Immersing homogeneous spaces in euclidean space

Pub . Ma . UAB Vol . 26 N2 3 Des . 1982 IMMERSING HOMOGENEOUS SPACES IN EUCLIDEANSPACE Howa d Hille * In [4], Lam p o es, amongo he hings, an imme sion e- sul o he eal lag mani old G IR (ni 1 . . .,n S )  = 0(n1+ . . .+ns)/0(n1)x . . .x0(nS) whe e 0(n) is he ealo hogonalg oup . His esul is a specialcaseo he ollowingmo egene al obse a ion . Pnopobí íon 1 . Le G be a compac , connec ed semisimple Lie g oup and H a closedsubg oup . Then ei he G/H is a i -mani old (anal so imme ses in codimension one) o G/H imme _ ses in ~dim,(S) whe e  is heLie algeb a o G . Rema k 2 . The dimension o he ambien Euclidean space is independen o he subg oup H, so one expec s he s onges esul s o small H . Bu i H is a maximal o us (all n i = 1 in abo e example) hen G/H is a 7 -mani old [1] . See Rema k 4 below . P coob . A eal ec o bundle o e G/H is de e minedby an ac ion o H on a eal ec o space . The angen bundle T(G/H)  comes om he adjoin ac ion o H on g/~,g = Lie EL Le n deno e he bundle o e ' G/H coming om he adjoin ac ion o H on B . C ea ly n is i ial since he ac ion ex ends o all o G . The e is a bundle epimo phism n- .T(G/H) * Suppo ed by he Alexande onHumbold -S i ung . 43 which necessa ily spli s (see o example 5,p .461) . Since dim(71) = dim(9), he heo em o Hi sch 131 yields he desi ed imme sion . (2) in R  ; whe e n = nl+ .. .+ns' Cono .21any 3 .  (Lam)  The mani old  GIR(ni, . . . . ns) Pxoo4 . Obse e dim 0(n) = (2) . Cono .elany 5 . These esui s a e aao as s ong a  heseo Lam (41 . . inme ses Remank 4 . Tha hese imme sions a e eally in e es ing ollows om [21  whe e i was shown ha i s = 2, ni = n2, one ob ains a bes possible imme sion o he eal G assmannian . Simila ly, one ob ains in he complex and qua e nioniccases . 12 (a) Ga(nl, . . .,ns) imme ses in IR n (b) GIH(nl, . . .,ns) imme ses in IR 2 2 n + n REFERENCES 1 . A . Bo el and F . Hi zeb uch, Cha ac e is ic classes and homogeneous spaces, III, Ame . J . Ma h . .82(1960)491-504 . 2 . H . Hille and R . S ong . Imme sion dimension o eal G assmannians,Ma h . Ann . 255(1981), 361-367 . 3 . M . Hi sch, Imme sions o mani olds,T ans . Ame . Ma h . Soc . 93(1959), 242-276 . 4 . K .Y . Lam, A o mula o he angen bundleo lag mani olds and ela ed mani olds, T ans .Ame .Ma h .Soc . 213(1975)305-314 . 5 . P . Lb le and L . Smi h, Line bundles o e amed mani olds, Ma h . Z . 138(1974), 35-52 . Ma hema isches Ins i u de Geo g-Augus -Uni e si á Bunsens . 3-5 3400 GS ingen