arXiv subjects

Marcos Salvai

Publications and source records attributed to Marcos Salvai.

17 recordsLinked to original sources

arXiv2025

Outer billiards in the complex hyperbolic plane

Given a quadratically convex compact connected oriented hypersurface $N$ of the complex hyperbolic plane, we prove that the characteristic rays of the symplectic form restricted to $N$ determine a double geodesic foliation of the exterior $U$ of $N$. This induces an outer billiard map $B$ on $U$. We prove that $B$ is a diffeomorphism (notice that weaker notions of strict convexity may allow the billiard map to be well-defined and invertible, but not smooth) and moreover, a symplectomorphism. These results generalize known geometric properties of the outer billiard maps in the hyperbolic plane and complex Euclidean space.

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arXiv2025

Outer billiards in the spaces of oriented geodesics of the three dimensional space forms

Let $M_{κ}$ be the three-dimensional space form of constant curvature $κ=0,1,-1$, that is, Euclidean space $\mathbb{R}^{3}$, the sphere $S^{3} $, or hyperbolic space $H^{3}$. Let $S$ be a smooth, closed, strictly convex surface in $M_{κ}$. We define an outer billiard map $B$ on the four dimensional space $\mathcal{G}_{κ}$ of oriented complete geodesics of $M_{κ}$, for which the billiard table is the subset of $\mathcal{G}_{κ}$ consisting of all oriented geodesics not intersecting $S$. We show that $B$ is a diffeomorphism when $S$ is quadratically convex. For $κ=1,-1$, $\mathcal{G}_{κ}$ has a Kähler structure associated with the Killing form of $\operatorname{Iso}(M_{κ})$. We prove that $B$ is a symplectomorphism with respect to its fundamental form and that $B$ can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in $\mathbb{R}^{2n}$ defined in terms of the standard symplectic structure. We show that $B$ does not preserve the fundamental symplectic form on $\mathcal{G}_{κ}$ associated with the cross product on $M_{κ}$, for $κ=0,1,-1$. We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.

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arXiv2023

The sub-Riemannian length spectrum for screw motions of constant pitch on flat and hyperbolic 3-manifolds

Let M be an oriented three-dimensional Riemannian manifold of constant sectional curvature k = 0,1,-1 and let SO(M) be its direct orthonormal frame bundle (direct refers to positive orientation), which may be thought of as the set of all positions of a small body in M. Given lambda in R, there is a three-dimensional distribution D^lambda on SO(M) accounting for infinitesimal rototranslations of constant pitch lambda. When lambda is different from k^2, there is a canonical sub-Riemannian structure on D^lambda. We present a geometric characterization of its geodesics, using a previous Lie theoretical description. For k = 0,-1, we compute the sub-Riemannian length spectrum of (SO(M),D^lambda) in terms of the complex length spectrum of M (given by the lengths and the holonomies of the periodic geodesics) when M has positive injectivity radius. In particular, for two complex length isospectral closed hyperbolic 3-manifolds (even if they are not isometric), the associated sub-Riemannian metrics on their direct orthonormal bundles are length isospectral.

math.DG↗

arXiv2023

A split special Lagrangian calibration associated with frame vorticity

Let M be an oriented three-dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle SO(M) --> M of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on Iso_o (M) \cong SO(M): A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal frame vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases.

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arXiv2023

The sub-Riemannian geometry of screw motions with constant pitch

We consider a family of Riemannian manifolds M such that for each unit speed geodesic gamma of M there exists a distinguished bijective correspondence L between infinitesimal translations along gamma and infinitesimal rotations around it. The simplest examples are R^3, S^3 and hyperbolic 3-space, with L defined in terms of the cross product. More generally, M is a connected compact semisimple Lie group, or its non-compact dual, or Euclidean space acted on transitively by some group which is contained properly in the full group of rigid motions. Let G be the identity component of the isometry group of M. A curve in G may be thought of as a motion of a body in M. Given lambda in R, we define a left invariant distribution on G accounting for infinitesimal roto-translations of M of pitch lambda. We give conditions for the controllability of the associated control system on G and find explicitly all the geodesics of the natural sub-Riemannian structure. We also study a similar system on R^7 rtimes SO(7) involving the octonionic cross product. In an appendix we give a friendly presentation of the non-compact dual of a compact classical group, as a set of "small rotations".

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arXiv2022

Infinitesimally helicoidal motions with fixed pitch of oriented geodesics of a space form

Let L be the manifold of all (unparametrized) oriented lines of R^3. We study the controllability of the control system in L given by the condition that a curve in L describes at each instant, at the infinitesimal level, an helicoid with prescribed angular speed alpha. Actually, we pose the analogous more general problem by means of a control system on the manifold G_kappa of all the oriented complete geodesics of the three dimensional space form of curvature kappa: R^3 for kappa = 0, S^3 for kappa = 1 and hyperbolic 3-space for kappa = -1. We obtain that the system is controllable if and only if alpha ^2 not equal kappa. In the spherical case with alpha = (+/-) 1, an admissible curve remains in the set of fibers of a fixed Hopf fibration of S^3. We also address and solve a sort of Kendall's (aka Oxford) problem in this setting: Finding the minimum number of switches of piecewise continuous curves joininig two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.

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arXiv2022

Tangent ray foliations and their associated outer billiards

Let $v$ be a unit vector field on a complete, umbilic (but not totally geodesic) hypersurface $N$ in a space form; for example on the unit sphere $S^{2k-1} \subset \mathbb{R}^{2k}$, or on a horosphere in hyperbolic space. We give necessary and sufficient conditions on $v$ for the rays with initial velocities $v$ (and $-v$) to foliate the exterior $U$ of $N$. We find and explore relationships among these vector fields, geodesic vector fields, and contact structures on $N$. When the rays corresponding to each of $\pm v$ foliate $U$, $v$ induces an outer billiard map whose billiard table is $U$. We describe the unit vector fields on $N$ whose associated outer billiard map is volume preserving. Also we study a particular example in detail, namely, when $N \simeq \mathbb{R}^3$ is a horosphere of the four-dimensional hyperbolic space and $v$ is the unit vector field on $N$ obtained by normalizing the stereographic projection of a Hopf vector field on $S^{3}$. In the corresponding outer billiard map we find explicit periodic orbits, unbounded orbits, and bounded nonperiodic orbits. We conclude with several questions regarding the topology and geometry of bifoliating vector fields and the dynamics of their associated outer billiards.

math.GT↗

arXiv2022

Mobius fluid dynamics on the unitary groups]{Möbius fluid dynamics on the unitary groups

We study the nonrigid dynamics induced by the standard birational actions of the split unitary groups $G=O_{o}\left( n,n\right) $, $SU\left( n,n\right) $ and $Sp\left( n,n\right) $ on the compact classical Lie groups $M=SO_{n}$, $% U_{n}$ and $Sp_{n}$, respectively. More precisely, we study the geometry of $% G$ endowed with the kinetic energy metric associated with the action of $G$ on $M,$ assuming that $M$ carries its canonical bi-invariant Riemannian metric and has initially a homogeneous distribution of mass. By the least action principle, force free motions (thought of as curves in $G$) correspond to geodesics of $G$. The geodesic equation may be understood as an inviscid Burgers equation with Möbius constraints. We prove that the kinetic energy metric on $G$ is not complete and in particular not invariant, find symmetries and totally geodesic submanifolds of $G$ and address the question under which conditions geodesics of rigid motions are geodesics of $G$. Besides, we study equivalences with the dynamics of conformal and projective motions of the sphere in low dimensions.

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arXiv2021

Harmonic unit normal sections of Grassmannians associated with cross products

Let G(k,n) be the Grassmannian of oriented subspaces of dimension k of R^n with its canonical Riemannian metric. We study the energy of maps assigning to each P \in G(k,n) a unit vector normal to P. They are sections of a sphere bundle E_{k,n}^1 over G(k,n). The octonionic double and triple cross products induce in a natural way such sections for k=2, n=7 and k=3, n=8, respectively. We prove that they are harmonic maps into E_{k,n}^1 endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. In a second instance we analyze the energy of maps assigning an orthogonal complex structure J(P) on P^{\bot} to each P\in G(2,8). We prove that the one induced by the octonionic triple product is a harmonic map into a suitable sphere bundle over G(2,8). This generalizes the harmonicity of the canonical almost complex structure of S^6.

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arXiv2018

Polar factorization of conformal and projective maps of the sphere in the sense of optimal mass transport

Let M be a compact Riemannian manifold and let $μ$,d be the associated measure and distance on M. Robert McCann obtained, generalizing results for the Euclidean case by Yann Brenier, the polar factorization of Borel maps S : M -> M pushing forward $μ$ to a measure $ν$: each S factors uniquely a.e. into the composition S = T \circ U, where U : M -> M is volume preserving and T : M -> M is the optimal map transporting $μ$ to $ν$ with respect to the cost function d^2/2. In this article we study the polar factorization of conformal and projective maps of the sphere S^n. For conformal maps, which may be identified with elements of the identity component of O(1,n+1), we prove that the polar factorization in the sense of optimal mass transport coincides with the algebraic polar factorization (Cartan decomposition) of this Lie group. For the projective case, where the group GL_+(n+1) is involved, we find necessary and sufficient conditions for these two factorizations to agree.

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arXiv2016

Centro-affine invariants and the canonical Lorentz metric on the space of centered ellipses

We consider smooth plane curves which are convex with respect to the origin. We describe centro-affine invariants (that is, GL_+(2,R)-invariants), such as centro-affine curvature and arc length, in terms of the canonical Lorentz structure on the three dimensional space of all the ellipses centered at zero, by means of null curves of osculating ellipses. This is the centro-affine analogue of the approach to conformal invariants of curves in the sphere introduced by Langevin and O'Hara, using the canonical pseudo Riemannian metric on the space of circles.

math.DG↗

arXiv2016

A dynamical presentation of the better than nice metric on the disc

This is a postprint of our paper "Force free Moebius motions of the circle" (J. Geom. Symmetry Phys. 27 (2012) 59-65), which we hadn't uploaded to arXiv previously. We would like to draw attention to the relationship with the article "A geometry where everything is better than nice", by Larry Bates and Peter Gibson (to appear in Proc. Amer. Math. Soc.). In our note we treat thoroughly a simple particular case of two previous, more substantial articles. We describe the force free Moebius motions of the circle, that is, the geodesics of the Lie group M ~ PSL_2(R) ~ O_o(1,2) of Moebius transformations of the circle, equipped with the Riemannian metric given by the kinetic energy induced by the action. It turns up that M decomposes as a Riemannian product S x D, where D is the unit disc endowed with a certain metric, which we now recognize as being essentially the one that is better than nice. Concerning geodesics, we only give the differential equation for their trajectories (using Clairaut's Theorem); we took pleasure in learning from the article by Bates and Gibson that they are actually hypocycloids.

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arXiv2015

Interpolation of geometric structures compatible with a pseudo Riemannian metric

Let (M, g) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g: two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is a product structure and symmetric with respect to g, then r induces a pseudo Riemannian product structure on M. Sometimes the integrability condition is expressed by the closedness of an associated two-form: if j is almost complex on M and ω(x, y) = g(jx, y) is symplectic, then M is almost pseudo Kähler. Now, product, complex and symplectic structures on M are trivial examples of generalized (para)complex structures in the sense of Hitchin. We use the latter in order to define the notion of interpolation of geometric structures compatible with g. We also compute the typical fibers of the twistor bundles of the new structures and give examples for M a Lie group with a left invariant metric.

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arXiv2015

Generalized geometric structures on complex and symplectic manifolds

On a smooth manifold M, generalized complex (generalized paracomplex) structures provide a notion of interpolation between complex (paracomplex) and symplectic structures on M. Given a complex manifold (M,j), we define six families of distinguished generalized complex or paracomplex structures on M. Each one of them interpolates between two geometric structures on M compatible with j, for instance, between totally real foliations and Kahler structures, or between hypercomplex and C-symplectic structures. These structures on M are sections of fiber bundles over M with typical fiber G/H for some Lie groups G and H. We determine G and H in each case. We proceed similarly for symplectic manifolds. We define six families of generalized structures on (M,omega), each of them interpolating between two structures compatible with omega, for instance, between a C-symplectic and a para-Kahler structure (aka bi-Lagrangian foliation).

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arXiv2014

Calibrated geodesic foliations of the hyperbolic space

Let H be the hyperbolic space of dimension n+1. A geodesic foliation of H is given by a smooth unit vector field on H all of whose integral curves are geodesics. Each geodesic foliation of H determines an n-dimensional submanifold M of the 2n-dimensional manifold L of all the oriented geodesics of H (up to orientation preserving reparametrizations). The space L has a canonical split semi-Riemannian metric induced by the Killing form of the isometry group of H. Using a split special Lagrangian calibration, we study the volume maximization problem for a certain class of geometrically distinguished geodesic foliations, whose corresponding submanifolds of L are space-like.

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arXiv2014

Global smooth geodesic foliations of the hyperbolic space

We consider foliations of the whole three dimensional hyperbolic space H^3 by oriented geodesics. Let L be the space of all the oriented geodesics of H^3, which is a four dimensional manifold carrying two canonical pseudo-Riemannian metrics of signature (2,2). We characterize, in terms of these geometries of L, the subsets M in L that determine foliations of H^3. We describe in a similar way some distinguished types of geodesic foliations of H^3, regarding to which extent they are in some sense trivial in some directions: On the one hand, foliations whose leaves do not lie in a totally geodesic surface, not even at the infinitesimal level. On the other hand, those for which the forward and backward Gauss maps are local diffeomorphisms. The subject of this article is within the framework of foliations by congruent submanifolds, and follows the spirit of the paper by Gluck and Warner where they understand the infinite dimensional manifold of all the great circle foliations of the three sphere.

math.DG↗

arXiv2007

On the geometry of the space of oriented lines of the hyperbolic space

Let H be the n-dimensional hyperbolic space of constant sectional curvature -1 and let G be the identity component of the isometry group of H. We find all the G-invariant pseudo-Riemannian metrics on the space OG_n of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, time- and space-like curves, providing a relationship between the geometries of OG_n and H. Moreover, we show that OG_3 is Kähler and find an orthogonal almost complex structure on OG_7.

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