For any ε > 0, we construct an explicit smooth Riemannian metric on the sphere Sn, n ≥ 3, that is within ε of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is ε-dense in the unit tangent bundle. Moreover, for any ε > 0, we construct a smooth Riemannian metric on S, n ≥ 3, that is within ε of the round metric and has a geodesic for which the complemen...

We study an inverse problem for a non-compact Riemannian manifold whose ends have the following properties : On each end, the Riemannian metric is assumed to be a short-range perturbation of the metric of the form (dy) + h(x, dx), h(x, dx) being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to (dy) +h(x, dx). Given two su...

We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is quadratic in curvature and study the resulting system of Euler–Lagrange equations. In the first part of the paper we lo...

Conformal geometry is at the core of pure mathematics. Conformal structure is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields. This work introduces a theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces—disc...

A non-autonomous sub-Riemannian problem is considered: Since periodicity with respect to the independent variable is assumed, one can define the averaged problem. In the case of the minimization of the energy, the averaged Hamiltonian remains quadratic in the adjoint variable. When it is non-degenerate, a Riemannian problem and the corresponding metric can be uniquely associated to the averaged...

Conformal geometry is in the core of pure mathematics. It is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields. This work introduces a theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces – discrete surface Ri...

We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian opt...

We propose a novel Riemannian preconditioning approach for the tensor completion problem with rank constraint. A Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manif...

Generalized tensor analysis in the sense of Colombeau’s construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a “Fun...

We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case Emb(R, R) which turns out to be Burgers’ equation. Then we derive the geodesic equation, the curvature, and the Jacobi equation of a right invariant Riemannian metric on an infinite dimensional Lie group, which we...