Parallel transport

Once more, we use the sphere to illustrate how parallel transport works (with the kernel).

f = {r Cos[x_2] Sin[x_1 + π - 1], r Cos[x_1 + π - 1] Cos[x_2], r Sin[x_2]}/.r→1 ;

pS = ParametricPlot3D[Evaluate[.99 f], {x_1, 0, 2π}, {x_2, 0, 1.4}, PlotPoints→ {19, 11}, Axes→False, Boxed→False, PolygonIntersections→True] ;

[Graphics:../HTMLFiles/index_165.gif]

The metric in (x_1,x_2) coordinates is

J = Μd[f, 1, 2] ;

MF[ℊ = [J] . J//]

( {{Cos[x_2]^2, 0}, {0, 1}} )

The essential computation is carried out in line 1 of the following code. The command is ΧP[g,{t,h},{0,2 π}]. The input to ΧP is the metric g, a curve γ(t):I→M, and the interval I⊂R.

We plot parallel transport of an orthonormal frame along several meridians.

pcr[#1] &/@{0, 1, 1.3} ;

[Graphics:../HTMLFiles/index_173.gif]

[Graphics:../HTMLFiles/index_174.gif]

[Graphics:../HTMLFiles/index_175.gif]

An animation is sampled by "AnimationDisplayTime".

mov = pcr[#1] &/@Range[0, 1.4, .03] ;

Export["C:/parallel_sphere.gif", mov, "gif", ConversionOptions→ {"Loop"→True, "AnimationDisplayTime"→1/4}] ;

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