1,478 research outputs found
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 of the adjoint bundle \mc{A}
gives rise, given certain algebraic conditions, to a unique preferred affine
connection with covariantly constant rho-tensor ,
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 . The paper concludes with some holonomy
constructions for free -distributions for .Comment: a corrected and improved version, 24 page
Free -distributions: holonomy, sub-Riemannian structures, Fefferman constructions and dual distributions
This paper analyses the parabolic geometries generated by a free
-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
.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 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 -graded geometries I: general theory
This paper analyses non-regular -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 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 be a Lie subalgebra of a semisimple Lie algebra and be
the corresponding pair of connected Lie groups. A Cartan geometry of type
associates to a smooth manifold a principal -bundle and a Cartan
connection, and a parabolic geometry is a Cartan geometry where is
parabolic. We show that if is parabolic, the adjoint tractor bundle of a
Cartan geometry, which is isomorphic to the Atiyah algebroid of the principal
-bundle, admits the structure of a (pre-)Courant algebroid, and we identify
the topological obstruction to the bracket being a Courant bracket. For
semisimple , the Atiyah algebroid of the principal -bundle associated to
the Cartan geometry of admits a pre-Courant algebroid structure if and
only if is parabolic.Comment: 24 pages. v3: Added Section 4.3. v4: Modified abstract, added
concluding remark
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