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Relative generic singularities of the exponential map. (English) Zbl 0836.58034
Authors’ abstract: “The authors investigate generic properties of the exponential map defined as \(\text{Exp}(v)= h^v_1\), for a vector field \(v\in \Gamma(g)\) (where \(\Gamma(g)\) denotes the Lipschitz sections of a subset \(g\) of vector subspaces of the sheaf of all smooth vector fields on a smooth manifold \(M\) and \(h^v_t\) is the flow generated by \(v\)).
They study restrictions of Exp to a suitable class of germs of submanifolds of \(M\), and find necessary and sufficient conditions for a subsheaf \(h\subset z\) such that for a generic vector field \(v\in \Gamma(h)\) the singularities of the flow of \(v\) arise as singularities of the flow of a generic vector field belonging to \(\Gamma(g)\). Applications of these results to Riemannian and sub-Riemannian geometry are presented and the context is chosen to include a theorem of A. Weinstein concerning the Riemannian exponential map”.

MSC:
37C10 Dynamics induced by flows and semiflows
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References:
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