Download Differential Geometry and Differential Equations: by Robert L. Bryant (auth.), Chaohao Gu, Marcel Berger, Robert PDF

By Robert L. Bryant (auth.), Chaohao Gu, Marcel Berger, Robert L. Bryant (eds.)

The DD6 Symposium was once, like its predecessors DD1 to DD5 either a learn symposium and a summer season seminar and focused on differential geometry. This quantity encompasses a choice of the invited papers and a few extra contributions. They disguise fresh advances and valuable traits in present learn in differential geometry.

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Additional info for Differential Geometry and Differential Equations: Proceedings of a Symposium, held in Shanghai, June 21 – July 6, 1985

Example text

H. S. , The functions of Schl~fli and Lobatschefsky, Quart. J. Math. 6 (1935), 13-29. The Imbedding P r o b l e m of R i e m a n n i a n G l o b a l l y Symmetric Spaces of the Compact Type Hu Yi ( ~ ~ )* Nankai University Introduction A compact simple Lie group G with the t w o - s i d e d invariant metric which is induced by the K i l l i n g form of its Lie algebra is n a t u r a l l y isometric to a compact irreducible R i e m a n n i a n g l o b a l l y symmetric space of type II. Suppose ~ to be the c o v e r i n g group of G, it is w e l l - k n o w n that there is o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n the set of the equivalent classes of irreducible complex r e p r e s e n t a t i o n s of G and the set of the highest weights of these representations.

T4~Z + G2 A=t111+t212 tl,t2sZ + Part II. Minimal imbeddings of compact Riemannian symmetric spaces of type I into spheres ~i. Class 1 representations Now let G be a connected o an involutive VII, simply connected compact a u t o m o r p h i s m of G, and K=Go={x~Gl~x=x}. [i], L is connected and G/K is a simply connected Riemannian globally representation X:M=G/K--~S isometric of symmetric space. immersion (see Prop. 1, that X is an imbedding the identity component ~eV, irreducible class 1 then the map X(gK)=p(g)v is a minimal [4]).

Sentation o f ~ , (dP~)e=p. , (A I i ) 2 (~i,~i) ~0 Proof. (mod ~), for any i, ni=l. It is enough to prove that p~(idv)={e,exp~ 2z,F-~M[I (A,M~) £0(mod ~) , ni:l} Consider now p~(exp2z/L-[M[)=eP(2~/L~M[ ) , its characteristic the form of roots have e(l'2~/~fM[ ) , where I is any weight of p. But we have l=i-~il-~i2 ..... ~is ' where ~il,''',els are some simple roots, therefore, if (I,M[) E0 (mod ~), then (I,2Z/L-[M~)H 0 (mod 2z/-c[~), SO, e (i' 2~/C~M~ ) =i. Since p ~ ( G ) c U (V~), pd(exp2~/~fM~)=idv~ , it follows that the right- hand side is included in p~1 (id).

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