Parallel Transport
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To parallel
transport a vector means to move the vector along a curve while keeping the
vector, defined on each point on the curve, pointing in the same direction. For
simplicity we will restrict our focus to a flat plane. Let
represent the Cartesian coordinates
of a point P in the plane. Let xa represent the coordinates of the same point P
but as expressed in terms of generalized coordinates. Assume that the
transformation
has a non-vanishing Jacobian, which
means that the transformation is invertible at any point where the Jacobian does
not vanish. According to the definition of a vector,
in the generalized coordinate system transforms from its value Ab
as
A
vector is, parallel displaced if, when moved from one point to another, the
components in Cartesian coordinates remain unchanged, i.e.
. This is illustrated below in Fig. 1
We
seek the relation that
must satisfy for parallel
displacement. Taking the
differential of Eq. (1) we get
Multiply
Eq. (4) through by
. The following relation will be needed in the subsequent derivation
The
affine connection or connection coefficients and are defined as
Substituting
the affine connection into Eq. (3) symbols gives
The affine connection is sometimes referred to as the Christoffel symbols and
denoted
since when a metric is defined on the manifold they have the same value at any
given point P and are related by
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