Volume and energy of vector fields on spheres. A survey. (English) Zbl 1029.53074

Gil-Medrano, Olga

Gil-Medrano, Olga (ed.) et al., Differential geometry. Proceedings of the international conference held in honour of the 60th birthday of A. M. Naveira, Valencia, Spain, July 8-14, 2001. Singapore: World Scientific. 167-178 (2002).

This survey article is concerned with volume and energy of vector fields on Riemannian manifolds.
Given a Riemannian manifold \((M, g_0)\), a vector field \(V\) on \(M\) can be considered as a map from \(M\) into its tangent bundle \(TM\) with Sasaki metric \(g_0^S\). Thus given another Riemannian metric \(g\) on \(M\), one can consider the energy of the vector field \(V\), the volume of the immersion \(V : M \to (TM, g_0^S)\) as the integral of the tension field of \(V\) and the integral of the induced metric, respectively. Usually, critical points of the energy functional are called a harmonic map and the critical points of the volume functional are called a minimal immersion.
The author surveys some recent results related with the conditions that vector fields become harmonic or minimal as maps from \(M\) into its tangent bundle \(TM\) or unit tangent bundle \(T^1M\). For instance, given a unit vector field \(V\) in a Riemannian manifold \((M, g_0)\) and a metric \(g\) on \(M\), the map \(V : (M, g) \to (T^1M, g_0^S)\) is harmonic if and only if \(R(\nabla V)(E_i), V, E_i) + \tau_g(Id) = 0\) and \(\nabla V (\tau_g(Id)) + \nabla_{E_i}(\nabla V)(E_i)\) is collinear to \(V\), where \(\{E_i\}\) is a \(g\)-orthonormal local frame and \(\tau_g(Id)\) denotes the tension of \(Id: (M,g) \to (M, g_0)\). In case \(g_0 = g\), \(V : M \to (T^1M, g_0^S)\) is minimal if and only if \((\nabla V) (\tau_{V^*g_0^S}(Id)) + \sum \nabla_{E_i}(\nabla V)(E_i)\) is collinear to \(V\).
On the other hand, the author also enumerates several results about critical unit vector fields on round spheres and discusses the infimum of the energy functional and volume functional of vector fields on spheres.
For the entire collection see [Zbl 0995.00012].

Reviewer: Gabjin Yun (Kyung-Ki, Yong-in)

MSC:

53C43	Differential geometric aspects of harmonic maps
53-02	Research exposition (monographs, survey articles) pertaining to differential geometry

Keywords:

energy functional and volume functional; harmonic maps; Hopf vector fields; Killing vector fields; minimal immersions; Sasaki metric; vector fields