Harmonicity of vector fields on four-dimensional generalized symmetric spaces
Let \((M=G/H,g)\) denote a four-dimensional pseudo-Riemannian generalized symmetric space and \(\mathfrak{g}=\mathfrak{m}+\mathfrak{h}\) the corresponding decomposition of the Lie algebra \(\mathfrak{g}\) of \(G\). Let \(g^s\) be the Sasaki metric on \(TM\). Given a smooth vector field \(V\) on \(M\), the energy of \(V\) as the energy of the map \(V:(M,g)\to (TM,g^s)\), i.e. \[ E(V)=\frac 12\int_M(\text{tr}_gV^\ast g^s)\text{d}v=\frac n2\text{vol}(M,g)+\frac 12\int_M||\nabla V||^2\text{d}v. \] If \(V:(M,g)\to (TM,g^s)\) is a critical point for the energy functional, then \(V\) is said to define a harmonic map. A smooth vector field \(V\) is said to be a harmonic section if and only if it is a critical point of \(\displaystyle E^v=\frac 12\int_M||\nabla V||^2\text{d}v\). In this paper the author completely determine the harmonicity properties of vector fields belonging to \(\mathfrak{m}\). In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. Vector fields defining harmonic maps are also classified. For the classification of generalized symmetric spaces, see the related paper [\textit{C. J. Kowalski} and \textit{J. Cerny}, Tensor, New Ser. 38, 255--267 (1982; Zbl 0505.53024)].
scientific article; zbMATH DE number 6039959
scientific article