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Ehresmann and Cartan
The founding members were all connected to the École Normale Supérieure in Paris and included Henri Cartan, Claude Chevalley, Jean Coulomb, Jean Delsarte, Jean Dieudonné, Charles Ehresmann, René de Possel, Szolem Mandelbrojt and André Weil.
Cartan connections are Ehresmann connections with additional structure which allows the parallel transport to be though of as a map " rolling " a certain model space along a curve in the manifold.
Ehresmann connections were, strictly speaking, not a generalization of Cartan connections.
Ehresmann connections were rather a solid framework for viewing the foundational work of other geometers of the time, such as Shiing-Shen Chern, who had already begun moving away from Cartan connections to study what might be called gauge connections.
Yet another way to define a Cartan connection is with an Ehresmann connection on the bundle E = Q ×< sub > G </ sub > G / H of the preceding section.
Historically, jet bundles are attributed to Ehresmann, and were an advance on the method ( prolongation ) of Élie Cartan, of dealing geometrically with higher derivatives, by imposing differential form conditions on newly-introduced formal variables.

Ehresmann and connection
More generally, the same construction allows one to construct a vector field for any Ehresmann connection on the tangent bundle.
In the context of Ehresmann connections, where the connection depends on a special notion of " horizontal lifting " of tangent spaces, one can define parallel transport via horizontal lifts.
An Ehresmann connection is a connection in a fibre bundle or a principal bundle by specifying the allowed directions of motion of the field.
In that same year, Charles Ehresmann, a student of Cartan's, presented a variation on the connection as a differential form view in the context of principal bundles and, more generally, fibre bundles.
* An approach to connections which makes direct use of the notion of transport of " data " ( whatever that may be ) is the Ehresmann connection.
A connection on E is also determined equivalently by a linear Ehresmann connection on E. This provides one method to construct the associated principal connection.
If x < sub > t </ sub > is a curve in M, then the Ehresmann connection on E supplies an associated parallel transport map τ < sub > t </ sub >: E < sub > x < sub > t </ sub ></ sub > → E < sub > x < sub > 0 </ sub ></ sub > from the fibre over the endpoint of the curve to the fibre over the initial point.
To show that this definition is equivalent to the others above, one must introduce a suitable notion of a moving frame for the bundle E. In general, this is possible for any G-connection on a fibre bundle with structure group G. See Ehresmann connection # Associated bundles for more details.
A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection.
A principal G-connection ω on P determines an Ehresmann connection on P in the following way.
This is an Ehresmann connection.
Conversely, an Ehresmann connection H ⊂ TP ( or v: TP → V ) on P defines a principal G-connection ω if and only if it is G-equivariant in the sense that.
Let ω be an Ehresmann connection on P ( which is a-valued one-form on P ).
** Ehresmann connection, gives a manner for differentiating sections of a general fibre bundle

Ehresmann and bundle
It gives rise to ( Ehresmann ) connections on any fiber bundle associated to P via the associated bundle construction.

Ehresmann and .
The concept is due to Ehresmann and Hopf in the 1940s.

Cartan and connection
The Einstein – Cartan – Sciama – Kibble theory of gravity extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a dynamical variable.
Moreover, even though there is no Levi-Civita connection on a conformal manifold, one can instead work with a conformal connection, which can be handled either as a type of Cartan connection modelled on the associated Möbius geometry, or as a Weyl connection.
A Cartan connection is a way of formulating some aspects of connection theory using differential forms and Lie groups.
As the twentieth century progressed, Élie Cartan developed a new notion of connection.
In these investigations, he found that a certain infinitesimal notion of connection ( a Cartan connection ) could be applied to these geometries and more: his connection concept allowed for the presence of curvature which would otherwise be absent in a classical Klein geometry.
# Generalised spaces with structure groups and connections, Cartan connection, holonomy, Weyl tensor
* Cartan connection, Cartan connection applications
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection.
In particular, the high point of his remarkable 1910 paper on Pfaffian systems in five variables is the construction of a Cartan connection modelled on a 5-dimensional homogeneous space for the exceptional Lie group G < sub > 2 </ sub >, which he and Engels had discovered independently in 1894 .</ ref > The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand.
For instance, in relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection.
For Lie groups, Maurer – Cartan frames are used to view the Maurer – Cartan form of the group as a Cartan connection.

Cartan and supplies
However, a Cartan connection supplies a way of connecting the infinitesimal model spaces within the manifold by means of parallel transport.

Cartan and curves
Henri Cartan, Joachim and Hermann Weyl and Lars Ahlfors extended Nevanlinna theory to holomorphic curves.
Such spheres can again be rolled along curves on S, and this equips S with another type of Cartan connection called a conformal connection.
The Cartan geometry of S consists of a copy of the model space G / H at each point of S ( with a marked point of contact ) together with a notion of " parallel transport " along curves which identifies these copies using elements of G. This notion of parallel transport is generic in the intuitive sense that the point of contact always moves along the curve.

Cartan and from
On the advice of Cartan and Weil, he moved to the University of Nancy where he wrote his dissertation under Laurent Schwartz in functional analysis, from 1950 to 1953.
The designation E < sub > 6 </ sub > comes from the Cartan – Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled A < sub > n </ sub >, B < sub > n </ sub >, C < sub > n </ sub >, D < sub > n </ sub >, and five exceptional cases labeled E < sub > 6 </ sub >, E < sub > 7 </ sub >, E < sub > 8 </ sub >, F < sub > 4 </ sub >, and G < sub > 2 </ sub >.
Let be the Cartan matrix of the Kac-Moody algebra, and let q be a nonzero complex number distinct from 1, then the quantum group, U < sub > q </ sub >( G ), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators ( where λ is an element of the weight lattice, i. e. for all i ), and and ( for simple roots, ), subject to the following relations:
* Eilenberg's biography − from the National Academies Press, by Hyman Bass, Henri Cartan, Peter Freyd, Alex Heller and Saunders Mac Lane.
In all of these the theory of differential forms plays a part, and Kähler counts as a major developer of the theory from its formal genesis with Élie Cartan.
Rosenfeld ( 1993 ) Élie Cartan ( 1869 – 1951 ), translated from Russian original by V. V.
He was a follower of Élie Cartan, working on the ' theory of equivalence ' in his time in China from 1937 to 1943, in relative isolation.
Typical of Cartan, one motivation for introducing the notion of a Cartan connection was to study the properties of pseudogroups from an infinitesimal point of view.
The fibre of the tautological bundle G → G / H of the Klein geometry at the point of contact is then identified with the fibre of the bundle P. Each such fibre ( in G ) carries a Maurer-Cartan form for G, and the Cartan connection is a way of assembling these Maurer-Cartan forms gathered from the points of contact into a coherent 1-form η defined on the whole bundle.
The covariant derivative can also be constructed from the Cartan connection η on P. In fact, constructing it in this way is slightly more general in that V need not be a fully fledged representation of G. Suppose instead that that V is a (, H )- module: a representation of the group H with a compatible representation of the Lie algebra.
The main conclusion ( that general relativity plus spin-orbit coupling implies nonzero torsion and Einstein – Cartan theory ) is derived from classical general relativity and classical differential geometry without recourse to quantum mechanical spin or spinor fields.
Also, Adamowicz does not show that Einstein – Cartan theory follows from general relativity plus spin.
For completeness, below are references to some speculative physical theories that employ torsion in ways that are different from Einstein – Cartan theory.
More general affine connections were then studied around 1920, by Hermann Weyl, who developed a detailed mathematical foundation for general relativity, and Élie Cartan, who made the link with the geometrical ideas coming from surface theory.
The designation E < sub > 8 </ sub > comes from the Cartan – Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled A < sub > n </ sub >, B < sub > n </ sub >, C < sub > n </ sub >, D < sub > n </ sub >, and five exceptional cases labeled E < sub > 6 </ sub >, E < sub > 7 </ sub >, E < sub > 8 </ sub >, F < sub > 4 </ sub >, and G < sub > 2 </ sub >.
The designation E < sub > 7 </ sub > comes from the Cartan – Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled A < sub > n </ sub >, B < sub > n </ sub >, C < sub > n </ sub >, D < sub > n </ sub >, and five exceptional cases labeled E < sub > 6 </ sub >, E < sub > 7 </ sub >, E < sub > 8 </ sub >, F < sub > 4 </ sub >, and G < sub > 2 </ sub >.
Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan.
One can also view the Maurer – Cartan form as being constructed from a Maurer – Cartan frame.
The initial construction by Élie Cartan and Wilhelm Killing of finite dimensional simple Lie algebras from the Cartan integers was type dependent.

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