arXiv subjects

P. Piu

Publications and source records attributed to P. Piu.

7 recordsLinked to original sources

arXiv2021

Bour's theorem and helicoidal surfaces with constant mean curvature in the Bianchi-Cartan-Vranceanu spaces

In this paper we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space R^3 to the case of helicoidal surfaces in the Bianchi-Cartan-Vranceanu (BCV) spaces, i.e. in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension $4$ or $6$, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.

math.DG↗

arXiv2018

Loxodromes on invariant surface in three-manifolds

In this paper, we prove important results concerning the loxodromes on an invariant surface in a three-dimensional Riemannian manifold, some of which generalize classical results about loxodromes on rotational surfaces in $\mathbb{R}^3$. In particular, we show how to parametrize a loxodrome on an invariant surface of $\mathbb{H}^2\times \mathbb{R}$ and $\mathbb{H}_3$ and we exhibit the loxodromes of some remarkable minimal invariant surfaces of these spaces. In addition, we give an explicit description of the loxodromes on an invariant surface with constant Gauss curvature.

math.DG↗

arXiv2017

Constant angle surfaces in the Lorentzian Heisenberg group

In this paper, we define and, then, we characterize constant angle spacelike and timelike surfaces in the three-dimensional Heisenberg group, equipped with a 1-parameter family of Lorentzian metrics. In particular, we give an explicit local parametrization of these surfaces and we produce some examples.

math.DG↗

arXiv2012

Surfaces in three-dimensional space forms with divergence-free stress-bienergy tensor

We introduce the notion of biconservative hypersurfaces, that is hypersurfaces with conservative stress-energy tensor with respect to the bienergy. We give the (local) classification of biconservative surfaces in 3-dimensional space forms.

math.DG↗

arXiv2010

On the three-dimensional homogeneous SO(2) - isotropic Riemannian manifolds

In this paper we consider some properties of the three-dimensional homogeneous SO(2)-isotropic Riemannian manifolds. In particular, we determine the geodesics, the totally geodesic surfaces, the totally umbilical surfaces and the geodesics of the rotational surfaces.

math.DG↗

arXiv2005

The Classification of Biharmonic Curves of Cartan-Vranceanu 3-Dimensional Spaces

In this article we characterize all biharmonic curves of the Cartan-Vranceanu 3-dimensional spaces and we give their explicit parametrizations.

math.DG↗

arXiv2003

Explicit Formulas for Non-Geodesic Biharmonic Curves of the Heisenberg Group

We consider the biharmonicity condition for maps between Riemannian manifolds (see [BK]), and study the non-geodesic biharmonic curves in the Heisenberg group H_3. First we prove that all of them are helices, and then we obtain explicitly their parametric equations.

math.DG↗