Immersing homogeneous spaces in euclidean space

Pub . Mat . UAB Vol . 26 N2 3 Des . 1982 IMMERSING HOMOGENEOUS SPACES IN EUCLIDEANSPACE Howard Hiller* In [4], Lam proves, amongother things, an immersion result for the real flag manifold G IR (ni 1 . . .,n S )  = 0(n1+ . . .+ns)/0(n1)x . . .x0(nS) where 0(n) is the realorthogonalgroup . His result is a specialcaseof the followingmoregeneral observation . Pnopobítíon 1 . Let G be a compact, connected semisimple Lie group and H a closedsubgroup . Then either G/H is a ir-manifold (anal so immerses in codimension one) or G/H immer_ ses in ~dim,(S) where  is theLie algebra of G . Rematk 2 . The dimension of the ambient Euclidean space is independent of the subgroup H, so one expects the strongest results for small H . But if H is a maximaltorus (all n i = 1 in above example) then G/H is a 7r-manifold [1] . See Remark 4 below . Prcoob . A realvectorbundle over G/H is determinedby an action of H on a realvectorspace . The tangent bundle T(G/H)  comes from the adjoint action of H on g/~,g = Lie EL Let n denote the bundle over ' G/H comingfrom the adjoint action of H on B . Crearly n is trivial since the action extends to all of G . There is a bundle epimorphism n- .T(G/H) * Supported by the Alexander vonHumboldt-Stiftung . 43 which necessarily splits (see for example f5,p .461) . Since dim(71) = dim(9), the theorem of Hirsch 131 yields the desired immersion . (2) in R  ; where n = nl+ .. .+ns' Cono .21any 3 .  (Lam)  The manifold  GIR(ni, . . . . ns) Pxoo4 . Observe dim 0(n) = (2) . Cono .elany 5 . Theseresuits are aaot as strong a  theseof Lam (41 . . inmerses Remank 4 . Thatthese immersions are really interesting follows from [21  where it was shownthat if s = 2, ni = n2, one obtains a bestpossible immersion for the real Grassmannian . Similarly, one obtains in the complex and quaternioniccases . 12 (a) Ga(nl, . . .,ns) immerses in IR n (b) GIH(nl, . . .,ns) immerses in IR 2 2 n + n REFERENCES 1 . A . Borel and F . Hirzebruch, Characteristic classes and homogeneous spaces, III, Amer . J . Math . .82(1960)491-504 . 2 . H . Hiller and R . Stong . Immersion dimension for real Grassmannians,Math . Ann . 255(1981), 361-367 . 3 . M . Hirsch, Immersions of manifolds,Trans . Amer . Math . Soc . 93(1959), 242-276 . 4 . K .Y . Lam, A formula forthe tangentbundleof flag manifolds and related manifolds, Trans .Amer .Math .Soc . 213(1975)305-314 . 5 . P . Lbffler and L . Smith, Line bundles over framed manifolds, Math . Z . 138(1974), 35-52 . Mathematisches Institut der Georg-August-Universitát Bunsenstr . 3-5 3400 GSttingen