250 research outputs found
On the harmonicity of normal almost contact metric structures
We consider normal almost contact structures on a Riemannian manifold and,
through their associated sections of an ad-hoc twistor bundle, study their
harmonicity, as sections or as maps. We rewrite these harmonicity equations in
terms of the Riemann curvature tensor and find conditions relating the
harmonicity of the almost contact and almost complex structures of the total
and base spaces of the Morimoto fibration.Comment: 14 page
Biminimal immersions
We study biminimal immersions, that is immersions which are critical points
of the bienergy for normal variations with fixed energy. We give a geometrical
description of the Euler-Lagrange equation associated to biminimal immersions
for: i) biminimal curves in a Riemannian manifold, with particular care to the
case of curves in a space form ii) isometric immersions of codimension one in a
Riemannian manifold, in particular for surfaces of a three-dimensional
manifold. We describe two methods to construct families of biminimal surfaces
using both Riemannian and horizontally homothetic submersions.Comment: Dedicated to Professor Renzo Caddeo on his 60th birthday. 2 figure
The biharmonic homotopy problem for unit vector fields on 2-tori
The bienergy of smooth maps between Riemannian manifolds, when restricted to
unit vector fields, yields two different variational problems depending on
whether one takes the full functional or just the vertical contribution. Their
critical points, called biharmonic unit vector fields and biharmonic unit
sections, form different sets. Working with surfaces, we first obtain general
characterizations of biharmonic unit vector fields and biharmonic unit sections
under conformal change of the metric. In the case of a 2-dimensional torus,
this leads to a proof that biharmonic unit sections are always harmonic and a
general existence theorem, in each homotopy class, for biharmonic unit vector
fields
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