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.

**Read Online or Download Differential Geometry and Differential Equations: Proceedings of a Symposium, held in Shanghai, June 21 – July 6, 1985 PDF**

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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).