  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.
