TY - JOUR
T1 - On a class of vector fields with discontinuity of divide-by-zero type and its applications
JF - Journal of dynamical and control systems 18, nr. 1 (2012) 135-158
Y1 - 2012
A1 - Roberta Ghezzi
A1 - Alexey O. Remizov
AB - We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three of them in the framework of differential geometry: singularities of geodesic flows in various singular metrics on surfaces.
PB - Springer
U1 - 7038
U2 - Mathematics
U4 - -1
ER -
TY - JOUR
T1 - The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry
JF - Journal of Dynamical and Control Systems 17 (2011) 141-161
Y1 - 2011
A1 - Bernard Bonnard
A1 - GrĂ©goire Charlot
A1 - Roberta Ghezzi
A1 - Gabriel Janin
AB - We study the tangential case in 2-dimensional almost-Riemannian geometry. We\\r\\nanalyse the connection with the Martinet case in sub-Riemannian geometry. We\\r\\ncompute estimations of the exponential map which allow us to describe the\\r\\nconjugate locus and the cut locus at a tangency point. We prove that this last\\r\\none generically accumulates at the tangency point as an asymmetric cusp whose branches are separated by the singular set.
PB - Springer
UR - http://hdl.handle.net/1963/4914
U1 - 4692
U2 - Mathematics
U3 - Functional Analysis and Applications
U4 - -1
ER -
TY - THES
T1 - Almost-Riemannian Geometry from a Control Theoretical Viewpoint
Y1 - 2010
A1 - Roberta Ghezzi
PB - SISSA
UR - http://hdl.handle.net/1963/4705
U1 - 4482
U2 - Mathematics
U3 - Functional Analysis and Applications
U4 - -1
ER -
TY - JOUR
T1 - Two-dimensional almost-Riemannian structures with tangency points
JF - Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 793-807
Y1 - 2010
A1 - Andrei A. Agrachev
A1 - Ugo Boscain
A1 - GrĂ©goire Charlot
A1 - Roberta Ghezzi
A1 - Mario Sigalotti
AB - Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss-Bonnet formula for almost-Riemannian structures with tangency points.

PB - Elsevier
UR - http://hdl.handle.net/1963/3870
U1 - 839
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -