1,344 research outputs found
Mathematical general relativity: a sampler
We provide an introduction to selected recent advances in the mathematical
understanding of Einstein's theory of gravitation.Comment: Some updates. A shortened version, to appear in the Bulletin of the
AMS, is available online at
http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2010-01304-
Compact Riemannian 7-manifolds with holonomy G2. I
This is the second of two papers about metrics of holonomy G2 on compact 7-manifolds. In our first paper [15] we established the existence of a family of metrics of holonomy G2 on a single, compact, simply-connected 7-manifold M, using three general results, Theorems A, B and C. Our purpose in this paper is to explore th
The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature
The harmonic oscillator as a distinguished dynamical system can be defined
not only on the Euclidean plane but also on the sphere and on the hyperbolic
plane, and more generally on any configuration space with constant curvature
and with a metric of any signature, either Riemannian (definite positive) or
Lorentzian (indefinite). In this paper we study the main properties of these
`curved' harmonic oscillators simultaneously on any such configuration space,
using a Cayley-Klein (CK) type approach, with two free parameters \ki, \kii
which altogether correspond to the possible values for curvature and signature
type: the generic Riemannian and Lorentzian spaces of constant curvature
(sphere , hyperbolic plane , AntiDeSitter sphere {\bf
AdS}^{\unomasuno} and DeSitter sphere {\bf dS}^{\unomasuno}) appear in this
family, with the Euclidean and Minkowski spaces as flat limits.
We solve the equations of motion for the `curved' harmonic oscillator and
obtain explicit expressions for the orbits by using three different methods:
first by direct integration, second by obtaining the general CK version of the
Binet's equation and third, as a consequence of its superintegrable character.
The orbits are conics with centre at the potential origin in any CK space,
thereby extending this well known Euclidean property to any constant curvature
configuration space. The final part of the article, that has a more geometric
character, presents those results of the theory of conics on spaces of constant
curvature which are pertinent.Comment: 29 pages, 6 figure
A numerical domain decomposition method for solving elliptic equations on manifolds
A new numerical domain decomposition method is proposed for solving elliptic
equations on compact Riemannian manifolds. The advantage of this method is to
avoid global triangulations or grids on manifolds. Our method is numerically
tested on some -dimensional manifolds such as the unit sphere , the
complex projective space and the product manifold .Comment: Final version. To appear in SIAM Journal on Scientific Computin
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