Harmonic and minimal unit vector fields on Riemannian symmetric spaces. (English) Zbl 1045.53036

Berndt, Jürgen; Vanhecke, Lieven; Verhóczki, László

Ill. J. Math. 47, No. 4, 1273-1286 (2003).

A connected closed submanifold \(F\) of a complete Riemannian manifold \(M\) is said to be reflective if the geodesic reflection of \(M\) in \(F\) is a well-defined global isometry. In this paper the authors give new examples of harmonic and minimal unit vector fields on semisimple symmetric spaces. These vector fields are constructed from isometric cohomogeneity one actions with a reflective singular orbit. Namely, if \(F\) is a reflective submanifold such that the rank of \(F^\bot\) equals one, then \(F\) determines a cohomogeneity one action on the symmetric space \(M\) such that \(F\) is one of the orbits. Then the principal orbits can be viewed as tubes about \(F\) provided that the codimension of \(F\) is greater than one. The geodesics emanating perpendicularly from \(F\) intersect each principal orbit of this action orthogonally. The unit tangent vectors of these geodesics yield the radial unit vector field associated to \(F\), which is defined on the open and dense subset formed by the union of principal orbits.
The main result is the following theorem:
“Let \(M\) be a Riemannian symmetric space of compact or non-compact type, and let \(F\) be a reflective submanifold of \(M\) such that its codimension is greater than one and the rank of \(F^\bot\) is equal to one. Then the radial unit vector field associated to \(F\) is harmonic and minimal.”
From this result one obtains new examples of minimal and harmonic unit vector fields, since there are several cohomogeneity one actions with a reflective singular orbit.

Reviewer: Rosa Anna Marinosci (Lecce)

MSC:

53C35	Differential geometry of symmetric spaces
53C20	Global Riemannian geometry, including pinching
53C40	Global submanifolds
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
57S15	Compact Lie groups of differentiable transformations

Keywords:

geodesic reflection; reflective submanifold; principal orbit; minimal and harmonic unit vector fields