Projective Geometry II: Cones and Complete Classifications

The aim of this paper and its prequel is to introduce and classify the irreducible holonomy algebras of the projective Tractor connection. This is achieved through the construction of a `projective cone', a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying manifold. This paper uses the result to enable the construction of manifolds with each possible holonomy algebra.Comment: 30 pages, paper 2 of 2. Slight corrections, as well as diffirent division with paper

Generalised Einstein condition and cone construction for parabolic geometries

This paper attempts to define a generalisation of the standard Einstein condition (in conformal/metric geometry) to any parabolic geometry. To do so, it shows that any preserved involution $\sigma$ of the adjoint bundle \mc{A} gives rise, given certain algebraic conditions, to a unique preferred affine connection $\nabla$ with covariantly constant rho-tensor $\mathsf{P}$, compatible with the algebraic bracket on \mc{A}. These conditions can reasonably be considered the generalisations of the Einstein condition, and recreate the standard Einstein condition in conformal geometry. The existence of such an involution is implies by some simpler structures: preserved metrics when the overall algebra \mf{g} is \mf{sl}(m,\mbb{F}), preserved complex structures anti-commuting with the skew-form for \mf{g}=\mf{sp}(2m,\mbb{F}), and preserved subundles of the tangent bundle, of a certain rank, for all the other non-exceptional simple Lie algebras. Examples of Einstein involutions are constructed or referenced for several geometries. The existence of cone constructions for certain Einstein involutions is then demonstrated.Comment: Newest version, with SO*(2m) include

Free 3-distributions: holonomy, Fefferman constructions and dual distributions

This paper analyses the parabolic geometries generated by a free 3-distribution in the tangent space of a manifold. It shows the existence of normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4,2) and SO(3,3), respectively. There is also a fascinating construction of a `dual' distribution when the holonomy reduces to $G_2'$. The paper concludes with some holonomy constructions for free $n$-distributions for $n>3$.Comment: a corrected and improved version, 24 page

Free nn-distributions: holonomy, sub-Riemannian structures, Fefferman constructions and dual distributions

This paper analyses the parabolic geometries generated by a free $n$-distribution in the tangent space of a manifold. It shows that certain holonomy reductions of the associated normal Tractor connections, imply preferred connections with special properties, along with Riemannian or sub-Riemannian structures on the manifold. It constructs examples of these holonomy reductions in the simplest cases. The main results, however, lie in the free 3-distributions. In these cases, there are normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4,2) and SO(3,3), respectively. There is also a fascinating construction of a `dual' distribution when the holonomy reduces to $G_2'$.Comment: First Draf

Addendum to "Ricci-flat holonomy: a Classification": the case of Spin(10)

This note fills a hole in the author's previous paper ``Ricci-Flat Holonomy: a Classification'', by dealing with irreducible holonomy algebras that are subalgebras or real forms of \mbb{C} \oplus \mf{spin}(10,\mbb{C}). These all turn out to be of Ricci-type.Comment: Corrections to a previous pape

Note on pre-Courant algebroid structures for parabolic geometries

This note aims to demonstrate that every parabolic geometry has a naturally defined per-Courant algebro\"id structure. This structure is a Courant algebro\"id if and only if the the curvature $\kappa$ of the Cartan connection vanishes. In all other cases, if the parabolic geometry is regular, there does not exist a natural universal expression for a Courant bracket.Comment: A short note on Courant brackets on parabolic geometries. 2nd Version. Removed an erronous proo

Definite signature conformal holonomy: a complete classification

This paper aims to classify the holonomy of the conformal Tractor connection, and relate these holonomies to the geometry of the underlying manifold. The conformally Einstein case is dealt with through the construction of metric cones, whose Riemmanian holonomy is the same as the Tractor holonomy of the underlying manifold. Direct calculations in the Ricci-flat case and an important decomposition theorem complete the classification for definitive signature.Comment: 31 Pages, sections and introduction to Cartan connection reworked and typos correcte

Non-regular ∣2∣|2|-graded geometries I: general theory

This paper analyses non-regular $|2|$-graded geometries, and show that they share many of the properties of regular geometries -- the existence of a unique normal Cartan connection encoding the structure, the harmonic curvature as obstruction to flatness of the geometry, the existence of the first two BGG splitting operators and of (in most cases) invariant prolongations for the standard Tractor bundle \mc{T}. Finally, it investigates whether these geometries are determined entirely by the distribution $H = T_{-1}$ and concludes that this is generically the case, up to a finite choice, whenever H^1(\mf{g}^1,\mf{g}) vanishes in non-negative homogeneity

Good and safe uses of AI Oracles

It is possible that powerful and potentially dangerous artificial intelligence (AI) might be developed in the future. An Oracle is a design which aims to restrain the impact of a potentially dangerous AI by restricting the agent to no actions besides answering questions. Unfortunately, most Oracles will be motivated to gain more control over the world by manipulating users through the content of their answers, and Oracles of potentially high intelligence might be very successful at this \citep{DBLP:journals/corr/AlfonsecaCACAR16}. In this paper we present two designs for Oracles which, even under pessimistic assumptions, will not manipulate their users into releasing them and yet will still be incentivised to provide their users with helpful answers. The first design is the counterfactual Oracle -- which choses its answer as if it expected nobody to ever read it. The second design is the low-bandwidth Oracle -- which is limited by the quantity of information it can transmit.Comment: 11 pages, 2 figure

Courant Algebroids in Parabolic Geometry

Let $p$ be a Lie subalgebra of a semisimple Lie algebra $g$ and $(G,P)$ be the corresponding pair of connected Lie groups. A Cartan geometry of type $(G,P)$ associates to a smooth manifold $M$ a principal $P$-bundle and a Cartan connection, and a parabolic geometry is a Cartan geometry where $P$ is parabolic. We show that if $P$ is parabolic, the adjoint tractor bundle of a Cartan geometry, which is isomorphic to the Atiyah algebroid of the principal $P$-bundle, admits the structure of a (pre-)Courant algebroid, and we identify the topological obstruction to the bracket being a Courant bracket. For semisimple $G$, the Atiyah algebroid of the principal $P$-bundle associated to the Cartan geometry of $(G,P)$ admits a pre-Courant algebroid structure if and only if $P$ is parabolic.Comment: 24 pages. v3: Added Section 4.3. v4: Modified abstract, added concluding remark