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Identifier

uch.math.phd//1989charitos

Title

1. Επιφάνειες με ισομετρικές σκιογραμμές 2. Επιφάνειες με ισομετρικές γεωδαισιακές

Alternative Title

Surfaces with concruent shadow-lines and surfaces with concruent geodesics

Creator

Charitos, Charalambos

Abstract

In this work a new characterization of A 2-Dimensional sphere in terms of its shadow-lines or geodesics, is given, contained in what we here after call theorem A and B. Theorem A: Let M be a compact and strictly convex surface embedded in the Euclidean space E3 or in the hyperbolic space H3. We suppose that all shadow-lines of M are congruent. Then M is a Euclidean 2-sphere or a hyperbolic 2-sphere respectively. Roughly speaking, to each point E of the sphere S2 corresponds a different shadow-line ΣΕ of M. So the idea of the proof is to construct a mapping Z which maps the point Z which maps the point E of S2 to a tangent vector ZE of ΣΕ at a fixed special point of ΣΕ if it is not a circle. There are certain difficulties related to the fact that Z is in general a multiple - valued function, depending on the possible symmetries of ΣΕ.This problem is handled by showing that the possible values of Z form a covering space of S2. In this way, an everywhere non-zero vector field Ξ, tangent to S2, can be constructed from Z. But it is well known that this is impossible ([M]). So we conclude that the shadow-lines of M are equal circles, which implies easily that M is a sphere. Theorem B: Let M be a surface in the Euclidean space E3, which is diffeomorphic to the sphere S2. We suppose that all geodesics of M are congruent, then M is a Euclidean 2-sphere. In order to prove this theorem we consider a curve Γ0 in E3 such that each geodesic of M is congruent to Γ0. Let K(S) be the curvature function of Γ0 by supposing that K(S) is not constant we find a surface S in the unit sphere bundle S1(M) of M such that the projection II: S M with II(VP)=P is a covering map of M. But in this case, an everywhere nonzero vector field, tangent to M, can be constricted which is impossible. So the function K(S) is constant and we get easily that M is a Euclidean sphere.

Language

Greek

Issue date

1989-09-01

Date available

1997-06-6

Collection

School/Department--School of Sciences and Engineering--Department of Mathematics--Doctoral theses

Type of Work--Doctoral theses

Permanent Link

https://elocus.lib.uoc.gr//dlib/a/a/f/metadata-dlib-1989charitos.tkl