By K. Leichtweiß (auth.), Dirk Ferus, Wolfgang Kühnel, Udo Simon, Bernd Wegner (eds.)
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Extra info for Global Differential Geometry and Global Analysis: Proceedings of the Colloquium Held at the Technical University of Berlin, November 21 – 24, 1979
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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).
Global Differential Geometry and Global Analysis: Proceedings of the Colloquium Held at the Technical University of Berlin, November 21 – 24, 1979 by K. Leichtweiß (auth.), Dirk Ferus, Wolfgang Kühnel, Udo Simon, Bernd Wegner (eds.)