{"id":{"repo_id":"greece","oai_identifier":"oai:10442/1290"},"canonical_url":"https://search.dev.ndltd.org/etd/greece/oai:10442/1290","repository":{"repo_id":"greece","name":"Greek National Archive of PhD Theses","base_url":"https://phdtheses.ekt.gr/eadd_oai/request"},"display":{"title":"1. ΕΠΙΦΑΝΕΙΕΣ ΜΕ ΙΣΟΜΕΤΡΙΚΕΣ ΣΚΙΟΓΡΑΜΜΕΣ 2. ΕΠΙΦΑΝΕΙΕΣ ΜΕ ΙΣΟΜΕΤΡΙΚΕΣ ΓΕΩΔΑΙΣΙΑΚΕΣ","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 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.","abstract_html":"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.","abstract_has_math":false,"creators":["Χαρίτος, Χαράλαμπος"],"institution":"Πανεπιστήμιο Κρήτης","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1989,"date_issued":"1989","date_published":"1989","updated_at":"2026-07-24T02:24:49Z","subjects":["GEODESICS","SHADOW LINES","Surfaces","THEOREME OF POINCARE-HOPF","ΓΕΩΔΑΙΣΙΑΚΕΣ","Επιφάνειες","ΘΕΩΡΗΜΑ POINCARE-HOPF","ΣΚΙΟΓΡΑΜΜΕΣ","Φυσικές Επιστήμες","Natural Sciences","Μαθηματικά","Mathematics"],"languages":["gre"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["10.12681/eadd/1290"],"render_values":[{"text":"10.12681/eadd/1290","href":"https://doi.org/10.12681/eadd/1290","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10442/hedi/1290","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Χαρίτος, Χαράλαμπος"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["1989"]},{"key":"dc:publisher","label":"Institution","values":["Πανεπιστήμιο Κρήτης","University of Crete (UOC)"]},{"key":"dc:type","label":"Dc Type","values":["PhD Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["GEODESICS","SHADOW LINES","Surfaces","THEOREME OF POINCARE-HOPF","ΓΕΩΔΑΙΣΙΑΚΕΣ","Επιφάνειες","ΘΕΩΡΗΜΑ POINCARE-HOPF","ΣΚΙΟΓΡΑΜΜΕΣ","Φυσικές Επιστήμες","Natural Sciences","Μαθηματικά","Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["gre"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["10.12681/eadd/1290","http://hdl.handle.net/10442/hedi/1290"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","ΔΙΝΟΥΜΕ Σ' ΑΥΤΗ ΤΗΝ ΕΡΓΑΣΙΑ ΕΝΑ ΚΑΙΝΟΥΡΓΙΟ ΧΑΡΑΚΤΗΡΙΣΜΟ ΜΙΑΣ ΣΦΑΙΡΑΣ ΔΙΑΣΤΑΣΗΣ 2 ΑΝΑΦΟΡΙΚΑ ΜΕ ΤΙΣ ΓΕΩΔΑΙΣΙΑΚΕΣ ΚΑΙ ΣΚΙΟΓΡΑΜΜΕΣ . ΣΥΓΚΕΚΡΙΜΕΝΑ ΑΠΟΔΕΙΚΝΥΟΥΜΕ ΤΑΕΞΗΣ ΘΕΩΡΗΜΑΤΑ. ΘΕΩΡΗΜΑ Α: ΕΣΤΩ Μ ΜΙΑ ΣΥΜΠΑΓΗΣ ΚΑΙ ΓΝΗΣΙΑ ΚΥΡΤΗ ΕΠΙΦΑΝΕΙΑ ΣΤΟΝΕΥΚΛΕΙΔΙΟ ΧΩΡΟ IF3 'Η ΣΤΟΝ ΥΠΕΡΒΟΛΙΚΟ ΧΩΡΟ ΙΗ3. ΥΠΟΘΕΤΟΥΜΕ ΟΤΙ ΟΛΕΣ ΟΙ ΣΚΙΟΓΡΑΜΜΕΣ ΤΗΣ Μ ΕΙΝΑΙ ΙΣΟΜΕΤΡΙΚΕΣ ΜΕΣΩ ΣΤΕΡΕΩΝ ΚΙΝΗΣΕΩΝ ΤΟΥ IF3 'Η ΙΣΟΜΕΤΡΙΩΝ ΤΟΥ ΙΗ3 ΑΝΤΙΣΤΟΙΧΑ. ΤΟΤΕ Η Μ ΕΙΝΑΙ ΜΙΑ ΓΕΩΜΕΤΡΙΚΗ ΣΦΑΙΡΑ ΣΤΟΝ IF3 'Η IH3. ΘΕΩΡΗΜΑ Β: ΕΣΤΩ Μ ΜΙΑ ΕΠΙΦΑΝΕΙΑ ΣΤΟΝ ΕΥΚΛΕΙΔΙΟ ΧΩΡΟ IF3. ΥΠΟΘΕΤΟΥΜΕ ΟΤΙ ΟΛΕΣ ΟΙ ΓΕΩΔΑΙΣΙΑΚΕΣ ΤΗΣ Μ ΕΙΝΑΙ ΙΣΟΜΕΤΡΙΚΕΣ ΜΕΤΑΞΥ ΤΟΥΣ ΜΕΣΩ ΣΤΕΡΕΩΝ ΚΙΝΗΣΕΩΝ ΤΟΥ IF3. ΓΙΑ ΤΗΝ ΑΠΟΔΕΙΞΗ ΤΩΝ ΠΑΡΑΠΑΝΩ ΘΕΩΡΗΜΑΤΩΝ ΑΠΟΔΕΙΚΝΥΟΥΜΕ ΟΤΙ ΟΙ ΣΚΙΟΓΡΑΜΜΕΣ (ΓΕΩΔΑΙΣΙΑΚΕΣ ΣΤΟ ΘΕΩΡΗΜΑ Β) ΕΧΟΥΝ ΩΣ ΚΑΜΠΥΛΕΣ ΤΟΥ IF3 'Η ΙΗ3 ΣΤΑΘΕΡΗ ΚΑΜΠΥΛΟΤΗΤΑ ΚΑΙ ΣΤΡΙΨΗ.ΤΟΥΤΟ ΙΣΧΥΕΙ, ΔΙΟΤΙ ΣΕ ΑΝΤΙΘΕΤΗ ΠΕΡΙΠΤΩΣΗ ΘΑ ΗΤΑΝ ΔΥΝΑΤΟΝ ΝΑ ΚΑΤΑΣΚΕΥΑΣΟΥΜΕ ΕΝΑ ΣΥΝΕΧΕΣ ΔΙΑΝΥΣΜΑΤΙΚΟ ΠΕΔΙΟ ΕΠΙ ΤΗΣ Μ ΤΟ ΟΠΟΙΟΝ ΕΙΝΑΙ ΑΔΥΝΑΤΟΝ ΔΙΟΤΙ Η Μ ΕΙΝΑΙΑΜΦΙΔΙΑΦΟΡΙΣΙΜΗ ΜΕ ΤΗΝ ΣΦΑΙΡΑ S2."]},{"key":"dc:title","label":"Title","values":["1. ΕΠΙΦΑΝΕΙΕΣ ΜΕ ΙΣΟΜΕΤΡΙΚΕΣ ΣΚΙΟΓΡΑΜΜΕΣ 2. ΕΠΙΦΑΝΕΙΕΣ ΜΕ ΙΣΟΜΕΤΡΙΚΕΣ ΓΕΩΔΑΙΣΙΑΚΕΣ","SURFACES WITH CONCRUENT SHADOW-LINES AND SURFACES WITH CONCRUENT GEODESICS"]}]}],"canonical_facts":{"dc:creator":["Χαρίτος, Χαράλαμπος"],"dc:date":["1989"],"dc:description":["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.","ΔΙΝΟΥΜΕ Σ' ΑΥΤΗ ΤΗΝ ΕΡΓΑΣΙΑ ΕΝΑ ΚΑΙΝΟΥΡΓΙΟ ΧΑΡΑΚΤΗΡΙΣΜΟ ΜΙΑΣ ΣΦΑΙΡΑΣ ΔΙΑΣΤΑΣΗΣ 2 ΑΝΑΦΟΡΙΚΑ ΜΕ ΤΙΣ ΓΕΩΔΑΙΣΙΑΚΕΣ ΚΑΙ ΣΚΙΟΓΡΑΜΜΕΣ . ΣΥΓΚΕΚΡΙΜΕΝΑ ΑΠΟΔΕΙΚΝΥΟΥΜΕ ΤΑΕΞΗΣ ΘΕΩΡΗΜΑΤΑ. ΘΕΩΡΗΜΑ Α: ΕΣΤΩ Μ ΜΙΑ ΣΥΜΠΑΓΗΣ ΚΑΙ ΓΝΗΣΙΑ ΚΥΡΤΗ ΕΠΙΦΑΝΕΙΑ ΣΤΟΝΕΥΚΛΕΙΔΙΟ ΧΩΡΟ IF3 'Η ΣΤΟΝ ΥΠΕΡΒΟΛΙΚΟ ΧΩΡΟ ΙΗ3. ΥΠΟΘΕΤΟΥΜΕ ΟΤΙ ΟΛΕΣ ΟΙ ΣΚΙΟΓΡΑΜΜΕΣ ΤΗΣ Μ ΕΙΝΑΙ ΙΣΟΜΕΤΡΙΚΕΣ ΜΕΣΩ ΣΤΕΡΕΩΝ ΚΙΝΗΣΕΩΝ ΤΟΥ IF3 'Η ΙΣΟΜΕΤΡΙΩΝ ΤΟΥ ΙΗ3 ΑΝΤΙΣΤΟΙΧΑ. ΤΟΤΕ Η Μ ΕΙΝΑΙ ΜΙΑ ΓΕΩΜΕΤΡΙΚΗ ΣΦΑΙΡΑ ΣΤΟΝ IF3 'Η IH3. ΘΕΩΡΗΜΑ Β: ΕΣΤΩ Μ ΜΙΑ ΕΠΙΦΑΝΕΙΑ ΣΤΟΝ ΕΥΚΛΕΙΔΙΟ ΧΩΡΟ IF3. ΥΠΟΘΕΤΟΥΜΕ ΟΤΙ ΟΛΕΣ ΟΙ ΓΕΩΔΑΙΣΙΑΚΕΣ ΤΗΣ Μ ΕΙΝΑΙ ΙΣΟΜΕΤΡΙΚΕΣ ΜΕΤΑΞΥ ΤΟΥΣ ΜΕΣΩ ΣΤΕΡΕΩΝ ΚΙΝΗΣΕΩΝ ΤΟΥ IF3. ΓΙΑ ΤΗΝ ΑΠΟΔΕΙΞΗ ΤΩΝ ΠΑΡΑΠΑΝΩ ΘΕΩΡΗΜΑΤΩΝ ΑΠΟΔΕΙΚΝΥΟΥΜΕ ΟΤΙ ΟΙ ΣΚΙΟΓΡΑΜΜΕΣ (ΓΕΩΔΑΙΣΙΑΚΕΣ ΣΤΟ ΘΕΩΡΗΜΑ Β) ΕΧΟΥΝ ΩΣ ΚΑΜΠΥΛΕΣ ΤΟΥ IF3 'Η ΙΗ3 ΣΤΑΘΕΡΗ ΚΑΜΠΥΛΟΤΗΤΑ ΚΑΙ ΣΤΡΙΨΗ.ΤΟΥΤΟ ΙΣΧΥΕΙ, ΔΙΟΤΙ ΣΕ ΑΝΤΙΘΕΤΗ ΠΕΡΙΠΤΩΣΗ ΘΑ ΗΤΑΝ ΔΥΝΑΤΟΝ ΝΑ ΚΑΤΑΣΚΕΥΑΣΟΥΜΕ ΕΝΑ ΣΥΝΕΧΕΣ ΔΙΑΝΥΣΜΑΤΙΚΟ ΠΕΔΙΟ ΕΠΙ ΤΗΣ Μ ΤΟ ΟΠΟΙΟΝ ΕΙΝΑΙ ΑΔΥΝΑΤΟΝ ΔΙΟΤΙ Η Μ ΕΙΝΑΙΑΜΦΙΔΙΑΦΟΡΙΣΙΜΗ ΜΕ ΤΗΝ ΣΦΑΙΡΑ S2."],"dc:identifier":["10.12681/eadd/1290","http://hdl.handle.net/10442/hedi/1290"],"dc:language":["gre"],"dc:publisher":["Πανεπιστήμιο Κρήτης","University of Crete (UOC)"],"dc:subject":["GEODESICS","SHADOW LINES","Surfaces","THEOREME OF POINCARE-HOPF","ΓΕΩΔΑΙΣΙΑΚΕΣ","Επιφάνειες","ΘΕΩΡΗΜΑ POINCARE-HOPF","ΣΚΙΟΓΡΑΜΜΕΣ","Φυσικές Επιστήμες","Natural Sciences","Μαθηματικά","Mathematics"],"dc:title":["1. ΕΠΙΦΑΝΕΙΕΣ ΜΕ ΙΣΟΜΕΤΡΙΚΕΣ ΣΚΙΟΓΡΑΜΜΕΣ 2. ΕΠΙΦΑΝΕΙΕΣ ΜΕ ΙΣΟΜΕΤΡΙΚΕΣ ΓΕΩΔΑΙΣΙΑΚΕΣ","SURFACES WITH CONCRUENT SHADOW-LINES AND SURFACES WITH CONCRUENT GEODESICS"],"dc:type":["PhD Thesis"]},"updated_at":"2026-07-24T02:24:49Z"}