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Distinguished curves and submanifolds in conformal geometry
No full textConformal geometry is a weakening of Riemannian geometry where one works with a
smooth manifold equipped with an equivalence class of Riemannian metrics, where two
metrics are equivalent if and only if they define the same angles between curves. Early
interest in conformal equivalence included the question of biholomorphic equivalence of
domains in the complex plane. Interest has also been driven by physics and general relativity, since light in spacetime follows null geodesics, and these only depend on the conformal structure. Recently there has been considerable progress in the study of codimension
one submanifolds in conformal manifolds. At the other extreme, there has also been an
increased understanding of the distinguished curves in conformal manifolds. In this thesis,
we develop a complete basic tractor theory of conformal submanifolds of any codimension
and use this to define a notion of distinguished conformal submanifolds. These distinguished submanifolds coincide with conformal circles and totally umbilic hypersurfaces in
the extremal cases. We emphasize three conformal tractor objects which we show encode
equivalent submanifold data. Our notion of distinguished submanifolds admits characterizations in terms of all three invariants. Our definition immediately leads to a procedure
for proliferation of conserved quantities along these submanifolds. We also obtain a theorem which characterizes our distinguished conformal submanifolds in terms of an incidence
relation and a parallel condition. We use this to show that zero loci of certain solutions
to a conformally invariant equation are, if nonempty, distinguished submanifolds. These
results extend existing results for conformal circles
Projectively Compact Pseudo-Riemannian Manifolds: Boundary Calculus and Natural Operators
No full textWe define a projectively Klein manifold to be a projective manifold with boundary where the interior is equipped with a pseudo-Riemannian metric that is projectively compact such that the boundary value of the extension of its scalar curvature is nowhere vanishing. As a consequence,
there is a conformal structure on the boundary along which the conformal and projective tractor
bundles agree.
A Klein manifold may also be interpreted to be a projective manifold with a boundary
given by the degeneracy locus of a solution of the metrisability equation, which is a first-order
Bernstein-Gelfand-Gelfand equation. From this it follows that there exists a projectively invariant
differential operator called the splitting operator and this defines a symmetric 2-cotractor
that we term the structure tractor. This gives a metric on the projective tractor bundle.
We introduce a notion of special asymptotic boundary scales which make the splittings of
the conformal and projective tractor bundles compatible along the boundary of Klein manifold.
This important observation simplifies the construction of a boundary calculus which relates
ambient projective quantities to their boundary conformal counterparts. By modifying the
Eastwood-Matveev equation, which expresses the prolongation of the metrisability equation
using the tractor calculus, the restriction of the ambient projective tractor connection along the
boundary is explicitly linked to the boundary conformal tractor connection.
An sl(2) algebra is shown to exist on a Klein manifold which is generated by a projective
tractor Laplacian-type operator, the determinant of the solution of the metrisability equation,
and a weight operator. This algebra enables the construction of tangential Laplacian-type operators
along the boundary of the Klein manifold. The generalised conformal Yamabe operator
and generalised Paneitz operator along the boundary of a Klein manifold are calculated, and we
show that generalised Graham-Jenne-Mason-Sparling (GJMS) operators can be constructed on
the boundary of Klein manifolds more generally. These are conformally invariant Laplacian
power operators that include in their coefficients extrinsic embedding data
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Conformal geometry, holography, and boundary calculus
No full textNon UBCUnreviewedAuthor affiliation: University of AucklandFacult
Boundary operators on conformally compact manifolds, and a boundary LoewnerNirembergYamabe problem
No full textNon UBCUnreviewedAuthor affiliation: University of AucklandFacult
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