The subelliptic heat kernel on SU(2): representations,asymptotics and gradient bounds

The Lie group SU(2) endowed with its canonical subriemannian structure appears as a three-dimensional model of a positively curved subelliptic space. The goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities.     

Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover necessary and sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.     

Équations différentielles stochastiques conduites par des lacets dans les groupes de Carnot

The subriemannian geometry of stochastic differential equations driven by processes generating loops in free Carnot groups are studied. To cite this article: F. Baudoin, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Controllability on Infinite-Dimensional Manifolds: A Chow–Rashevsky Theorem

One of the fundamental problems in control theory is that of controllability, the question of whether one can drive the system from one point to another with a given class of controls. A classical result in geometric control theory of finite-dimensional (nonlinear) systems is Chow–Rashevsky theorem that gives a sufficient condition for controllability on any connected manifold of finite dimension. In other words, the classical Chow–Rashevsky theorem, which is in fact a primary theorem in subriemannian geometry, gives a global connectivity property of a subriemannian manifold. In this paper, following the unified approach of Kriegl and Michor (The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, vol. 53, Am. Math. Soc., Providence, 1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Chow–Rashevsky theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable. To indicate an application of our approach to the infinite-dimensional geometric control problems, we conclude the paper with a novel controllability result on the group of orientation-preserving diffeomorphisms of the unit circle.     

A Semple-type approach to a problem of Goursat: The multi-flag case

We consider the problem of classifying the orbits within a tower of fibrations with fibers diffeomorphic to projective planes and we generalize the tower of fiber bundles due to J. Semple. This tower, which was rediscovered by Montgomery and Zhitomirskii in the context of subriemannian geometry, admits a natural action of the diffeomorphism group of affine 3-space, and these orbits correspond to classes of Goursat multi-flags. We demonstrate that it is possible to classify many of these orbits by elementary means by appealing to some basic tools in projective geometry, and the combinatorics of spatial curves.     

Problems and progress in microswimming

Summary Stokesian swimming is a geometric exercise, a collective game. In Part I, we review Shapere and Wilczek's gauge-theoretical approach for a single organism. We estimate the speeds of organisms moving by propagating small amplitude waves, and we make a conjecture regarding a new inequality for the Stokes' curvature. In Part II, we extend the gauge theory to collective motions. We advocate the influx of nonlinear control theory and subriemannian geometry. Computationally, parallel algorithms are natural, each microorganism representing a separate processor. In the final section, open questions motivated by biology are presented. Dedicated to the memory of Juan C. Simo, a pioneer in the use of geometry to produce better analytical and numerical methods in mechanics This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan C. Simo.     

The Subelliptic Heat Kernels on SL(2, ℝ) and on its Universal Covering \widetilde{\mathbf{SL}(2,\mathbb{R})}: Integral Representations and Some Functional Inequalities

In this paper, we study a subelliptic heat kernel on the Lie group SL(2, ℝ) and on its universal covering [(SL(2,\mathbbR))\tilde]\widetilde{\mathbf{SL}(2,\mathbb{R})}. The subelliptic structure on SL(2,ℝ) comes from the fibration SO(2)→SL(2,ℝ) →H ² and it can be lifted to [(SL(2,\mathbbR))\tilde]\widetilde{\mathbf{SL}(2,\mathbb{R})}. First, we derive an integral representation for these heat kernels. These expressions allow us to obtain some asymptotics in small times of the heat kernels and give us a way to compute the subriemannian distance. Then, we establish some gradient estimates and some functional inequalities like a Li-Yau type estimate and a reverse Poincaré inequality that are valid for both heat kernels.