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In this paper we prove a general stability result for higher order geometric flows on the circle, which basically states that if the initial condition is close to a round circle, the curve evolves smoothly and exponentially fast towards a…

In this paper we formulate some conjectures about algebraic flows on Shimura varieties. In the first part of the paper we prove the `logarithmic Ax-Lindemann theorem'. We then prove a result concerning the topological closure of the images…

We obtain necessary and sufficient conditions for the integrability in quadratures of geodesic flows on homogeneous spaces $M$ with invariant and central metrics. The proposed integration algorithm consists in using a special canonical…

We establish a framework, namely, nuclear bounded Fr\'{e}chet manifolds endowed with Riemann-Finsler structures to study geodesic curves on certain infinite dimensional manifolds such as the manifold of Riemannian metrics on a closed…

We consider geodesic flows between hypersurfaces in $\R^n$. However, rather than consider using geodesics in $\R^n$, which are straight lines, we consider an induced flow using geodesics between the tangent spaces of the hypersurfaces…

We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $\Sigma$ that we construct are either Birkhoff sections, meaning that they…

We consider some generalizations of the classical nonholonomic integrator and give a geometric approach to characterize controllability for these systems. We use Stokes' theorem and results from complex analysis to obtain necessary and…

We present a new and direct proof of Grothendieck's generic freeness lemma in its general form. Unlike the previously published proofs, it does not proceed in a series of reduction steps and is fully constructive, not using the axiom of…

We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration,…

Physicists face major challenges in modelling multi-scale phenomena that are observed in geophysical flows (e.g. in the Earth's oceans and atmosphere, or liquid planetary cores). In particular, complexities arise because geophysical fluids…

We construct Riemannian manifolds with completely integrable geodesic flows, in particular various nonhomogeneous examples. The methods employed are a modification of Thimm's method, Riemannian submersions and connected sums.

We present control-oriented models for transient dynamics of isothermal one-dimensional gas flow through multiple pipes in series and intersecting pipe geometries. These composite models subsume algebraic constraints that would otherwise…

In this article we prove that the Hausdorff dimension of geodesic directions that are recurrent and diverge on average coincides with the entropy at infinity of the geodesic flow for any complete, pinched negatively curved Riemannian…

We prove a Schwarz type lemma for harmonic mappings between the unit and a geodesic line in a Riemenn surface.

An explanation of stability of fireballs is proposed based on quantum effects in a thin surface layer of negatively charged plasma surrounding a positive kernel of a fireball. We construct a quantization of the geodesic flow on the sphere…

We extend results of parametric geometry of numbers to a general diagonal flow on the space of lattices. Moreover, we compute the Hausdorff dimension of the set of trajectories with every given behavior, with respect to a nonstandard metric…

We study the geodesic problem on the group of diffeomorphism of a domain M$\subset$Rd, equipped with the H(div) metric. The geodesic equations coincide with the Camassa-Holm equation when d=1, and represent one of its possible…

Let (T^2, g) be a two-dimensional Riemannian torus. In this paper we prove that the topological entropy of the geodesic flow restricted to the set of initial conditions of minimal geodesics vanishes, independent of the choice of the…

In this paper, we investigate the embeddings for topological flows. We prove an embedding theorem for discrete topological system. Our results apply to suspension flows via constant function, and for this case we show an embedding theorem…

Using quadratic forms, we stablish a criteria to relate the curvature of a Riemannian manifold and partial hyperbolicity of its geodesic flow. We show some examples which satisfy the criteria and another which does not satisfy it but still…