• Bastiani calculus
• Frechet spaces
• infinite dimensional manifolds
• weak metrics
• curvature

The goal of this work is to provide a solid foundation for the computation of the sectional curvature in Frechet manifolds. To begin with, we review the basics of calculus for functions between Frechet spaces, in the framework originally considered by Michal and Bastiani. Then we introduce manifolds locally modelled on Frechet spaces, with a focus on weak Riemannian metrics, which have not been systematically treated by the literature yet. In particular we provide two useful formulae for the sectional curvature, known for classical finite dimensional manifolds, which extend to our Frechet setting under suitable assumptions. Finally, we apply the formulae to compute the curvature of some manifolds with vanishing geodesic distance, in order to provide additional details on a conjecture of Michor and Mumford connecting vanishing distances with locally positively unbounded sectional curvature.