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By McCoy J. A.

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Extra resources for A Bernstein property of solutions to a class of prescribed affine mean curvature equations

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2 (3) analoge globale (Definitions-) Gleichungen f~r ~7xY. (iii) Seien ~Z,~ zu T , ~ ^ (M) g e m ~ L~(TM;TM) (ii) gegeben. In Verallgemei- nerung des am SchluS von (ii) Festgestellten gilt mit den dortigen Argumenten (~) g(TxY,Z) = g(~xY,Z) + 89 + g(Y,(T-~)(Z,Tf-X)) - - g (Tf- ~(T-~) (Y, Z) ~ fHr alle Morphismen f:N ---~M und alle X ~ ( N ) , Y , Z ~ ( f ) . ))E~L 3 (M), sogar Schnitt in L (M) (TM) ist. Ist dies erf~llt, so gilt nach (~) ~ X Y = VxY + (T-~)(X,Y). 7(iv) betrachteten affinen Raumes ~ aller Zusammenh~nge auf M.

H. ~ und S,exp keine Beziehung vorausgesetzt wird). 3 impliziert dar~berhinaus, dab U(p) sogar so gew~hlt werden kann, dab f~r ein ~ ~ + mit o ~ < i n f ~(p) gilt: FUr alle q ~U(p) liegt U(p) P~Cr~ -1 ganz im Definitionsbereich der nat~rlichen Karte expq :Bs >Bg(Oq). 3 gibt es zu jedem p ~ M eine Umgebung V(p) yon p und ~ > o , so daS (~,exp): q ~ J B~(o ) >~_~ ~q,B~(q)) Diffeomorohismus zwischen diesen -1 eXpq : B ~ ( q ) q~V(p) ~ ~ ~eV(p) o f f e n e n Mengen a u s 'I'M bzw M~M i s t >B[(Oq) W~hle nun %g(o,~) U(p)• yon ( p , p ) Dieses da~ fur und e i n e U(p) e r f G l l t tion (q,U(p))c (q,Bs jedes Umgebung U ( p ) yon p , i n dem o s die (insbesondere qgV(p) n a t [ r l i c h e Behauptung, ist K a r t e um q ) .

Es gibt genau einen Zusammen- hang ~,K auf M, der riemannsch und torsionsfrei hang heist der Levi-Civita-Zusa~nenhan~ ist. Dieser Zusammen- yon (H,g). : Dieser Satz ben~tigt keine Partitionen der Eins, falls man die dayon unabh~ngigen Charakterisierungen zugrundelegt yon riemannsch und torsionsfrei (also die mittels Christoffelsymbolen oder mittels der "NatHrlichkeit offene Teilmenge U von M i s t oder mittels Kurven hinsichtlich Einschr~nkungen": K/T2~der Levi-Civita-Zusammenhang FHr jede yon (U,glu), vgl.

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A Bernstein property of solutions to a class of prescribed affine mean curvature equations by McCoy J. A.


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