Geodesic Normal Coordinates Research Articles

On Momentum Operators Given by Killing Vectors Whose Integral Curves Are Geodesics

The paper considers momentum operators on intrinsically curved manifolds. Given that momentum operators are Killing vector fields whose integral curves are geodesics, the corresponding manifold is flat or of the compact type with positive constant sectional curvature and dimensions equal to 1, 3, or 7. Explicit representations of momentum operators and the associated Casimir element are discussed for the 3-sphere S3. It is verified that the structural constants of the underlying Lie algebra are proportional to 2 ℏ/R, where R is the curvature radius of S3 and ℏ is the reduced Planck’s constant. This results in a countable energy and momentum spectrum of freely moving particles in S3. The maximal resolution of the possible momenta is given by the de Broglie wave length, λR=πR, which is identical to the diameter of the manifold. The corresponding covariant position operators are defined in terms of geodesic normal coordinates, and the associated commutator relations of position and momentum are established.

A convergent finite difference method for optimal transport on the sphere

We introduce a convergent finite difference method for solving the optimal transportation problem on the sphere. The method applies to both the traditional squared geodesic cost (arising in mesh generation) and a logarithmic cost (arising in the reflector antenna design problem). At each point on the sphere, we replace the surface PDE with a Generated Jacobian equation posed on the local tangent plane using geodesic normal coordinates. The discretization is inspired by recent monotone methods for the Monge-Ampère equation, but requires significant adaptations in order to correctly handle the mix of gradient and Hessian terms appearing inside the nonlinear determinant operator, as well as the singular logarithmic cost function. Numerical results demonstrate the success of this method on a wide range of challenging problems involving both the squared geodesic and the logarithmic cost functions.

Reconstruction of piecewise smooth wave speeds using multiple scattering

Let $c$ be a piecewise smooth wave speed on $\mathbb R^n$, unknown inside a domain $\Omega$. We are given the solution operator for the scalar wave equation $(\partial_t^2-c^2\Delta)u=0$, but only outside $\Omega$ and only for initial data supported outside $\Omega$. Using our recently developed scattering control method, we prove that piecewise smooth wave speeds are uniquely determined by this map, and provide a reconstruction formula. In other words, the wave imaging problem is solvable in the piecewise smooth setting under mild conditions. We also illustrate a separate method, likewise constructive, for recovering the locations of interfaces in broken geodesic normal coordinates using scattering control.

NORMAL FORM OF THE METRIC FOR A CLASS OF RIEMANNIAN MANIFOLDS WITH ENDS

In many problems of PDE involving the Laplace-Beltrami operator on manifolds with ends, it is often useful to introduce radial or geodesic normal coordinates near infinity. In this paper, we prove the existence of such coordinates for a general class of manifolds with ends, which includes asymptotically conical and hyperbolic manifolds. We study the decay rate to the metric at infinity associated to radial coordinates and also show that the latter metric is always conformally equivalent to the metric at infinity associated to the original coordinate system. We finally give several examples illustrating the sharpness of our results.

Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounded geometry and, moreover, can be generalized to Triebel‐Lizorkin spaces on manifolds, improving [11].

On Strong Unique Continuation of Coupled Einstein Metrics

The strong unique continuation property for Einstein metrics can be concluded from the well-known fact that Einstein metrics are analytic in geodesic normal coordinates. Here we give a proof of the same result that given two Einstein metrics with the same Ricci curvature on a fixed manifold, if they agree to infinite order around a point, then they must coincide, up to a local diffeomorphism, in a neighborhood of the point. The novelty of our method lies in the use of a Carleman inequality and thus circumventing the use of analyticity; thus the method is robust under certain non-analytic perturbations. As an example, we also show the strong unique continuation property for the Riemannian Einstein-scalar-field system with cosmological constant.

Uniqueness of the time coordinate for certain singular space-times in general relativity

Many singular space-times are described locally in geodesic normal coordinates with the singularity described by the time coordinate equal to zero. Under appropriate conditions the time coordinate in this description is uniquely determined. Furthermore, every timelike curve in the coordinate patch is inextendible and has the same limit point as exactly one time coordinate line. The proof of uniqueness of the time coordinate employs methods from the theory of first order partial differential equations in a new way.

Generalization of the Activated Complex Theory of Reaction Rates. II. Classical Mechanical Treatment

In its usual classical form, activated-complex theory assumes a particular expression for the kinetic energy of the reacting system, one being associated with a rectilinear motion along the reaction coordinate. The derivation of the rate expression given in the present paper is based on the general kinetic-energy expression. A rate equation of the customary form is obtained: krate=(kT/h)exp[−(F‡−Fr)/kT], where F‡ is the free energy of a system constrained to exist on a hypersurface in n-dimensional space and Fr is the free energy of the reactants. The usual derivation is then reinterpreted, in terms of geodesic normal coordinates, to be somewhat more general than it appears. Normally, rotation—vibration interaction is neglected, as in the above derivation, although not in treatments of some special reactions in the literature for which the centrifugal potential is important. A derivation is given which includes the influence of this centrifugal potential but which omits Coriolis effects.