The L-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian manifold (N, g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the L-metric.

There has been new interest in the successful application of differential geometric methods in the control of p.d.e.’s. (See for example Contemporary Mathematics #268, AMS 2000, particularly the article by the present authors). Here we describe those results, and some newer results using the methods of curvature flows. We also present an example for which control is possible but cannot be prove...

Let (M, g) be a compact Riemannian spin manifold. The AtiyahSinger index theorem yields a lower bound for the dimension of the kernel of the Dirac operator. We prove that this bound can be attained by changing the Riemannian metric g on an arbitrarily small open set.

We show with a new variational approach that any Riemannian metric on a multiply connected schlicht domain in R can be represented by globally conformal parameters. This yields a “Riemannian version” of Koebe’s mapping theorem. Mathematics Subject Classification (2000): 30C20, 49J45, 49Q05, 49Q10, 53A10

In Theorem 1, we generalize some results of Szabó [Sz1, Sz2] for Berwald metrics that are not necessarily strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F . As an application we show (Corollary 3) that every Berwald projectively flat metric is a Minkowski metric; this statement is a “Berwald” version of Hilbert’s 4th problem...

The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function φ in the form K D(H,K) = ∑ i,j φ(λi, λj) −1TrPiHPjK when ∑ i λiPi is the spectral decomposition of the foot point D and the Hermitian matrices H,K are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping D 7→ ...

The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function φ in the form K D(H,K) = ∑ i,j φ(λi, λj) −1TrPiHPjK when ∑ i λiPi is the spectral decomposition of the foot point D and the Hermitian matrices H,K are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping D 7→ ...

The Yamabe problem, solved by Trudinger [14], Aubin [1], and Schoen [12], asserts that any Riemannian metric on a closed manifold is conformal to a metric with constant scalar curvature. Escobar [8], [9] has studied analogous questions on manifolds with boundary. To fix notation, let (M,g) be a compact Riemannian manifold of dimension n ≥ 3 with boundary ∂M . We denote by Rg the scalar curvatur...

Gradient flows in the Sobolev space H1 have been shown to enjoy favorable regularity properties. We propose a generalization of prior approaches for Sobolev active contour segmentation by changing the notion of distance in the Sobolev space, which is achieved through treatment of the function and its derivative in Riemannian manifolds. The resulting generalized Riemannian Sobolev space provides...

We study well-posedness of a class of nonlocal interaction equations with spatially dependent mobility. We also allow for the presence of boundaries and external potentials. Such systems lead to the study of nonlocal interaction equations on subsets M of Rd endowed with a Riemannian metric g. We obtain conditions, relating the interaction potential and the geometry, which imply existence, uniqu...