geodesic flow in non-compact manifolds with negative curvature.
method Proving the geodesic flow is Anosov by showing average sectional curvature is negative and uniformly away from zero.
result Constructed a non-compact manifold with Anosov geodesic flow.
Complete integrability proved for SR geodesic flow on S^7.
problem Complete integrability of subriemannian geodesic flow on S^7.
method Adapting a method by A. Thimm, constructing functionally independent first integrals.
result Complete integrability in the sense of Liouville proved for SR geodesic flow.
By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics.
Study on integrability of geodesic flows on Heisenberg group.
problem Integrability of geodesic flows on Heisenberg group.
method Investigation of two classes of normal geodesic flows associated with left-invariant sub-Riemannian metric.
result Left-left configuration is completely integrable in non-commutative sense, while left-right configuration exhibits non-commutative integrability in dimensions > 5.
Study geodesics on spheres with constant curvature, showing integrability and invariant properties.
problem Characterizing geodesics on spheres with constant flag curvature.
method Analyzing non-reversible Finsler metrics on S2 with constant flag curvature 1. result Geodesic flow is conjugate to Katok's examples and length of shortest closed geodesic is invariant.
The paper proves rigidity for Anosov geodesic flows on certain manifolds.
problem Rigidity of Anosov geodesic flows on specific manifolds.
method Analyzing sectional curvatures and applying rigidity results.
result Rigidity holds for Anosov geodesic flows on manifolds with bounded negative sectional curvatures.
The paper proves conditions for Anosov geodesic flows on non-compact manifolds.
problem Conditions for Anosov geodesic flows on non-compact manifolds.
method Analyzing sectional curvature and focal points to determine Anosov geodesic flows.
result A sufficient condition for Anosov geodesic flows on non-compact manifolds.
Condition for Anosov flows on non-compact manifolds without conjugate points.
problem Conditions for Anosov flows on non-compact manifolds.
method Formulated a condition for complete, connected and non-compact Riemannian manifolds.
result No conjugate points in geodesic flow if Anosov.
We study the geodesic flow on the global holomorphic sections of the bundle π:TS2→S2 induced by the neutral Kähler metric on the space of oriented lines of R3, which we identify with TS2. This flow is shown to be completely integrable when the sections are symplectic and the behaviour of the …
Geodesic completeness for Riemannian metrics on smooth probability densities is studied.
problem None of the studied Riemannian metrics are geodesically complete.
method Analysis of Hamilton--Jacobi-like partial differential equations, providing order conditions for global existence and uniqueness.
result Geodesic completeness is established for a class of higher-order Sobolev type metrics.
The paper studies geodesic completeness for Lie groups and their metrics.
problem Geodesic completeness of pseudo and holomorphic Riemannian metrics on Lie groups.
method Euler-Arnold formalism, detailed study of geodesics, classification of metrics.
result Full classification of geodesic completeness for the Lie group SL(2, C).
The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T∗Q∖Q. If the geodesic flow is toric integrable, the cosphere bundle admit…
Study integrability of conformal geodesics on gravitational instantons.
problem Integrability of conformal geodesic flow on gravitational instantons.
method Analyzing conformal geodesic flow equations on SO(3)-invariant gravitational instantons, separating Hamilton-Jacobi equations, finding commuting first integrals, and using conformal Killing-Yano tensors. result First example of an integrable conformal geodesic flow on a non-symmetric four-manifold.
We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. F…
Study geodesic curves on Heisenberg group, classify them, and compute first step of quadrature.
problem Classifying geodesic curves on the Heisenberg group.
method Completely integrable Hamiltonian system, classification of geodesic curves.
result Complete classification of geodesic curves on the Heisenberg group.
Study shows how brain completes missing parts of curves and constructs surfaces of negative curvature.
problem How the brain completes missing parts of curves and constructs surfaces of negative curvature.
method Solving variational problems to find sub-Riemannian geodesics and constructing surfaces of constant negative curvature.
result There is a one-to-one correspondence between sub-Riemannian geodesics used by the brain and rotational surfaces of constant negative curvature.
In this paper we study the geodesic flow on nilmanifolds equipped with a left-invariant metric. We write the underlying definitions and find general formulas for the Poisson involution. As an example we develop the Heisenberg Lie group equipped with its canonical metric. We prove that a family of first integrals giving…
Study X-ray transform on Anosov manifolds with improved stability estimates.
problem Analyzing the geodesic X-ray transform on Anosov manifolds.
method Refined Livsic theorem for Anosov flows, new quantitative finite time Livsic theorem.
result New stability estimates for the X-ray transform.
Study finds conjugate points in geodesics of Kolmogorov flows on torus.
problem Characterizing pairs of integers (m,n) for which geodesics have conjugate points.
method Analysis of geodesics in the group of volume-preserving diffeomorphisms of a torus using stream functions.
result Existence of conjugate points for all pairs of strictly positive integers (m,n).
Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic 3-manifolds…
Non-ergodic geodesic flow on Cantor tree surfaces found.
problem Determining when geodesic flow on Cantor tree surfaces is non-ergodic.
method Interpolating between two rates of convergence of cuff lengths to zero to prove non-ergodicity.
result Cantor tree surfaces with certain rates of cuff length convergence are non-parabolic.
Study geometric and analytical properties of ρ-Einstein solitons.
problem Characterize geometric and analytical features of ρ-Einstein solitons. method Analyze the spectrum of the drifted Laplacian operator and prove volume growth estimates.
result Establish new volume growth estimates for geodesic balls of complete noncompact ρ-Einstein solitons. The paper solves a flow problem on surfaces with boundary to converge to the hyperbolic metric.
problem Solving a normalized Ricci flow on surfaces with boundary to converge to the complete hyperbolic metric.
method Introduced a unique solution for the normalized Ricci flow with prescribed geodesic curvature on the boundary, using a Cauchy-Dirichlet problem.
result The flow converges locally uniformly to the complete hyperbolic metric.
We study the geodesic flow on the normal line congruence of a minimal surface in R3 induced by the neutral Kähler metric on the space of oriented lines. The metric is lorentz with isolated degenerate points and the flow is shown to be completely integrable. In addition, we give a new holomorphic description …
We introduce a combinatorial curvature flow for PL metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow…
Lower bound found for volumes of modular link complements.
problem Finding a lower bound for the volumes of modular link complements.
method Analyzing the volumes of link complements associated with geodesics in the modular surface.
result First linear lower volume bound in terms of exponents of code words.
Developed a random walk analog of geodesic flow on hyperbolic groups.
problem Geodesic flow on hyperbolic groups due to non-uniqueness of geodesics.
method Introduced a new framework using random walks and bi-infinite trajectories.
result Established ergodicity of the randomized geodesic flow and exponential mixing.
We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports s…
Study on Yamabe flow on manifolds with singularities, proving removability.
problem Yamabe flow on manifolds with submanifold singularities.
method Analyzing the Yamabe flow on Riemannian manifolds of dimension m≥3 minus a closed submanifold of dimension n. result Removability of singularities preserved along the Yamabe flow in certain cases.
A criterion in terms of differential invariants for a metric on a surface to be Liouville is established. Moreover, in this paper we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals does the geodesic flow of a given metric poss…
The group of real 4 by 4 upper triangular matrices with 1s on the diagonal has a left-invariant subRiemannian (or Carnot-Caratheodory) structure whose underlying distribution corresponds to the superdiagonal. We prove that the associated subRiemannian geodesic flow is not completely integrable. This provides the first …
Compactifies geodesic flows on hyperbolic surfaces, revealing attractive circles at infinity.
problem Geodesic flows on non-compact hyperbolic surfaces without cusps.
method Constructs a geometrical compactification using one-dimensional distributions tangent to stable and unstable horocycles.
result Existence of attractive circles at infinity in the compactified flow.
Study shows volumes of knot complements are bounded by linear functions of geodesic periods.
problem Volume calculation of knot complements associated with geodesics on modular surfaces.
method Analyzes geodesics on modular surfaces, their associated knots, and their complements' volumes.
result Volumes of knot complements are bounded linearly by the period of geodesic continued fractions.
In this paper, we systemally study the long time behavior of the curve shortening flow in a closed or non-compact complete locally Riemannian symmetric manifold. Assume that we have a global flow. Then we can exhibit a a limit for the global behavior of the flow. In particular, we show the following results. 1). Let $\…
We prove complete integrability of the Manakov-type SO(n)-invariant geodesic flows on homogeneous spaces SO(n)/SO(k1)×...×SO(kr), for any choice of k1,...,kr, k1+...+kr≤n. In particular, a new proof of the integrability of a Manakov symmetric rigid body motion around a fixed point is presented…
If X is a proper CAT(-1)-space and Γ a non-elementary discrete group of isometries acting properly discontinuously on X, it is shown that the geodesic flow on the quotient space Y=X/Γ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary ∂X and the non-wanderin…
The study counts geodesics in a specific type of space, finding a limit related to its entropy.
problem Counting geodesics in compact locally CAT(0) spaces with a rank one axis.
method Analyzes the geodesic flow on the space of geodesics and uses entropy.
result The limit of the number of geodesics of length ≤ t approaches a specific formula involving entropy.
The aim of this note is to present simple proofs of the completeness of Manakov's integrals for a motion of a rigid body fixed at a point in Rn, as well as for geodesic flows on a class of homogeneous spaces SO(n)/SO(n1)×⋯×SO(nr).
Proves short-time existence of Ricci flow on specific manifolds.
problem Existence and curvature estimates of Ricci flow on noncompact manifolds.
method Combining He's method with Simon and Topping's approach.
result Ricci flow on specified manifolds has nonnegative bisectional curvature.
Integrable geodesics found on special orthogonal group.
problem Analyzing normal geodesics on the special orthogonal group.
method Adapted Lax pair and bi-Hamiltonian structure.
result Almost all normal geodesics are completely integrable.
It has been proved that on 2-dimensional orientable compact manifolds of genus g>1 there is no integrable geodesic flow with an integral polynomial in momenta. There is a conjecture that all integrable geodesic flows on T2 possess an integral quadratic in momenta. All geodesic flows on S2 and T2 possessing i…
The paper studies algebraic relations of first integrals on specific Lie groups.
problem Algebraic relations of first integrals on step-two and step-three nilpotent Lie groups.
method Analysis of isometry algebra and invariant first integrals.
result Complete families of first integrals can be constructed with Killing vector fields and symmetric Killing 2-tensor fields in low dimensions.
Proves lower bounds on Hausdorff dimension of projections of invariant sets.
problem Lower bounds on Hausdorff dimension of projections of invariant sets.
method Transversal property of geodesics, (k+1)-linear curved Kakeya estimate, Bourgain-Guth argument. result Proves a lower bound on the Hausdorff dimension of projections of invariant sets.
New stability estimate for Anosov manifolds' metrics.
problem Locally determine the metric from the marked length spectrum.
method Anosov geodesic flow and non-positive curvature.
result Two close enough metrics with the same marked length spectrum are isometric.
Develops a lifting theory for exponential maps in semi-Riemannian geometry.
problem Overcoming singularities in exponential maps to prove geodesic connectivity.
method Lifting theory for semi-Riemannian manifolds with path-continuation property.
result General path-lifting theorem extending globally under certain conditions.
Geodesic completeness proven for certain symmetric spaces.
problem Geodesic completeness of compact locally symmetric pseudo-Riemannian manifolds of signature (2,2).
method Analysis of unique parallel lightlike vector fields and group actions.
result Geodesic completeness established for compact locally symmetric spaces modeled on specific spaces.