The problem of minimizing the cost functional of an Optimal Control System through the use of constrained Variational Calculus is a generalization of the geodetic problem in Riemannian geometry. In the framework of a geometric formulation of Optimal Control, we define a metric structure associated to the Optimal Control System on the enlarged space of state and time variables, such that the min...

We show that on every compact spin manifold admitting a Riemannian metric of positive scalar curvature Friedrich’s eigenvalue estimate for the Dirac operator can be made sharp up to an arbitrarily small given error by choosing the metric suitably.

The Virasoro-Bott group endowed with the right-invariant L2metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.

Given a compact Riemannian manifold on which a compact Lie group acts by isometries, it is shown that there exists a Riemannian foliation whose leaf closure space is naturally isometric (as a metric space) to the orbit space of the group action. Furthermore, this isometry (and foliation) may be chosen so that a leaf closure is mapped to an orbit with the same volume, even though the dimension o...

In this note we discuss conditions under which a linear connection on a manifold equipped with both a symmetric (Riemannian) and a skew-symmetric (almost-symplectic or Poisson) tensor field will preserve both structures. If (M, g) is a (pseudo-)Riemannian manifold, then classical results due to T. Levi-Civita, H. Weyl and E. Cartan [7] show that for any (1, 2) tensor field T i jk which is skew-...

Let M be a closed simply connected manifold and 0 < δ ≤ 1. Klingenberg and Sakai conjectured that there exists a constant i0 = i0(M, δ) > 0 such that the injectivity radius of any Riemannian metric g on M with δ ≤ Kg ≤ 1 can be estimated from below by i0. We study this question by collapsing and Alexandrov space techniques. In particular we establish a bounded version of the KlingenbergSakai co...

A mesh smoothing method based on Riemannian metric comparison is presented in this paper. This method minimizes a cost function constructed from a measure of metric non-conformity that compares two metrics: the metric that transforms the element into a reference element and a specified Riemannian metric, that contains the target size and shape of the elements. This combination of metrics allows...

We study the Dirichlet problem for a class of fully nonlinear elliptic equations related to conformal deformations of metrics on Riemannian manifolds with boundary. As a consequence we prove the existence of a conformal metric, given its value on the boundary as a prescribed metric conformal to the (induced) background metric, with a prescribed curvature function of the Schouten tensor.

Spaces with radially symmetric curvature at base point p are shown to be diffeomorphic to space forms. Furthermore, they are either isometric to Rn or Sn under a radially symmetric metric, to RPn with Riemannian universal covering of Sn equipped with a radially symmetric metric, or else have constant curvature outside a metric ball of radius equal to the injectivity

We show that, for each value of α ∈ (−1, 1), the only Riemannian metrics on the space of positive definite matrices for which the ∇ and ∇ connections are mutually dual are matrix multiples of the Wigner-Yanase-Dyson metric. If we further impose that the metric be monotone, then this set is reduced to scalar multiples of the WignerYanase-Dyson metric.