Search for: All records

Abstract Let be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points. 

We show that if an open set in $$\mathbb R^d$$ can be fibered by unit $$n$$-spheres, then $$d\le 2n+1$$, and if $d=2n+1$, then the spheres must be pairwise linked, and $$n\in \{0,1,3,7\}$$. For these values of $$n$$, we construct unit $$n$$-sphere fibrations in $$R^{2n+1}$$. 

https://arxiv.org/abs/2309.11237 

A great sphere fibration is a sphere bundle with total space S n S^n and fibers which are great k k -spheres. Given a smooth great sphere fibration, the central projection to any tangent hyperplane yields a nondegenerate fibration of R n \mathbb {R}^n by pairwise skew, affine copies of R k \mathbb {R}^k (though not all nondegenerate fibrations can arise in this way). Here we study the topology and geometry of nondegenerate fibrations, we show that every nondegenerate fibration satisfies a notion of Continuity at Infinity, and we prove several classification results. These results allow us to determine, in certain dimensions, precisely which nondegenerate fibrations correspond to great sphere fibrations via the central projection. We use this correspondence to reprove a number of recent results about sphere fibrations in the simpler, more explicit setting of nondegenerate fibrations. For example, we show that every germ of a nondegenerate fibration extends to a global fibration, and we study the relationship between nondegenerate line fibrations and contact structures in odd-dimensional Euclidean space. We conclude with a number of partial results, in hopes that the continued study of nondegenerate fibrations, together with their correspondence with sphere fibrations, will yield new insights towards the unsolved classification problems for sphere fibrations. 

We show that any embedding $$\mathbb {R}^d \to \mathbb {R}^{2d+2^{\gamma (d)}-1}$$ inscribes a trapezoid or maps three points to a line, where $$2^{\gamma (d)}$$ is the smallest power of $$2$$ satisfying $$2^{\gamma (d)} \geq \rho (d)$$ , and $$\rho (d)$$ denotes the Hurwitz–Radon function. The proof is elementary and includes a novel application of nonsingular bilinear maps. As an application, we recover recent results on the nonexistence of affinely $$3$$ -regular maps, for infinitely many dimensions $$d$$ , without resorting to sophisticated algebraic techniques. 

https://arxiv.org/abs/2301.00246