Editing Connection (vector bundle) (section)

=== Relationship to Christoffel symbols ===

In [[pseudo-Riemannian geometry]], the [[Levi-Civita connection]] is often written in terms of the [[Christoffel symbols]] <math>\Gamma_{ij}^{\ \ k}</math> instead of the connection one-form <math>A</math>. It is possible to define Christoffel symbols for a connection on any vector bundle, and not just the tangent bundle of a pseudo-Riemannian manifold. To do this, suppose that in addition to <math>U</math> being a trivialising open subset for the vector bundle <math>E\to M</math>, that <math>U</math> is also a [[Topological_manifold#Coordinate_charts|local chart]] for the manifold <math>M</math>, admitting local coordinates <math>\mathbf{x} = (x^1,\dots,x^n);\quad x^i: U \to \mathbb{R}</math>.

In such a local chart, there is a distinguished local frame for the differential one-forms given by <math>(dx^1,\dots,dx^n)</math>, and the local connection one-forms <math>A_i^{ j}</math> can be expanded in this basis as

:<math>A_i^{\ j} = \sum_{\ell=1}^n \Gamma_{\ell i}^{\ \ j} dx^\ell</math>

for a collection of local smooth functions <math>\Gamma_{\ell i}^{\ \ j} : U \to \mathbb{R}</math>, called the ''Christoffel symbols'' of <math>\nabla</math> over <math>U</math>. In the case where <math>E=TM</math> and <math>\nabla</math> is the Levi-Civita connection, these symbols agree precisely with the Christoffel symbols from pseudo-Riemannian geometry.

The expression for how <math>\nabla</math> acts in local coordinates can be further expanded in terms of the local chart <math>U</math> and the Christoffel symbols, to be given by

:<math> \nabla(s) = \sum_{i,j=1}^k \sum_{\ell=1}^n \left(\frac{\partial s^j}{\partial x^\ell} + \Gamma_{\ell i}^{\ \ j} s^i\right) dx^\ell \otimes e_j.</math>
Contracting this expression with the local coordinate tangent vector <math>\frac{\partial}{\partial x^\ell}</math> leads to
:<math> \nabla_{\frac{\partial}{\partial x^\ell}} (s) = \sum_{i,j=1}^k \left(\frac{\partial s^j}{\partial x^\ell} + \Gamma_{\ell i}^{\ \ j} s^i\right) e_j.</math>

This defines a collection of <math>n</math> locally defined operators

:<math>\nabla_\ell: \Gamma(U,E) \to \Gamma(U,E);\quad \nabla_\ell(s) := \sum_{i,j=1}^k \left(\frac{\partial s^j}{\partial x^\ell} + \Gamma_{\ell i}^{\ \ j} s^i\right)e_j,</math>

with the property that

:<math>\nabla(s) = \sum_{\ell=1}^n dx^\ell \otimes \nabla_\ell(s).</math>