Abstract Riemannian stratifications (Q1107175)

An abstract stratification A is considered, of which each stratum X is a Riemannian manifold. Conditions for the possibility to endow A with a metric are given. Stronger conditions (compatibility properties and uniqueness of the prolongation) which make the metric space A a G-space are also investigated. The collection of Riemannian metrics on strata satisfying the compatibility properties is called compatible and the abstract stratification endowed with compatible collection of Riemannian metrics on strata is called Riemannian. The final conclusion is that every abstract stratification has Riemannian structure.