Harmonic vector fields on Finsler manifolds (Q282726)

The current paper characterizes the harmonic vector vector fields on Finsler manifolds through the two scalar functions \(I\) and \(J\) with significant geometric interpretations. The authors show that a vector field on a compact Finsler manifold \(M\) without boundary becomes a harmonic vector field under some conditions. Also they show that for \(J=0\) this harmonic vector field is parallel in the Berwald connection and for \(J> 0\) there is no non-zero harmonic vector field. Further they give some results on the first horizontal de Rham cohomology group of the sphere bundle \(SM\) for \(J=0\) and \( J> 0\). Finally, they show that if the complete lift of a harmonic vector field \(X\) coincides with its canonical lift on \(SM\), then \(X\) is Killing.