Energy and Riemannian flows (Q952177)

Suppose \((M^{n+1},g)\) is a smooth, closed, connected oriented Riemannian manifold of dimension \(n+1,\;n\geq 2\), and \(\mathcal{L}\) is an oriented 1-foliation on \(M\) given by a nonsingular unit vector field \(N\). The energy density of \(\mathcal{L}\) at the point \(p\) is \(\displaystyle e(\mathcal{L})(p)=\frac 12|\nabla N|^2(p)\), where \(\nabla\) is the Levi-Civita connection on \(M\). The energy of the foliation \(\mathcal{L}\) is defined by \[ E(\mathcal{L})=\int_Me(\mathcal{L})\mu, \] where \(\mu\) is the volume form on \(M\). The foliation is said to be harmonic if it is a critical foliation for this energy functional under variation of \(\mathcal{L}\) through foliations \(\mathcal{L}_t,\;|t|<\epsilon\). The main result of this paper is Theorem 1.1. Let \(\mathcal{L}\) be an oriented Riemannian flow given by a nonsingular unit vector field \(N\) on a closed oriented Riemannian manifold of dimension \(n+1,\;n\geq 2\). Assume that \(g\) is bundle-like with respect to \(\mathcal{L}\) and that the mean curvature form of \(\mathcal{L}\) is basic with respect to \(g\). Then the flow \(\mathcal{L}\) is harmonic if and only if \[ g(Ric(N)-2S(\nabla_NN),E)=0, \quad\forall E\in Q, \] where \(Q\) is the normal bundle of \(\mathcal{L}\), \(S\) the shape operator on \(Q\) and \(Ric(N)\) is the Ricci curvature operator in the direction \(N\). Moreover, if \((M,g)\) has constant sectional curvature \(C\), then the flow \(\mathcal{L}\) is harmonic if and only if it is isometric, and in this case, \(\displaystyle E(\mathcal{L})=\frac{nC}{2}Vol(M)\) if \(n\) is even, and \(E(\mathcal{L})=0\) otherwise. For the case \(n=2\) this paper gives the following: Theorem 1.5. Let \(\mathcal{L}\) be an oriented isometric flow on a closed oriented Riemannian manifold \((M^3, g)\) of dimension 3. Assume that the metric \(g\) is bundle-like with respect to \(\mathcal{L}\). Then the following properties are equivalent: (i) \(\mathcal{L}\) is harmonic; (ii) \(\mathcal{L}\) is either transverse to a 2-dimensional foliation \(\mathcal{F}\) and \(E(\mathcal{L})=0\), or \(\mathcal{L}\) is transverse to a contact structure on \(M\) and \(E(\mathcal{L})=Vol(M,g)\).