Jc Gonzalez-Davila

Examples of minimal unit vector fields

We provide a series of examples of Riemannian manifoldsequipped with a minimal unit vector field.

Harmonicity and minimality of oriented distributions

We consider an oriented distribution as a section of the corresponding Grassmann bundle and, by computing the tension of this map for conveniently chosen metrics, we obtain the conditions which the distribution must satisfy in order to be critical for the functionals related to the volume or the energy of the map. We show that the three-dimensional distribution ofS4m+3 tangent to the quaternionic Hopf fibration defines a harmonic map and a minimal immersion and we extend these results to more general situations coming from 3-Sasakian and quaternionic geometry.

Energy and volume of unit vector fields on three-dimensional Riemannian manifolds

Abstract We study the stability and instability of harmonic and minimal unit vector fields and the existence of absolute minima for the energy and volume functional on three-dimensional compact manifolds, in particular on compact quotients of unimodular Lie groups.

Harmonic almost contact structures via the intrinsic torsion

We proceed further in the study of harmonicity for almost contact metric structures already initiated by Vergara-Díaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways and show conditions relating harmonicity and classes of almost contact metric structures. Additionally, we study the harmonicity of such structures as a map into the quotient bundle of the oriented orthonormal frames by the action of the structural group U(n)×1. Finally, by using a Bochner type formula proved by Bör and Hernández Lamoneda, we display some examples which give the absolute minimum for the energy.

Energy of Radial Vector Fields on Compact Rank One Symmetric Spaces

We consider the energy (or the total bending) of unit vector fields oncompact Riemannian manifolds for which the set of its singularitiesconsists of a finite number of isolated points and a finite number ofpairwise disjoint closed submanifolds. We determine lower bounds for theenergy of such vector fields on general compact Riemannian manifolds andin particular on compact rank one symmetric spaces. For this last classof spaces, we compute explicit expressions for the total bending whenthe unit vector field is the gradient field of the distance function toa point or to special totally geodesic submanifolds (i.e., for radialunit vector fields around this point or these submanifolds).

Locally symmetric immersions

We use reflections with respect to submanifolds and related geometric results to develop, inspired by the work of Ferus and other authors, in a unified way a local theory of extrinsic symmetric immersions and submanifolds in a general analytic Riemannian manifold and in locally symmetric spaces. In particular we treat the case of real and complex space forms and study additional relations with holomorphic and symplectic reflections when the ambient space is almost Hermitian. The global case is also taken into consideration and several examples are given.

New examples of weakly symmetric spaces

We provide some new examples of weakly symmetric spaces inside the class of complete, simply connected Riemannian manifolds equipped with a complete unit Killing vector field such that the reflections with respect to its flow lines are global isometries.

INVARIANT SUBMANIFOLDS IN FLOW GEOMETRY

We begin a study of invariant isometric immersions into Riemannian manifolds ( M, g ) equipped with a Riemannian flow generated by a unit Killing vector field ξ. We focus our attention on those ( M, g ) where ξ is complete and such that the reflections with respect to the flow lines are global isometries (that is, ( M, g ) is a Killing-transversally symmetric space) and on the subclass of normal flow space forms. General results are derived and several examples are provided.