A diffeomorphism-invariant metric on the space of vector-valued one-forms (Q2025307)

The article under review introduces a diffeomorphism-invariant Riemannian metric on the space of vector-valued one-forms on a compact (orientable) manifold. As this metric does not involve derivation of vector component, the authors call this an \(L^2\)-type metric. Moreover, the new metric is connected to the Ebin metric (cf. [\textit{D. G. Ebin}, in: Global Analysis, Proc. Sympos. Pure Math. 15, 11--40 (1970; Zbl 0205.53702)]) on the space of all Riemannian metrics via a Riemannian submersion. In a bit more detail, the authors consider the manifold of vector-valued one-forms of full rank (where a one form is of full rank if it is injective). Then the geodesic equations are calculated. Recall that on an infinite-dimensional manifold the geodesic distance may be degenerate and thus fail to be a metric (see [\textit{P. W. Michor} and \textit{D. Mumford}, Doc. Math. 10, 217--245 (2005; Zbl 1083.58010)]). In the case at hand, the geodesic equations admit explicit solution formulae and an explicit computation of the geodesics. As a consequence, it is proved that the geodesic distance is non-degenerate. Furthermore, the authors prove that the Riemannian metric leads to a geodesically and metrically incomplete space. Then completely geodesic subspaces are studied. Finally, the sectional curvature of the infinite-dimensional Riemannian metric is calculated. Here the main observation is that, depending on the dimension of the base manifold and the target space, the sectional curvature either has a semidefinite sign or admits both signs.