Isoparametric functions and harmonic and minimal unit vector fields. (English) Zbl 1004.53046

Boeckx, E.; Vanhecke, L.

Fernández, Marisa (ed.) et al., Global differential geometry: the mathematical legacy of Alfred Gray. Proceedings of the international congress on differential geometry held in memory of Professor Alfred Gray, Bilbao, Spain, September 18-23, 2000. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 288, 20-31 (2001).

Let \(M\) be a Riemannian manifold and \(T_1M\) the unit tangent sphere bundle of \(M\) equipped with the Sasaki metric. A smooth unit vector field on \(M\) is considered as a smooth map from \(M\) into \(T_1M\) and is said to be harmonic if it is critical for the corresponding energy functional and minimal if it is critical for the corresponding volume functional. The authors show that if \(f\) is an isoparametric function on an Einstein manifold \(M\) then the unit vector field \(\nabla f/|\nabla f|\) is a harmonic vector field on \(\{x \in M \mid (df)_x \neq 0\}\). The authors discuss several examples for real, complex and quaternionic space forms, which also provide examples of minimal vector fields. Finally they construct minimal and harmonic vector fields on generalized Heisenberg groups (these groups are not Einstein manifolds).
For the entire collection see [Zbl 0980.00033].

Reviewer: Jürgen Berndt (Hull)

MSC:

53C43	Differential geometric aspects of harmonic maps
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
58E20	Harmonic maps, etc.

Keywords:

minimal unit vector fields; harmonic unit vector fields; harmonic maps; Einstein manifolds; generalized Heisenberg groups