The paper proposes a Riemannian Manifold Hamiltonian Monte Carlo sampler to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The method provides a fully automated adaptation mechanism that circumvents the costly pilot runs required to tune proposal densities for Metropolis-Hastings or in...

We consider the problem Pcurve of minimizing ∫ ` 0 √ β 2 + |κ(s)|2ds for a planar curve having fixed initial and final positions and directions. Here κ is the curvature of the curve with free total length `. This problem comes from a 2D model of geometry of vision due to Petitot, Citti and Sarti. Here we will provide a general theory on cuspless sub-Riemannian geodesics within a sub-Riemannian ...

It was proved in [25] that for a contact Riemannian manifold with non-integrable almost complex structure, the Yamabe problem is subcritical in the sense that its Yamabe invariant is less than that of the Heisenberg group. In this paper we give a complete proof of the solvability of the contact Riemannian Yamabe problem in the subcritical case. These two results implies that the Yamabe problem ...

Harmonic maps are natural generalizations of harmonic functions and are critical points of the energy functional defined on the space of maps between two Riemannian manifolds. The Liouville type properties for harmonic maps have been studied extensively in the past years (Cf. [Ch], [C], [EL1], [EL2], [ES], [H], [HJW], [J], [SY], [S], [Y1], etc.). In 1975, Yau [Y1] proved that any harmonic funct...

In pseudo-Riemannian geometry the spaces of space-like and timelike geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generaliza...

The classification of the equilibrium shapes that a self-gravitating fluid can take in a Riemannian manifold is a classical problem in Mathematical Physics. In this paper it is proved that the equilibrium shapes are isoparametric submanifolds. Some geometric properties of them are also obtained, e.g. classification and existence for some Riemannian spaces and relationship with the isoperimetric...

In this article we extend the computational geometric curve reconstruction approach to curves in Riemannian manifolds. We prove that the minimal spanning tree, given a sufficiently dense sample set of a curve, correctly reconstructs the smooth arcs and further closed and simple curves in Riemannian manifolds. The proof is based on the behaviour of the curve segment inside the tubular neighbourh...

We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the ma...

In this article, we study the smoothness of Riemannian submersions for open manifolds with non-negative sectional curvature. Suppose thatM is a C-smooth, complete and non-compact Riemannian manifold with nonnegative sectional curvature. Cheeger-Gromoll [ChG] established a fundamental theory for such a manifold. Among other things, they showed that M admits a totally convex exhaustion {Ωu}u≥0 of...

We give results about the L-kernel and the spectrum of the Dirac operator on a complete Riemannian manifold which admits a conformal compactifation to a compact manifold.