Gradient horizontal de fonctions polynomiales

Patrice Orro (1) , Krzysztof Kurdyka (1) , Si Tiep Dinh

1 LAMA - Laboratoire de Mathématiques

We study trajectories of sub-Riemannian (also called horizontal) gradient of polynomials. In this setting Łojasiewicz's gradient inequality does not hold and a trajectory of a horizontal gradient may be of infinite length, moreover it may accumulate on a closed curve. We show that these phenomena are exceptional; for a generic polynomial function the behavior of the trajectories of horizontal gradients are similar to the he behavior of the trajectories of a Riemannian gradient. To obtain the finiteness of the length of trajectories we change suitably the sub- Riemannian metric. We consider a class of splitting distributions which contains those of Heisenberg and Martinet. For a generic polynomial f the set Vf of horizontal critical points, is a smooth algebraic set of dimension 1 or the empty set, moreover f|Vf is a Morse function. We show that for a generic polynomial function any trajectory of the horizontal gradient (which approaches Vf ) has a limit, as in the Riemannian case studied by S. Łojasiewicz.

Domaines: Géométrie algébrique [math.AG]; Géométrie différentielle [math.DG]

Patrice Orro, Krzysztof Kurdyka, Si Tiep Dinh. Gradient horizontal de fonctions polynomiales. Annales de l'Institut Fourier, 2009, 59. ⟨hal-00389016⟩