Mauricio Godoy Molina

Sub-Riemannian geometry of parallelizable spheres

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

The rolling problem: overview and challenges

In the present paper we give a historical account -ranging from classical to modern results– of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the development of intrinsic Riemannian geometry, from geometric control theory and sub-Riemannian geometry. We also mention how other areas -such as robotics and interpolation theory- have employed the rolling problem.

Rigidity of 2-Step Carnot Groups

In the present paper, we study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either always of infinite type or generically rigid, and we specify the bi-dimensions for each of the choices. Explicit criteria for rigidity of pseudo H- and J-type algebras are given. In particular, we establish the relation of the so-called

Sub-Riemannian Geodesics in the Octonionic H-type Group

J^2

Sub-semi-Riemannian geometry of general H-type groups

J2-condition to rigidity, and we explore these conditions in relation to pseudo H-type algebras.

AN INTRINSIC FORMULATION OF THE ROLLING MANIFOLDS PROBLEM

We study the control system of a Riemannian manifold M of dimension n rolling on the sphere

Geometric conditions for the existence of a rolling without twisting or slipping

S^n