Minimal unit vector fields. (English) Zbl 1006.53053

Gil-Medrano, Olga; Llinares-Fuster, Elisa

Tohoku Math. J., II. Ser. 54, No. 1, 71-84 (2002).

Let \((M,g)\) be a Riemannian manifold such that the set \({\mathcal X}^1(M)\) of unit vector fields is not empty, and \(T^1M\) the unit tangent bundle. The volume of an element \(V\in{\mathcal X}^1(M)\) is defined as the volume of the submanifold \(V(M)\), which is the image of the immersion \(V:M\to T^1M\), when \(T^1 M\) is equipped with the restriction of the Sasaki metric \(g^S\). The unit vector fields of minimum volume on \(M\), if they exist, are to be found among the critical points of the volume functional restricted to \({\mathcal X}^1(M)\).
In order to characterize the critical points, the authors compute the first variation of the functional. Result: An element \(V\in {\mathcal X}^1(M)\) is a critical point of the volume functional restricted to \({\mathcal X}^1(M)\) if and only if \(V:M\to (T^1M,g^S)\) is a minimal immersion. They also study minimal unit vector fields in constant curvature spaces. Let \(f:{\mathcal X}^1 (M)\to C^\infty (M)\) be defined as \(f(V)= \sqrt{\det L_V}\), where the volume functional \(F:{\mathcal X}^1(M) \to\mathbb{R}\) is given by \(F(V)=\int_M f(V)dv\), \(dv\) is the density on \(M\) defined by \(g\), \(L_V=\text{Id} +(\nabla V)^t_0 \nabla V\), and \(\nabla V\) is a \((1,1)\)-tensor field.
Theorem: Let \(M\) be an \(n\)-dimensional manifold of constant sectional curvature \(K\). Then every unit Killing vector field \(V\) on \(M\) is minimal. Moreover, \(f(V)=(K+1)^{(n-1)/2}\) and \(F(V)= (K+1)^{(n-1)/2} \text{Vol}(M)\).
Finally, the authors prove the following result: A necessary and sufficient condition for a unit Killing vector field \(V\) to be minimal is the vanishing of a certain 1-form given in terms of the covariant derivative of \(V\) and the curvature tensor. From this results are obtained some interesting corollaries.

Reviewer: Costache Apreutesei (Iaşi)

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C20	Global Riemannian geometry, including pinching
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

volume of vector fields; critical points; minimal vector fields; Killing vector fields; Hopf vector fields; minimal immersion

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