  Covariance operators are fundamental in functional data analysis, providing
the canonical means to analyse functional variation via the celebrated
Karhunen--Lo\`eve expansion. These operators may themselves be subject to
variation, for instance in contexts where multiple functional populations are
to be compared. Statistical techniques to analyse such variation are intimately
linked with the choice of metric on covariance operators, and the intrinsic
infinite-dimensionality of these operators. In this paper, we describe the
manifold geometry of the space of trace-class infinite-dimensional covariance
operators and associated key statistical properties, under the recently
proposed infinite-dimensional version of the Procrustes metric. We identify
this space with that of centred Gaussian processes equipped with the
Wasserstein metric of optimal transportation. The identification allows us to
provide a complete description of those aspects of this manifold geometry that
are important in terms of statistical inference, and establish key properties
of the Fr\'echet mean of a random sample of covariances, as well as generative
models that are canonical for such metrics and link with the problem of
registration of functional data.
