Jose Veloso | UFPA - Federal University of Pará (original) (raw)

Papers by Jose Veloso

Geometriae Dedicata, 2020

We define flag structures on a real three manifold M as the choice of two complex lines on the co... more We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted connection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are given by totally real submanifolds in the flag space SL(3, C)/B, where B is the subgroup of upper triangular matrices. We also define a global invariant which is analogous to the Chern-Simons secondary class invariant for three manifolds with a Riemannian structure and to the Burns-Epstein invariant in the case of CR structures. It turns out to be constant on homotopy classes of totally real immersions in flag space.

arXiv: Differential Geometry, 2020

The authors Balogh-Tyson-Vecchi in arXiv:1604.00180 utilize the Riemannian approximations scheme ... more The authors Balogh-Tyson-Vecchi in arXiv:1604.00180 utilize the Riemannian approximations scheme (mathbbH1,L)(\mathbb H^1, _L)(mathbbH1,L), in the Heisenberg group, introduced by Gromov, to calculate the limits of Gaussian and normal curvatures defined on surfaces of mathbbH1\mathbb H^1mathbbH1 when LrightarrowinftyL\rightarrow\inftyLrightarrowinfty. They show that these limits exist (unlike the limit of Riemannian surface area form or length form), and they obtain Gauss-Bonnet theorem in mathbbH1\mathbb H^1mathbbH1 as limit of Gauss-Bonnet theorems in (mathbbH1,L)(\mathbb H^1, _L)(mathbbH1,L) when LLL goes to infinity. This construction was extended by Wang-Wei in arXiv:1912.00302 to the affine group and the group of rigid motions of the Minkowski plane. We generalize constructions of both papers to surfaces in sub-Riemannian three dimensional manifolds following the approach of arXiv:1909.13341, and prove analogous Gauss-Bonnet theorem.

Journal of Dynamical and Control Systems, 2009

For a k-step sub-Riemannian manifold which admits a bracket generating vector at a point, we desc... more For a k-step sub-Riemannian manifold which admits a bracket generating vector at a point, we describe a region near the point where the exponential map is a local diffeomorphism. This is proved by taking the Taylor series of the exponential map and calculating the first nonzero term, which has order $ 2{\left( {{{\mathcal{D}}}_{{{\mathcal{H}}}} - n} \right)} $ , where n is the topological dimension and $ {{\mathcal{D}}}_{{{\mathcal{H}}}} $ is the Hausdorff dimension of the metric space associated to the sub-Riemannian manifold.

Journal of Geometry and Physics

Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure d... more Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description of the structure of these curvature invariants in the cases where the background structure is one of the parabolic geometries. As an illustration, constant curvature models are discussed for certain sub-Riemannian geometries.

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Journal of Dynamical and Control Systems

We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space H 1. The sub-Rieman... more We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space H 1. The sub-Riemannian distance makes H 1 a metric space and consenquently with a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transverse to the tangent space of S. If all points of S have this property, we prove a Gauss-Bonnet formula and for compact surfaces (which are topologically a torus) we obtain ∫ S K = 0.

Journal of Differential Geometry, 1990

Page 1. LIE'S THIRD THEOREM FOR INTRANSITIVE LIE EQUATIONS JOSÉ MM VELOSO Introduction In [4... more Page 1. LIE'S THIRD THEOREM FOR INTRANSITIVE LIE EQUATIONS JOSÉ MM VELOSO Introduction In [4], H. Goldschmidt used the formalism developed by B. Malgrange [9] to prove Lie's third theorem in the context of transitive ...

Journal of Dynamical and Control Systems, 2016

In this paper, we determine the necessary conditions for a non-horizontal submanifold of sub-Riem... more In this paper, we determine the necessary conditions for a non-horizontal submanifold of sub-Riemannian Heisenberg group H n to be of minimal (spherical) Hausdorff measure. We give the first variation of the (spherical) Hausdorff measure for a non-horizontal submanifold, and find that the minimality implies the tensor equation H + L = 0, where H is analogous to the mean curvature and L is a mean Lie bracket. The theorem is stated in the more general setting of a stratified Lie group with a constant metric factor.

Eprint Arxiv 1210 7110, Oct 26, 2012

We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space H1H^1H1. The sub-Riem... more We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space H1H^1H1. The sub-Riemannian distance makes H1H^1H1 a metric space and consenquently with a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transverse to the tangent space of S. If all points of S have this property, we prove a Gauss-Bonnet formula and for compact surfaces (which are topologically a torus) we obtain intSK=0\int_S K = 0intSK=0.