Minimal unit vector fields
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.
scientific article; zbMATH DE number 1775135