Biharmonic curves in Finsler spaces
The author extends the notion of bienergy functional to curves on Finsler spaces and deduces the equations of biharmonic curves in the Finslerian case. These equations involve, besides the flag curvature, two non-Riemannian objects: the Cartan tensor and the Landesberg tensor.NEWLINENEWLINEThe problem of existence of biharmonic curves is investigated and, as in Riemannian geometry, Finslerian biharmonic curves are shown to be of constant geodesic curvature. NEWLINENEWLINENEWLINEIt is found that all Finslerian geodesics are biharmonic but the converse is not true. For this converse, the author proves that any closed biharmonic curve is a geodesic. Moreover, in dimension two, for any Landsberg space with nonpositive flag curvature and any flat Finsler space (i.e., locally Minkowskian), biharmonic curves are geodesics. NEWLINENEWLINENEWLINEOn the other hand, the author shows that there exist biharmonic curves whish are not geodesics, namely, flat Finsler spaces of dimension higher than two. NEWLINENEWLINENEWLINEFinally, two concrete examples admitting proper biharmonic curves are given: a 2-dimensional projectively flat space and a 3-dimensional locally Minkowskian space.
scientific article; zbMATH DE number 6369558