# The parallel transporter

By definition, for a vector $${w}$$ at a point $${p}$$ of an $${n}$$-dimensional manifold $${M}$$, parallel transport assigns a vector $${\parallel_{C}\left(w\right)}$$ at another point $${q}$$ that is dependent upon a specific path $${C}$$ in $${M}$$ from $${p}$$ to $${q}$$.

To see that this dependence upon the path matches our intuition, we can consider a vector transported in what we might consider to be a “parallel” fashion along the edges of an eighth of a sphere. In this example, the sphere is embedded in $${\mathbb{R}^{3}}$$ and the concept of “parallel” corresponds to incremental vectors along the path having a projection onto the original tangent plane that is parallel to the original vector. The above depicts how a vector $${w}$$ transported in what we intuitively consider to be a “parallel” way along two different paths ($${B}$$ and $${C=C_{1}+C_{2}}$$) on a surface results in two different vectors.

The parallel transporter is therefore a map $${\parallel_{C}\colon T_{p}M\to T_{q}M}$$, where $${C}$$ is a curve in $${M}$$ from $${p}$$ to $${q}$$. To match our intuition we also require that this map be linear (i.e. parallel transport is assumed to preserve the vector space structure of the tangent space); that it be the identity for vanishing $${C}$$; that if $${C=C_{1}+C_{2}}$$ then $${\parallel_{C}=\parallel_{C_{2}}\parallel_{C_{1}}}$$; and that the dependence on $${C}$$ be smooth (this is most easily defined in the context of fiber bundles, which we will cover in a later chapter). If we then choose a frame on $${U\subset M}$$, we have bases for each tangent space that provide isomorphisms $${T_{p}U\cong\mathbb{R}^{n}}$$, $${T_{q}U\cong\mathbb{R}^{n}}$$. Thus the parallel transporter can be viewed as a map $${\parallel^{\lambda}{}_{\mu}\colon\left\{ C\right\} \to GL\left(n,\mathbb{R}\right)}$$ from the set of curves on $${U}$$ to the Lie group $${GL\left(n,\mathbb{R}\right)}$$; however, it is important to note that the values of $${\parallel^{\lambda}{}_{\mu}}$$ depend upon the choice of frame.
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