Editing Connection (vector bundle) (section)

==Motivation==

Let {{mvar|M}} be a [[differentiable manifold]], such as [[Euclidean space]]. A vector-valued function <math>M \to \mathbb{R}^n</math> can be viewed as a [[Section (fiber bundle)|section]] of the trivial [[vector bundle]] <math>M\times \mathbb{R}^n \to M.</math> One may consider a section of a general differentiable vector bundle, and it is therefore natural to ask if it is possible to differentiate a section, as a generalization of how one differentiates a function on {{mvar|M}}.

[[File:Bundle section.svg|thumb|A section of a bundle may be viewed as a generalized function from the base into the fibers of the vector bundle. This can be visualized by the graph of the section, as in the figure above.]]

The model case is to differentiate a function <math>X: \mathbb{R}^n \to \mathbb{R}^m</math> on Euclidean space <math>\mathbb{R}^n</math>. In this setting the derivative <math>dX</math> at a point <math>x\in \mathbb{R}^n</math> in the direction <math>v\in \mathbb{R}^n</math> may be defined by the standard formula
:<math>dX(v)(x) = \lim_{t\to 0} \frac{X(x+tv) - X(x)}{t}.</math>
For every <math>x\in \mathbb{R}^n</math>, this defines a new vector <math>dX(v)(x)\in\mathbb{R}^m.</math>

When passing to a section <math>X</math> of a vector bundle <math>E</math> over a manifold <math>M</math>, one encounters two key issues with this definition. Firstly, since the manifold has no linear structure, the term <math>x+tv</math> makes no sense on <math>M</math>. Instead one takes a path <math>\gamma: (-1,1) \to M</math> such that <math>\gamma(0) = x, \gamma'(0) = v</math> and computes 
:<math>dX(v)(x) = \lim_{t\to 0} \frac{X(\gamma(t)) - X(\gamma(0))}{t}.</math>
However this still does not make sense, because <math>X(\gamma(t))</math> and <math>X(\gamma(0))</math> are elements of the distinct vector spaces <math>E_{\gamma(t)}</math> and <math>E_x.</math> This means that subtraction of these two terms is not naturally defined.

The problem is resolved by introducing the extra structure of a '''connection''' to the vector bundle. There are at least three perspectives from which connections can be understood. When formulated precisely, all three perspectives are equivalent.

# (''[[Parallel transport]]'') A connection can be viewed as assigning to every differentiable path <math>\gamma</math> a [[linear isomorphism]] <math>P_t^{\gamma} : E_{\gamma(t)} \to E_{x}</math> for all <math>t.</math> Using this isomorphism one can transport <math>X(\gamma(t))</math> to the fibre <math>E_x</math> and then take the difference; explicitly, <math display="block">\nabla_vX = \lim_{t\to 0} \frac{P_{t}^{\gamma} X(\gamma(t)) - X(\gamma(0))}{t}.</math>In order for this to depend only on <math>v,</math> and not on the path <math>\gamma</math> extending <math>v,</math> it is necessary to place restrictions (in the definition) on the dependence of <math>P_t^{\gamma}</math> on <math>\gamma.</math> This is not straightforward to formulate, and so this notion of "parallel transport" is usually derived as a by-product of other ways of defining connections. In fact, the following notion of "Ehresmann connection" is nothing but an infinitesimal formulation of parallel transport.
# (''[[Ehresmann connection]]'') The section <math>X</math> may be viewed as a smooth map from the smooth manifold <math>M</math> to the smooth manifold <math>E.</math> As such, one may consider the [[pushforward (differential)|pushforward]] <math>dX(v),</math> which is an element of the [[tangent space]] <math>T_{X(x)}E.</math> In Ehresmann's formulation of a connection, one chooses a way of assigning, to each <math>x</math> and every <math>e\in E_x,</math> a direct sum decomposition of <math>T_{X(x)}E</math> into two linear subspaces, one of which is the natural embedding of <math>E_x.</math> With this additional data, one defines <math>\nabla_vX</math> by projecting <math>dX(v)</math> to be valued in <math>E_x.</math> In order to respect the linear structure of a vector bundle, one imposes additional restrictions on how the direct sum decomposition of <math>T_{e}E</math> moves as {{mvar|e}} is varied over a fiber.
# (''[[Covariant derivative]]'') The standard derivative <math>dX(v)</math> in Euclidean contexts satisfies certain dependencies on <math>X</math> and <math>v,</math> the most fundamental being linearity. A covariant derivative is defined to be any operation <math>(v,X)\mapsto\nabla_vX</math> which mimics these properties, together with a form of the [[product rule]]. 

Unless the base is zero-dimensional, there are always infinitely many connections which exist on a given differentiable vector bundle, and so there is always a corresponding ''choice'' of how to differentiate sections. Depending on context, there may be distinguished choices, for instance those which are determined by solving certain [[partial differential equation]]s. In the case of the [[tangent bundle]], any [[pseudo-Riemannian metric]] (and in particular any [[Riemannian metric]]) determines a canonical connection, called the [[Levi-Civita connection]].