Steepest descent on real flag manifolds

Among the compact homogeneous spaces, a very distinguished subclass is formed by the (generalized) real flag manifolds which by definition are the orbits of the isotropy representations of Riemannian symmetric spaces (sorbits). This class contains most compact symmetric spaces (e.g. all hermitian ones), all classical flag manifolds over real, complex and quaternionic vector spaces, all adjoint orbits of compact Lie groups (generalized complex flag manifolds) and many others. They form the main examples for isoparametric submanifolds and their focal manifolds (so called constant principal curvature manifolds); in fact for most codimensions these are the only such spaces (cf. [7], [9], [8]). Any real flag manifold M enjoys two very peculiar geometric properties: It carries a transitive action of a noncompact Lie group G, and it is embedded in euclidean space as a taut submanifold, i.e. almost all height or coordinate functions are perfect Morse functions (at least for Z/2-coefficients). The aim of our paper is to link these two properties by the following theorem: The gradient flow of any height function is a one-parameter subgroup of G where the gradient is defined with respect to a suitable homogeneous metric s on M ; in the case where M is an adjoint orbit s is a homogeneous Kähler metric. In other words, the lines of steepest descend for the height function (gradient flow lines) are obtained by applying a one-parameter subgroup of G. This is an elementary fact when M is a euclidean sphere and G its conformal group: The gradient of any height function is a conformal vector field. For adjoint orbits, the (generalized) complex flag manifolds, this fact was observed earlier by Guest and Ohnita [4]. Our more general result can be derived from their theorem since real flag manifolds are contained in complex flag manifolds