On generalization of Zermelo navigation problem on Riemannian manifolds

Abstract

We generalize the Zermelo navigation problem and its solution on Riemannian manifolds (M, h) admitting a space dependence of a ship's own speed |u(x)| h ≤ 1 in the presence of a perturbation W determined by a mild velocity vector field |W (x)| h < |u(x)| h , with application of Finsler metric of Randers type.M.S.C. 2010: 53B20, 53C21, 53C22, 53C60, 49J15, 49J53.

Generalized Zermelo navigation on Hermitian manifolds under mild wind

“…This standard assumption we have already dropped in purely geometric approach (cf. [14,15]). Hence, the generalization included the problem in the original setting with |u| h = 1 = const.…”

A note on generalization of Zermelo navigation problem on Riemannian manifolds with strong perturbation

“…This recalls the scenario in the generalized Randers case in the absence of a wind. Then, however, the resulting Randers metric is Riemannian and conformal to the corresponding background Riemannian metric h; see Proposition 2.5 in [11]. The adequate and wider investigation on conformal and weakly conformal Finsler geometry can be found, in particular, in [14,13].…”

“…with W R = 0 and h(u, u) = const. Then, however, Randers metric is reduced to the corrresponding background Riemannian metric h up to scaling; for more details see the study in [11]. Such a case also corresponds to a pair of conformal homothetic Finsler metrics, that is a special case of weakly conformally equivalent Finsler metrics considered in [14].…”

“…Such a scenario does not occur in the case of Randers metric including the generalized version of the problem in the presence of a weak wind where |W R | h < |u| h , cf. [11]. For the perturbation (16) we obtain the form of the resulting metric as follows .…”

“…This assumption we have already dropped considering the problem on Riemannian manifolds, however being in the case of a background weak wind which guarantees a full control of navigating ship (cf. [11]). Reviewing the bibliography in this scope one may find the paper by A. de Mira Fernandes [9] who accomodated shortly after Zermelo's contribution a varying magnitude ship's velocity.…”