SURFACES WITH CONCRUENT SHADOW-LINES AND SURFACES WITH CONCRUENT GEODESICS
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 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 F ...
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