Searcharxiv⌕ Search

arXiv subjects

Marian Ioan Munteanu

Publications and source records attributed to Marian Ioan Munteanu.

At least 19 recordsLinked to original sources
arXiv2024

Conformal trajectories in 3-dimensional space form

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds $M^3$. Given a conformal vector field $V\in\mathfrak{X}(M^3)$, a conformal trajectory of $V$ is a regular curve $γ$ in $M^3$ satisfying $\nabla_{γ'}γ'=q\, V\timesγ'$, for some fixed non-zero constant $q\in {\mathbb{R}}$. In this paper, we study conformal trajectories in the space forms ${\mathbb{R}}^3$, ${\mathbb{S}}^3$ and ${\mathbb{H}}^3$. For (non-Killing) conformal vector fields in ${\mathbb{S}}^3$ (respectively in ${\mathbb{H}}^3$), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively hyperbolic) functions on the arc-length parameter. In the case of Euclidean space ${\mathbb{R}}^3$, we obtain the same result for the radial vector field and characterising all conformal trajectories.

math.DG↗
arXiv2024

Translators of the mean curvature flow in the special linear group $SL(2,\mathbb{R})$

Translators in the special linear group $SL(2,\mathbb{R})$ are surfaces whose mean curvature $H$ and unit normal vector $N$ satisfy $H=\langle N,X\rangle$, where $X$ is a fixed Killing vector field. In this paper we study and classify those translators that are invariant by a one-parameter group of isometries. By the Iwasawa decomposition, there are three types of such groups. The dimension of the Killing vector fields is $4$ and an exhaustive discussion is done for each one of the Killing vector fields and each of the invariant surfaces. In some cases, explicit parametrizations of translators are obtained.

math.DG↗
arXiv2024

Solitons of the mean curvature flow in $\mathbb{s}^2\times\mathbb{R}$

A soliton of the mean curvature flow in the product space $\mathbb{s}^2\times\mathbb{R}$ as a surface whose mean curvature $H$ satisfies the equation $H=\langle N,X\rangle$, where $N$ is the unit normal of the surface and $X$ is a Killing vector field. In this paper we consider the vector field tangent to the fibers and the vector field associated to a rotations about an axis of $\mathbb{s}^2$, respectively. We give a classification of the solitons with respect to these vector fields assuming that the surface is invariant under a one-parameter group of vertical translations or under a group of rotations of $\mathbb{s}^2$.

math.DG↗
arXiv2023

Magnetic unit vector fields

We show that a unit vector field on an oriented Riemannian manifold is a critical point of the Landau Hall functional if and only if it is a critical point of the Dirichlet energy functional. Therefore, we provide a characterization for a unit vector field to be a magnetic map into its unit tangent sphere bundle. Then, we classify all magnetic left invariant unit vector fields on $3$-dimensional Lie groups.

math.DG↗
arXiv2018

Magnetic curves in the real special linear group

We investigate contact magnetic curves in the real special linear group of degree 2. They are geodesics of the Hopf tubes over the projection curve. We prove that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers. Finally, we study contact homogeneous magnetic trajectories in SL(2,R) and show that they project to horocycles in H2(-4).

math.DG↗
arXiv2013

Periodic magnetic curves in elliptic Sasakian space forms

It is an interesting question whether a given equation of motion has a periodic solution or not, and in the positive case to describe them. We investigate periodic magnetic curves in elliptic Sasakian space forms and we obtain a quantization principle for periodic magnetic flowlines on Berger spheres. We give a criterion for periodicity of magnetic curves on the unit sphere ${\mathbb{S}}^3$.

math.DG↗
arXiv2011

Geometry of PR-warped products in para-Kaehler manifolds

In this paper, we initiate the study of $\p R$-warped products in para-Kähler manifolds and prove some fundamental results on such submanifolds. In particular, we establish a general optimal inequality for $\p R$-warped products in para-Kähler manifolds involving only the warping function and the second fundamental form. Moreover, we completely classify $\p R$-warped products in the flat para-Kähler manifold with least codimension which satisfy the equality case of the inequality.

math.DG↗
arXiv2010

Minimal translation surfaces in $Sol_3$

In the homogeneous space Sol$_3$, a translation surface is parameterized by $x(s,t)=α(s)\astβ(t)$, where $α$ and $β$ are curves contained in coordinate planes and $\ast$ denotes the group operation of Sol$_3$. In this paper we study translation surfaces in Sol$_3$ whose mean curvature vanishes.

math.DG↗
arXiv2010

Semi-invariant $ξ^{\bot}$-submanifolds of generalized quasi-Sasakian manifolds

A structure on an almost contact metric manifold is defined as a generalization of well-known cases: Sasakian, quasi-Sasakian, Kenmotsu and cosymplectic. Then we consider a semi-invariant $ξ^{\bot}$-submanifold of a manifold endowed with such a structure and two topics are studied: the integrability of distributions defined by this submanifold and characterizations for the totally umbilical case. In particular we recover results of Kenmotsu, Eum and Papaghiuc.

math.DG↗
arXiv2010

On certain surfaces in the Euclidean space ${\mathbb{E}}^3$

In the present paper we classify all surfaces in $\E^3$ with a canonical principal direction. Examples of these type of surfaces are constructed. We prove that the only minimal surface with a canonical principal direction in the Euclidean space ${\mathbb{E}}^3$ is the catenoid.

math.DG↗
arXiv2009

Constant Angle Surfaces in a warped product

Let $I \subseteq \R$ be an open interval, $f : I \to \R$ a strictly positive function and denote by $\E^2$ the Euclidean plane. We classify all surfaces in the warped product manifold $I \times_f \E^2$ for which the unit normal makes a constant angle with the direction tangent to $I$.

math.DG↗