Loxodromes on Invariant Surfaces in Three-Manifolds

Abstract

In this paper, we prove important results concerning the loxodromes on an invariant surface in a three-dimensional Riemannian manifold, some of which generalize classical results about loxodromes on rotational surfaces in R 3 . In particular, we show how to parametrize a loxodrome on an invariant surface of H 2 × R and H3 and we exhibit the loxodromes of some remarkable minimal invariant surfaces of these spaces. In addition, we give an explicit description of the loxodromes on an invariant surface with constant Gauss curvature.

Time-like loxodromes on helicoidal surfaces in Minkowski 3-space

“…Thus, the loxodromes are a popular topic in navigation, see [1] and [12]. From mathematical point of view, geometers have studied loxodromes defined on different kind of surfaces in various ambient spaces and then many works have been found in literature such as [11,15,13,9,5,6,7,10,16].…”