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Published online by Cambridge University Press: 08 December 2022
In this chapter, we will discuss Riemannian metrics on infinite-dimensional spaces. Particular emphasis will be placed on the new challenges which arise on infinite-dimensional spaces. One new feature is that Riemannian metrics comes in several flavours on infinite-dimensional spaces. These are not present in the finite dimensional setting. The strongest flavour (as we shall see) is the notion of a strong Riemannian metric which is treated in classical monographs on infinite-dimensional geometry. It is also the most restrictive setting as it forces one to work on Hilbert manifolds. Of greater interest are for this reason the weak Riemannian metrics which are however possibly ill behaved. As an example we will discuss at length geodesics for Riemannian metrics on infinite-dimensional spaces. The aim is to exhibit examples of Riemannian manifolds for which the finite dimensional theory breaks down and the geodesic distance vanishes.
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