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Геометрија четвородимензионих нилпотентних Лијевих група

Geometry of four-dimensional nilpotent Lie groups

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2015
Disertacija3788.pdf (1.432Mb)
Sukilovic_Tijana.pdf (127.2Kb)
Author
Šukilović, Tijana Z.
Mentor
Vukmirović, Srđan
Committee members
Jovanović, Božidar
Đorić, Mirjana
Antić, Miroslava
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Abstract
U ovom radu izlažemo klasifikaciju levo-invarijantnih metrika proizvoljne signature na četvorodimenzionim nilpotentnim Lijevim grupama. Detaljno ispitujemo njihovu geometriju, sa posebnim naglaskom na grupe holonomija i dekompozabilnost metrika. Takođe, potpuno opisujemo grupe izometrija i nalazimo primere metrika za koje su zadovoljene stroge nejednakosti Isplit < Iaut < I: U sluqaju metrika neutralne signature na nilpotentnim Lijevim grupama sa degenerisanim centrom dobijamo Vokerove metrike. Formulišemo i dokazujemo potreban i dovoljan uslov da one dopuštaju nilpotentnu grupu izometrija. Na kraju, dajemo odgovor na pitanje egzistencije projektivno ekvivalentnih metrika. Pokazujemo da su na četvorodimenzionim nilpotentnim Lijevim grupama sve levo-invarijantne metrike ili geometrijski rigidne ili postoje njima projektivno ekvivalentne metrike koje su istovremeno i afino ekvivalentne. Iako su sve afino ekvivalentne metrike levo-invarijantne, Njihova signatura može biti različita.
In the present work we classify left invariant metrics of arbitrary signature on four-dimensional nilpotent Lie groups. Their geometry is extensively studied with special emphasis on holonomy groups and decomposability of metrics. Also, isometry groups are completely described and we give examples of metrics where strict inequalities Isplit < Iaut < I hold. It is interesting that Walker metrics appear as the underlying structure of neutral signature metrics on the nilpotent Lie groups with degenerate center. We fnd necessary and suffient condition for them to locally admit nilpotent group of isometries. Finally, we solve the problem of projectively equivalent metric on four-dimensional nilpotent Lie groups by showing that left invariant metric is either geometrically rigid or have projectively equivalent metrics that are also affinely equivalent. All affinely equivalent metrics are left invariant, while their signature may change.
Faculty:
Универзитет у Београду, Математички факултет
Date:
02-04-2015
Projects:
  • Geometry, Education and Visualization With Applications (RS-174012)
Keywords:
nilpotentne Lijeve grupe / nilpotent Lie group / grupe holonomija / grupe izometrija / geodezijski ekvivalentne metrike / holonomy groups / isometry groups / geodesically equivalent metrics
[ Google Scholar ]
Handle
https://hdl.handle.net/21.15107/rcub_nardus_5802
URI
http://eteze.bg.ac.rs/application/showtheses?thesesId=3233
https://nardus.mpn.gov.rs/handle/123456789/5802
https://fedorabg.bg.ac.rs/fedora/get/o:11614/bdef:Content/download
http://vbs.rs/scripts/cobiss?command=DISPLAY&base=70036&RID=47602959

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