Frederic P Miller

Cartan connection Frederic P Miller

Publisher Marketing: Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces. The theory of Cartan connections was developed by Elie Cartan, as part of (and a way of formulating) his method of moving frames (repere mobile). It operates with differential forms and so is computational in character. 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.