Projective Compactness and Conformal Boundaries

Let $\overline{M}$ be a smooth manifold with boundary $\partial M$ and interior $M$. Consider an affine connection $\nabla$ on $M$ for which the boundary is at infinity. Then $\nabla$ is projectively compact of order $\alpha$ if the projective structure defined by $\nabla$ smoothly extends to all of $\overline{M}$ in a specific way that depends on no particular choice of boundary defining function. Via the Levi--Civita connection, this concept applies to pseudo--Riemannian metrics on $M$. We study the relation between interior geometry and the possibilities for compactification, and then develop the tools that describe the induced geometry on the boundary. We prove that a pseudo--Riemannian metric on $M$ which is projectively compact of order two admits a certain asymptotic form. This form was known to be sufficient for projective compactness, so the result establishes that it provides an equivalent characterization. From a projectively compact connection on $M$, one obtains a projective structure on $\overline{M}$, which induces a conformal class of (possibly degenerate) bundle metrics on the tangent bundle to the hypersurface $\partial M$. Using the asymptotic form, we prove that in the case of metrics, which are projectively compact of order two, this boundary structure is always non--degenerate. We also prove that in this case the metric is necessarily asymptotically Einstein, in a natural sense. Finally, a non--degenerate boundary geometry gives rise to a (conformal) standard tractor bundle endowed with a canonical linear connection, and we explicitly describe these in terms of the projective data of the interior geometry.Comment: Substantially revised, including simpler arguments for many of the main results. 32 pages, comments are welcom

Scalar Curvature and Projective Compactness

Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior does not extend to the boundary (because for example the interior is complete) whereas its projective structure does, then the metric is projectively compact of order 2; this order is a measure of volume growth toward infinity. The result implies a host of results including that the metric satisfies asymptotic Einstein conditions, and induces a canonical conformal structure on the boundary. Underpinning this work is a new interpretation of scalar curvature in terms of projective geometry. This enables us to show that if the projective structure of a metric extends to the boundary then its scalar curvature also naturally and smoothly extends.Comment: Final version to be published in J. Geom. Phys. Includes minor typo corrections and a new summarising corollary. 10 page

CR-Tractors and the Fefferman Space

We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman's construction of a canonical conformal class on the total space of a circle bundle over a non--degenerate CR manifold of hypersurface type. In particular we construct and treat the basic objects that relate the natural bundles and natural operators on the two spaces. This is illustrated with several applications: We prove that a number of conformally invariant overdetermined systems admit non--trivial solutions on any Fefferman space. We show that the space of conformal Killing fields on a Fefferman space naturally decomposes into a direct sum of subspaces, which admit an interpretaion as solutions of certain CR invariant PDE's. Finally we explicitly analyze the relation between tractor calculus on a CR manifold and the complexified conformal tractor calculus on its Fefferman space, thus obtaining a powerful computational tool for working with the Fefferman construction.Comment: AMSLaTeX, 46 pages, v3: added link http://www.iumj.indiana.edu/IUMJ/fulltext.php?year=2008&volume=57&artid=3359 to published version, which has different numbering of statement

Conformally Invariant Operators via Curved Casimirs: Examples

We discuss a general scheme for a construction of linear conformally invariant differential operators from curved Casimir operators; we then explicitly carry this out for several examples. Apart from demonstrating the efficacy of the approach via curved Casimirs, this shows that this method applies both in regular and in singular infinitesimal character, and also that it can be used to construct standard as well as non--standard operators. The examples treated include conformally invariant operators with leading term, in one case, a square of the Laplacian, and in another case, a cube of the Laplacian.Comment: AMSLaTeX, 16 pages, v2: minor changes, final version to appear in Pure Appl. Math.

Holonomy reductions of Cartan geometries and curved orbit decompositions

We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modelling homogeneous space and carries a canonical induced Cartan geometry. The result can therefore be understood as a `curved orbit decomposition'. The theory is then applied to the study of several invariant overdetermined differential equations in projective, conformal and CR-geometry. This makes use of an equivalent description of solutions to these equations as parallel sections of a tractor bundle. In projective geometry we study a third order differential equation that governs the existence of a compatible Einstein metric. In CR-geometry we discuss an invariant equation that governs the existence of a compatible K\"{a}hler-Einstein metric.Comment: v2: major revision; 30 pages v3: final version to appear in Duke Math.

Solution of ordinary differential equations by means of Lie series

Solution of ordinary differential equations by Lie series - Laplace transformation, Weber parabolic-cylinder functions, Helmholtz equations, and applications in physic

Lie series for celestial mechanics, accelerators, satellite stabilization and optimization

Lie series applications to celestial mechanics, accelerators, satellite orbits, and optimizatio