conformal geometry
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
, the tractor bundle is a particular
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
constructed on a
conformal manifold
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
whose fibres form an
effective
Effectiveness is the capability of producing a desired result or the ability to produce desired output. When something is deemed effective, it means it has an intended or expected outcome, or produces a deep, vivid impression.
Etymology
The ori ...
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
of the
conformal group
In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space.
S ...
(see
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with ...
).
The term ''tractor'' is a
portmanteau
A portmanteau word, or portmanteau (, ) is a blend of wordsT. Y. Thomas as an alternative formulation of the
Cartan Cartan may refer to:
* Élie Cartan (1869–1951), French mathematician who worked with Lie groups
* Henri Cartan (1904-2008), French mathematician who worked in algebraic topology, son of Élie Cartan
* Anna Cartan (1878-1923), French mathematici ...
conformal connection In conformal differential geometry, a conformal connection is a Cartan connection on an ''n''-dimensional manifold ''M'' arising as a deformation of the Klein geometry given by the celestial ''n''-sphere, viewed as the homogeneous space
:O+(n+1 ...
, and later rediscovered within the formalism of
local twistor
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
s and generalized to
projective connection In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold.
The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to ...
s by
Michael Eastwood
Michael G. Eastwood is a mathematician at the University of Adelaide, known for his work in twistor theory, conformal differential geometry and invariant differential operators. In 1976 he received a PhD at Princeton University in several com ...
''et al.'' in Bailey, T. N.; Eastwood, M. G.; Gover, A. R., "Thomas's structure bundle for conformal, projective and related structures", ''Rocky Mountain J.'' 24 (1994), 1191–1217.