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Additional resources for Infinite Loop Spaces - Hermann Weyl Lectures the Institute for Advanced Study

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Let A(G),A(H),A(K) be the corresponding Lie algebras, and choose some ad(H)-invariant complement P of A(H) in A(G), some ad(K)-invariant complement Q of A(K) in A(H). Then any ad(H)-invariant scalar product on P gives rise to a G-invariant metric h on G/H; similarly any ad(K)-invariant scalar product on Q gives rise to a H-invariant metric on H/K. Now if g is the G-invariant metric on G/K corresponding to the scalar product on P @ Q given by the two preceding scalar products on P and Q and the condition that P and Q are orthogonal, then~rg is a Riemannian submersion from (G/K,g) onto (G/H,h) with totally geodesic fibres isometric to (H/K,k).

I-oj) i 0 Green's Assume theorem. (A) and let I max M Proof: (i) an Einstein (ai_oj)2 + nR . space, n > 3 , = fll g r a d Use do i__ n-1 Theorem. p > 0 be . f. let (M,g) be [6]. an E i n s t e i n space, n ~ 3 , R > 0 , equality . Then max ~ ~ DR . {p 6 M I f(P) M (ii) If in (i) sphere holds of iff = 0} (M,g) curvature is n o w h e r e is R and dense isometrically H ~ nR .

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Infinite Loop Spaces - Hermann Weyl Lectures the Institute for Advanced Study by Adams J. F.

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