arXiv subjects

Marcos M. Diniz

Publications and source records attributed to Marcos M. Diniz.

2 recordsLinked to original sources

arXiv2015

First variation of the Hausdorff measure of non-horizontal submanifolds in stratified Lie groups

We determine necessary conditions for a non-horizontal submanifold of a sub-Riemannian stratified Lie group to be of minimal measure. We calculate the first variation of the measure for a non-horizontal submanifold and find that the minimality condition implies the tensor equation $H+σ=0$, where $H$ is analogous to the mean curvature and $σ$ is the mean torsion. We also discuss new examples of minimal non-horizontal submanifolds in the Heisenberg group, in particular surfaces in $\mathbb H^2$.

math.DG↗

arXiv2012

Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$

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 $\int_S K = 0$.

math.DG↗