Surfaces with isometric geodesics

Abstract

The aim of the paper is to prove the Theorem: Let M be a surface in the euclidean space £ ' which is diffeomorphic to the sphere and suppose that all geodesies of M are congruent. Then M is a euclidean sphere.1980 Mathematics subject classification (1985 Revision): 53.53A05. IntroductionThe aim of this paper is to prove the following: Theorem. Let M be a surface in the euclidean space E 3 , which is diffeomorphic to the sphere S 2 . We suppose that all geodesies of M are congruent. Then M is a euclidean 2-sphere.The basic idea of the proof is the following: We consider a curve F o in £ 3 such that each geodesic of M is congruent to F o and let k(s) be the curvature function of T o . By supposing that k{s) is not constant, we find a surface S in the unit sphere bundle S i (M) of M such that the projection n: S-*M with n(v p ) = p is a covering map of M. But in this case, an everywhere non-zero vector field, tangent to M, can be constructed and it is well-known that this is impossible [5]. So the function k(s) is constant and we get easily that M is a euclidean 2-sphere.We would like to make the following remarks:(i) The hypothesis that M is diffeomorphic to S 2 is not an essential restriction. In fact, if M is compact and 7r 1 (M)#0, then there are geodesies on M which do not have the same length [1,2]. On the contrary, there exist surfaces in E 3 , diffeomorphic to the sphere S 2

Open surfaces with congruent geodesics

“…Compactness of M, which implies that M is a diffeomorphic to a sphere, is crucial in establishing the above result (see [3]). …”

“…In the present note following the principal ideas of [3] we are able, for the first time, to remove the compactness assumption. In fact we show:…”

“…Let M be a complete non-compact surface without boundary embedded in the euclidean space E 3 . We suppose that all geodesies of M are congruent.…”

“…In the course of the proof we will often refer to the compact case [3]. However, we will make this paper as self-contained as possible by introducing all necessary notation and definitions.…”

“…Proof. For the proof of (a) we remark that if k 0 is a non-critical value of fe(s) then r is of rank 1 on r~l(k 0 ) (for more details see the proof of Proposition 2 in [3]). …”