1. ΕΠΙΦΑΝΕΙΕΣ ΜΕ ΙΣΟΜΕΤΡΙΚΕΣ ΣΚΙΟΓΡΑΜΜΕΣ 2. ΕΠΙΦΑΝΕΙΕΣ ΜΕ ΙΣΟΜΕΤΡΙΚΕΣ ΓΕΩΔΑΙΣΙΑΚΕΣ

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 HEREAFTER CALL THEOREMA AND B. THEOREM A: LET M BE A COMPACT AND STRICTLY CONVEX SURFACE EMBEDDED INTHE 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 ΣE OF M . SO THE IDEA OF THE PROOF IS TO CONSTRUCT A MAPPING Z WHICH MAPS THE POINT E OF S2 TO A TANGENT VECTOR ZE OF ΣE 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 ONTHE 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 ASURFACE IN THE EUCLIDEAN SPACE E3, WHICH IS DIFFEOMORPHIC TO THE SPHERE S2. WESUPPOSE THAT ALL GEODESICS OF M AN 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 Π: S M WITH Π(VP)=P IS A COVERING MAP OF M. BUT IN THIS CASE, AN EVERYWHERE NONZERO VECTOR FIELD, TAGENT TANGENT TO M, CAN BE CONSTRUCTED WHICH IS IMPOSSIBLE. SO THE FUNCTION K(S) IS CONSTANT AND WE GETEASILY THAT M IS A EUCLIDEAN SPHERE.

Grantor dc:publisher

Πανεπιστήμιο Κρήτης

Year dc:date

1989

Author dc:creator

• Χαρίτος, Χαράλαμπος

• GEODESICS
• SHADOW LINES
• Surfaces
• THEOREME OF POINCARE-HOPF
• ΓΕΩΔΑΙΣΙΑΚΕΣ
• Επιφάνειες
• ΘΕΩΡΗΜΑ POINCARE-HOPF
• ΣΚΙΟΓΡΑΜΜΕΣ
• Φυσικές Επιστήμες
• Natural Sciences
• Μαθηματικά
• Mathematics

Language dc:language

gre

Identifier

10.12681/eadd/1290

OAI identifier oai:identifier

oai:10442/1290