arXiv subjects

Eduardo Hulett

Publications and source records attributed to Eduardo Hulett.

4 recordsLinked to original sources

arXiv2023

The sub-Riemannian geometry of screw motions with constant pitch

We consider a family of Riemannian manifolds M such that for each unit speed geodesic gamma of M there exists a distinguished bijective correspondence L between infinitesimal translations along gamma and infinitesimal rotations around it. The simplest examples are R^3, S^3 and hyperbolic 3-space, with L defined in terms of the cross product. More generally, M is a connected compact semisimple Lie group, or its non-compact dual, or Euclidean space acted on transitively by some group which is contained properly in the full group of rigid motions. Let G be the identity component of the isometry group of M. A curve in G may be thought of as a motion of a body in M. Given lambda in R, we define a left invariant distribution on G accounting for infinitesimal roto-translations of M of pitch lambda. We give conditions for the controllability of the associated control system on G and find explicitly all the geodesics of the natural sub-Riemannian structure. We also study a similar system on R^7 rtimes SO(7) involving the octonionic cross product. In an appendix we give a friendly presentation of the non-compact dual of a compact classical group, as a set of "small rotations".

math.DG↗

arXiv2015

Conformal geometry of marginally trapped surfaces in $\mathbb{S}^4_1$

A spacelike surface $S\subset \mathbb{S}^4_1$ is marginally trapped if its mean curvature vector is lightlike. On any oriented spacelike surface $S \subset \mathbb{S}^4_1$ we show that a choice of orientation of the normal bundle $ν(S)$ determines a smooth map $G: S \to \mathbb{S}^3$ which we call the null Gauss map of $S$. We show that if $S$ is marginally trapped then $G$ is a conformal immersion away the zeros of certain quadratic Hopf-differential of $S$ and so the surface $G(S)$ is uniquely determined up to conformal transformations of $\mathbb{S}^3$ by two invariants: the normal Hopf differential $κ$ and the Schwartzian derivative $s$. We show that these invariants plus an additional quadratic differential $δ$ are related by a differential equation and determine the geometry of $S$ up to ambient isometries of $\mathbb{S}^4_1$. This allows us to obtain a characterization of marginally trapped surfaces $S$ whose null Gauss image is a constrained Willmore surface in $\mathbb{S}^3$ in the sense of C.Bohle, G. Peters and U.Pinkall [arXiv:math/0411479]. As an application of these results we construct and study integrable non-trivial one-parameter deformations of marginally trapped surfaces with non-zero parallel mean curvature vector and those with flat normal bundle.

math.DG↗

arXiv2011

Surfaces in $\mathbb{S}^4$ with normal harmonic Gauss maps

We consider conformal immersions of Riemann surfaces in $\bb{S}^4$ and study their Gauss maps with values in the Grassmann bundle $\mathcal{F} = SO_5/T^2 \to \mathbb{S}^4$. The energy of maps from Riemann surfaces into $\mathcal{F}$ is considered with respect to the normal metric on the target and immersions with harmonic Gauss maps are characterized. We also show that the normal-harmonic map equation for Gauss maps is a completely integrable system, thus giving a partial answer of a question posed by Y. Ohnita in \cite{ohnita}. Associated $\mathbb{S}^1$-families of parallel mean curvature immersions in $\mathbb{S}^4$ are considered. A lower bound of the normal energy of Gauss maps is obtained in terms of the genus of the surface.

math.DG↗

arXiv2010

Spacelike surfaces in De Ditter 3-space and their twistor lifts

We deal here with the geometry of the twistor fibration $\mathcal{Z} \to \bb{S}^3_1$ over the De Sitter 3-space. The total space $\mathcal{Z}$ is a five dimensional reductive homogeneous space with two canonical invariant almost CR structures. Fixed the normal metric on $\mathcal{Z}$ we study the harmonic map equation for smooth maps of Riemann surfaces into $\mathcal{Z}$. A characterization of spacelike surfaces with harmonic twistor lifts to $\mathcal{Z}$ is obtained. It is also shown that the harmonic map equation for twistor lifts can be formulated as the curvature vanishing of an $\bb{S}^1$-loop of connections i.e. harmonic twistor lifts exist within $\bb{S}^1$-families. Special harmonic maps such as holomorphic twistor lifts are also considered and some remarks concerning (compact) vacua of the twistor energy are given.

math.DG↗