Sergey I. Agafonov
We present a geometric interpretation of integrability of geodesic flow by quadratic integrals in terms of the web theory and construct integrable billiards on surfaces admitting such integrals.
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Any smooth surface in R^3 may be flattened along the z-axis, and the flattened surface becomes close to a billiard table in R^2 . We show that, under some hypotheses, the geodesic flow of this surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersive and has finite horizon, then the geodesic flow of the corresponding surface is Anosov. We apply this result to the theory of mechanical linkages and their dynamics: we provide a new example of a simple linkage whose physical behavior is Anosov. For the first time, the edge lengths of the mechanism are given …
In this paper we show that in some cases the E.Hopf rigidity phenomenon admits quantitative interpretation. More precisely we estimate from above the measure of the set $\mathcal{M}$ swept by minimal orbits. These estimates are sharp, i.e. if $\mathcal{M}$ occupies the whole phase space we recover the E.Hopf rigidity. We give these estimates in two cases: the first is the case of convex billiards in the plane, sphere or hyperbolic plane. The second is the case of conformally flat Riemannian metrics on a torus. It seems to be a challenging question to understand such a quantitative bounds for B…
We consider a billiard in the sphere S^2 with circular obstacles, and give a sufficient condition for its flow to be uniformly hyperbolic. We show that the billiard flow in this case is approximated by an Anosov geodesic flow on a surface in the ambiant space S^3. As an application, we show that every orientable surface of genus at least 11 admits an isometric embedding into S^3 (equipped with the standard metric) such that its geodesic flow is Anosov. Finally, we explain why this construction cannot provide examples of isometric embeddings of surfaces in the Euclidean R^3 with Anosov geodesic…
We interpret magnetic billiards as Finsler ones and describe an analog of the string construction for magnetic billiards. Finsler billiards for which the law "angle of incidence equals angle of reflection" are described. We characterize the Finsler metrics in the plane whose geodesics are circles of a fixed radius. This is a magnetic analog of Hilbert's fourth problem asking to describe the Finsler metrics whose geodesics are straight lines.
A Correspondence between Billiards and Geodesics
本文通过构造折叠型曲面族，证明了在黎曼台球桌（温和假设下）中，测地线段序列可收敛至台球轨迹，从而建立了台球轨迹与测地线之间的双向逼近关系。
In this article we construct L--A representations of geodesic flows on quadrics and of billiard problems within ellipsoids in the pseudo--Euclidean spaces. A geometric interpretation of the integrability analogous to the classical Chasles theorem for symmetric ellipsoids is given. We also consider a generalization of the billiard within arbitrary quadric allowing virtual billiard reflections.
Many classical facts in Riemannian geometry have their pseudo-Riemannian analogs. For instance, the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. We discuss the geometry of these structures in detail, as well as introduce and study pseudo-Euclidean billiards. In particular, we prove pseudo-Euclidean analogs of the Jacobi-Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on …
Uniform hyperbolicity is a strong chaotic property which holds, in particular, for Sinai billiards. In this paper, we consider the case of a nonflat billiard, that is, a Riemannian manifold with boundary. Each trajectory follows the geodesic flow in the interior of the billiard, and bounces when it meets the boundary. We give a sufficient condition for a nonflat billiard to be uniformly hyperbolic. As a particular case, we obtain a new criterion to show that a closed surface has an Anosov geodesic flow.
Let $(M, g)$ be a Riemannian manifold with boundary, where $g$ is a non-trapping metric. Let $SM$ be the space of the spherical tangent to $M$ bundle, and $v^g$ the geodesic vector field on $SM$. We study the scattering maps $C_{v^g}: \partial^+_1SM \to \partial^-_1SM$, generated by the $v^g$-flow, and the dynamics of the billiard maps $B_{v^g, τ}: \partial^+_1SM \to \partial^+_1SM$, where $τ$ denotes an involution, mimicking the elastic reflection from the the boundary $\partial M$. We getting a variety of holography theorems that tackle the inverse scattering problems for $C_{v^g}$ and theor…
Outer billiards in the spaces of oriented geodesics of the three dimensional space forms
本文在三维常曲率空间形式的有向测地线空间中定义外台球映射，证明其在二次凸曲面下为微分同胚，并在曲率±1情形下证明其为辛同胚，同时引入周期点和乐概念以开展双曲情形的动力学研究。
Fractional-linear integrals of geodesic flows on surfaces and Nakai's geodesic 4-webs
证明了曲面上测地流的动量分式线性积分空间维数为3或5（后者对应常高斯曲率），并给出其存在的几何判据为曲面承载具有常交比的测地4-网。
Wire billiard is defined by a smooth embedded closed curve of non-vanishing curvature $k$ in $\mathbb{R}^n$ (a wire). For a class of curves, that we call nice wires, the wire billiard map is area preserving twist map of the cylinder. In this paper we are investigating whether the basic features of conventional planar billiards extend to this more general situation. In particular, we extend Lazutkin's KAM result, as well as Mather's converse KAM result, to wire billiards. We address the notion of caustics: for wire billiards, it corresponds to striction curve of the ruled surface spanned by the…
We present a complete description of a class of linearizable planar geodesic webs which contain a parallelizable 3-subweb.
We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x_{1},...,x_{n}) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d > n) of hypersurfaces to be hyperplanar webs. These conditions are systems of generalized Euler equations, and for flat connections we give an explicit construction of their solutions.
Integrable geodesic flow in 3D and webs of maximal rank
本文以测地4-网刻画三维空间中具有两个交换二次积分的可积测地流，并证明在自然几何条件下度量为 Stäckel 型。
In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic we…
We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar results are obtained for planar d-webs, d > 4, provided that additional d-4 second-order invariants vanish.
We prove that a surface carries a hexagonal 3-web of geodesics if and only if the geodesic flow on the surface admits a cubic first integral and show that the system of partial differential equations, governing metrics on such surfaces, is integrable by generalized hodograph transform method. We present some new local examples of such metrics, discuss known ones, and establish an analogue of the celebrated Graf and Sauer Theorem for Darboux superintegrable metrics.
By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they form a Liouville net. Let us say that a geodesically convex curve on a Riemannian surface has the Poritsky property if it can be parametrized in such a way that all of its string diffeomorphisms are shifts with respect to this parameter. In 1950, Poritsky has shown that the only closed pl…
We characterize fundamental domains of affine reflection groups as those polyhedral convex bodies which support a continuous billiard dynamics. We interpret this characterization in the broader context of Alexandrov geometry and prove an analogous characterization for isosceles tetrahedra in terms of continuous quasigeodesic flows. Moreover, we show an optimal regularity result for convex bodies: the billiard dynamics is continuous if the boundary is of class $\mathcal{C}^{2,1}$. In particular, billiard trajectories converge to geodesics on the boundary in this case. Our proof of the latter co…
For a four-dimensional (nonisoclinicly geodesic) three-web W (3, 2, 2), a transversal distribution $Δ$ is defined by the torsion tensor of the web. In general, this distribution is not integrable. The authors find necessary and sufficient conditions of its integrability and prove the existence theorem for webs W (3, 2, 2) with integrable distributions $Δ$. They prove that for a web W (3, 2, 2) with the integrable distribution $Δ$, the integral surfaces $V^2$ of $Δ$ are totally geodesic in an affine connection of a certain bundle of affine connections. They also consider a class of webs W (3, 2…
We present old and recent results on rank problems and linearizability of geodesic planar webs.
Integrable Geodesic Flows on Cones over Riemannian Manifolds
本文研究了锥面上的测地流，证明其超积分性及可积性，受Birkhoff弹道研究启发。
We present a projectively invariant description of planar linear 3-webs. For a non-hexagonal 3-web, we introduce family of projective torsion-free Cartan connections, the web leaves being geodesics for each member of the family, and give a web linearization criterion. Finally, we propose an algorithm for resolving the Gronwall conjecture and illustrate this approach by proving the conjecture for 3-webs whose 2 foliations are 2 pencils of lines.
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