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By Slavenas P.

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Extra resources for A Note on the Triple System, lambda Tauri

Example text

Die Zuordnung -multilinear und total anti- symmetrisch. Auf diese Weise haben wir jedem ( ~ E / ~ , ( ~ in natOrlicher Weise ein Element der ~usseren Algebra ~ber dew ~ - m o d u l ~ zugeordnet. Letztere bezeichnen wir nit / ~ 4 ~ und ~ @ ~ 3 " bezeichne die pte Stufe. Man kann zeigen, dass die betrachtete Zuordnung ein Ieomorphismus auf ist, der alle algebraischen Strukturen respektiert. ~ ~, U , dann kann so dargestellt werden co , -" ~'i' 4^ (7) Hier sind ~ ® i , . i p ~ die Komponenten yon (7) eind d i e ~ total antieymmetriech.

E~i~-. ~ X ~ ~-.. ~. _ (alle oberen Indizes) X;,,~@ ~- (alle unteren Indizes) (6) SpeziBll fur ~ E ~ $ (7) 26 ~4. Differentialformen Der Cartaneche Kalk~l der Differentialformen kann in der ART besonders nOtzlich eingesetzt warden. FUr seine Entwicklung beginnen wir mit einigen algebraischen Vorbereitungen. 1. Aeussere Alqebra Es seien A eine kommutative, aseoziative, unit~re 1~-Algebra und E ein A-Modul. Im folgenden ist entweder A = ~ und E ein l~-Vektorraum endlicher Dimension, oder ~ ~ ;14diffb.

Deshalb sind in einer Umgebung V won ~ T x ( ~ die zugeh~rigen Geod~ten f~r ~ [ 0 , 1 ] definiert. 0ie Abbildung exp x bildet a J ~ N J in den munkt ~(i) ab, wobei ~ O ~ - ~ J " und ~ eine Geod~te ist. 0ffensichtlich ist ~ j ~ ~ ~ . Setzen wir darin s = i , so folgt ~ - J q ~ 5 " ~ - - ~ . - 5~ f~r jedes 43-~'~-~ ] . Als Anwendung des impliziten Funktionentheorems erhalten wit deshalb d~n Satz 2. Die Abbildung exp x ist ein Diffeomorphismus von einer Umgebung yon ~ ~T~-{ auf sine Umgebung von ~ ~ - ~ .

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