Abstract
In this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.
Acknowledgements
The authors are deeply grateful to the referees for valuable comments and helpful suggestions.
Funding: This work was supported by NSFC (no. 11271198, 51535008, 11671212) and SRFDP of China.
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