Stratifications of spaces with nonnegative sectional curvature and their relation to global structures and invariants

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Stratifications of Riemannian manifolds arise in various ways. There is a strong interplay between the global structures and invariants on the stratified manifold and the corresponding structures and invariants on subobjects given by the stratification. Our goal is to use this interaction for selected spaces of nonnegative sectional curvature in either of the two or even in both directions depending on the situation: Somtimes a suitable stratification is the key to solve a global question for the stratified space. Sometimes a global question for a Riemannian manifold is hard to answer when it is considered intrinsically, but it may become solvable when one employs a certain ambient structure. The problem always is to find appropriate stratifications. In order to do this we study a large amount of stratifications in detail.