Editing Connection (mathematics) (section)

===Resolution===
The problem observed above is that the usual [[directional derivative]] of [[vector calculus]] does not behave well under changes in the coordinate system when applied to the components of vector fields.  This makes it quite difficult to describe how to translate vector fields in a parallel manner, if indeed such a notion makes any sense at all.  There are two fundamentally different ways of resolving this problem.

The first approach is to examine what is required for a generalization of the directional derivative to "behave well" under coordinate transitions.  This is the tactic taken by the [[covariant derivative]] approach to connections: good behavior is equated with [[covariance and contravariance of vectors|covariance]].  Here one considers a modification of the directional derivative by a certain [[linear operator]], whose components are called the [[Christoffel symbols]], which involves no derivatives on the vector field itself.  The directional derivative ''D''<sub>'''u'''</sub>'''v''' of the components of a vector '''v''' in a coordinate system ''φ'' in the direction '''u''' are replaced by a ''covariant derivative'':

:<math>\nabla_{\mathbf u} {\mathbf v} = D_{\mathbf u} {\mathbf v} + \Gamma(\varphi)\{{\mathbf u},{\mathbf v}\}</math>

where Γ depends on the coordinate system ''φ'' and is [[Bilinear form|bilinear]] in '''u''' and '''v'''.  In particular, Γ does not involve any derivatives on '''u''' or '''v'''.  In this approach, Γ must transform in a prescribed manner when the coordinate system ''φ'' is changed to a different coordinate system.  This transformation is not [[tensor]]ial, since it involves not only the ''first derivative'' of the coordinate transition, but also its ''second derivative''.  Specifying the transformation law of Γ is not sufficient to determine Γ  uniquely.  Some other normalization conditions must be imposed, usually depending on the type of geometry under consideration.  In [[Riemannian geometry]], the [[Levi-Civita connection]] requires compatibility of the [[Christoffel symbols]] with the [[Riemannian metric|metric]] (as well as a certain symmetry condition).  With these normalizations, the connection is uniquely defined.

The second approach is to use [[Lie group]]s to attempt to capture some vestige of symmetry on the space.  This is the approach of [[Cartan connection]]s.  The example above using rotations to specify the parallel transport of vectors on the sphere is very much in this vein.