arXiv subjects

Mauricio Godoy Molina

Publications and source records attributed to Mauricio Godoy Molina.

21 records · Page 2Linked to original sources

arXiv2010

Sub-Riemannian geodesics and heat operator on odd dimensional spheres

In this article we study the sub-Riemannian geometry of the spheres $S^{2n+1}$ and $S^{4n+3}$, arising from the principal $S^1-$bundle structure defined by the Hopf map and the principal $S^3-$bundle structure given by the quaternionic Hopf map respectively. The $S^1$ action leads to the classical contact geometry of $S^{2n+1}$, while the $S^3$ action gives another type of sub-Riemannian structure, with a distribution of corank 3. In both cases the metric is given as the restriction of the usual Riemannian metric on the respective horizontal distributions. For the contact $S^7$ case, we give an explicit form of the intrinsic sub-Laplacian and obtain a commutation relation between the sub-Riemannian heat operator and the heat operator in the vertical direction.

math.DG↗

arXiv2010

An intrinsic formulation of the rolling manifolds problem

We present an intrinsic formulation of the kinematic problem of two $n-$dimensional manifolds rolling one on another without twisting or slipping. We determine the configuration space of the system, which is an $\frac{n(n+3)}2-$dimensional manifold. The conditions of no-twisting and no-slipping are decoded by means of a distribution of rank $n$. We compare the intrinsic point of view versus the extrinsic one. We also show that the kinematic system of rolling the $n$-dimensional sphere over $\mathbb R^n$ is controllable. In contrast with this, we show that in the case of $SE(3)$ rolling over $\mathfrak{se}(3)$ the system is not controllable, since the configuration space of dimension 27 is foliated by submanifolds of dimension 12.

math.DG↗

arXiv2009

Sub-Riemannian geometry of parallelizable spheres

The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere $S^3$ originating from different constructions. Namely, we describe the sub-Riemannian geometry of $S^3$ arising through its right Lie group action over itself, the one inherited from the natural complex structure of the open unit ball in $\comp^2$ and the geometry that appears when considering the Hopf map as a principal bundle. The main result of this comparison is that in fact those three structures coincide. In the second place, we present two bracket generating distributions for the seven dimensional sphere $S^7$ of step 2 with ranks 6 and 4. These yield to sub-Riemannian structures for $S^7$ that are not present in the literature until now. One of the distributions can be obtained by considering the CR geometry of $S^7$ inherited from the natural complex structure of the open unit ball in $\comp^4$. The other one originates from the quaternionic analogous of the Hopf map.

math.DG↗