Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks
From the abstract: This paper deals with the computation of the sectional curvature for the manifolds of \(N\) landmarks in \(D\) dimensions with respect to the Riemannian metric induced by the diffeomorphism group. The inverse metric tensor in local coordinates for these manifolds has the property that each of its elements depends on at most \(2D\) points out of \(ND\) coordinates. This makes the matrices of partial derivatives of the cometric very sparse in nature, and so it solves the highly nontrivial problem of developing a formula, called the Mario's formula, that expresses the sectional curvature in terms of the cometric and its first and second partial derivatives. We apply such a formula to the manifolds of landmarks, explore the case of geodesics on which only two points have nonzero momenta, and compute the sectional curvature of \(2\)-planes spanned by the tangents to such geodesics.
scientific article; zbMATH DE number 6082565