Abstract
We present the notion of weakly metrically regular functions on manifolds. Then, a sufficient condition for a real valued function defined on a complete Rie- mannian manifold to be weakly metrically regular is obtained, and two optimization problems on Riemannian manifolds are considered. Moreover, we present a general- ization of the Palais-Smale condition for lower semicontinuous functions defined on manifolds. Then, we use this notion to obtain necessary conditions of optimality for a general minimization problem on complete Riemannian manifolds.
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