P(A'(E)) P(A'(fO) +p(E) 7T h-* A ' ( 7 r ) 3) One has the following embeddings: N= + 'P) -- 1l ) (Z' +7 "*)/2 Gl>l+p ,(E)^E», ( B ) ^ ^w, Hh w i t h tfiV= = ( ( (Z' + )V/ 22 G IIll+ l l + ,(B)^^,with N N G+ ,with Gttl+p (E)-+E ,with *^ == ( ' t, P ) +p(E)^E 22 PROOF.

Let us consider, for example, the embedding G*l+ (E) -> SN C E, dimE = N + 1 , where N = (l+p) - 1 . The Lie group of isometries of the manifold Gjl+ (E) as a symmetric space is SO(l+p) . Let X — {xij)1<-*
*

*The corresponding submanifold is the flat torus e s , where s C m can be taken to be ( s = - 0 -X* X 0 \ xi ,X: o o O Xj G R \ with p < I (similarly when p > I). | | Similarly we get the results listed in the following table. TAB. 13 - ( S t a n d a r d e m b e d d i n g s of G r a s s m a n n manifolds in E u clidean space). 1) Let L be a connected real non-compact semisimple Lie group. Let A(L) be the Lie algebra of L and K a maximal compact subgroup, with Lie algebra A(K). Let A(L) = A{K) 0 p be a Cart an decomposition. *

### Geometry of PDEs and mechanics by Agostino Prastaro

by George

4.4