A diffeomorphism-invariant metric on the space of vector-valued one-forms

Abstract

In this article we introduce a diffeomorphism-invariant Riemannian metric on the space of vector valued one-forms. The particular choice of metric is motivated by potential future applications in the field of functional data and shape analysis and by connections to the Ebin metric on the space of all Riemannian metrics. In the present work we calculate the geodesic equations and obtain an explicit formula for the solutions to the corresponding initial value problem. Using this we show that it is a geodesically and metrically incomplete space and study the existence of totally geodesic subspaces. Furthermore, we calculate the sectional curvature and observe that, depending on the dimension of the base manifold and the target space, it either has a semidefinite sign or admits both signs.