index_15.html

Möbius band

immersion via f: the domain of f:D→ is a subset D⊂ around 0

$assum = {-ε<x_1<ε, -ε<x_2<ε, 10<r} ;$

$f = {(r + x_2 Cos[x_1]) Cos[x_1], (r + x_2 Cos[x_1]) Sin[x_1], x_2 Sin[x_1]} ;$

$ParametricPlot3D[Evaluate[{f/.r→3}], {x_1, -0, 2 π}, {x_2, -1, 1}, PlotPoints→ {30, 9}] ;$

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The metric tensor g on D⊂ is the pullback of the standard metric tensor Id[3] on via f.

$( {{r^2 + 2 r Cos[x_1] x_2 + 1/2 (3 + Cos[2 x_1]) x_2^2, 0}, {0, 1}} )$

The following expression is the (euclidean) length of "going once around". The length is minimal "walking in the middle", i.e. when =0. In this case, the length is simply 2, the circumferrence of a circle.

Have a look at the standard tensors of the sphere: g, Γ, R, Ric, S

$metric ( {{r^2 + 2 r Cos[x_1] x_2 + 1/2 (3 + Cos[2 x_1]) x_2^2, 0}, {0, 1}} )$

$Γ_ (i, j)^2 = ( {{-r Cos[x_1] - 1/2 (3 + Cos[2 x_1]) x_2, 0}, {0, 0}} )$

${1, 2, ( {{0, -(4 r^2)/(2 r^2 + 4 r Cos[x_1] x_2 + (3 + Cos[2 x_1]) x_2^2)^2}, {(2 r^2)/(2 r^2 + 4 r Cos[x_1] x_2 + (3 + Cos[2 x_1]) x_2^2), 0}} )}$

$ricci ( {{-(2 r^2)/(2 r^2 + 4 r Cos[x_1] x_2 + (3 + Cos[2 x_1]) x_2^2), 0}, {0, -(4 r^2)/(2 r^2 + 4 r Cos[x_1] x_2 + (3 + Cos[2 x_1]) x_2^2)^2}} )$

$metric ( {{r^2, 0}, {0, 1}} )$

$Γ_ (i, j)^1 = ( {{0, 1/r}, {1/r, 0}} )$

$Γ_ (i, j)^2 = ( {{-r, 0}, {0, 0}} )$

${1, 2, ( {{0, -1/r^2}, {1, 0}} )}$

$ricci ( {{-1, 0}, {0, -1/r^2}} )$

For some reason the following line takes forever to execute. In the next line we introduce a new coordinate , which parametrizes the local normal vector field.

$ParametricPlot3D[Evaluate[{φ/.{x_2→t, x_3→0}, φ/.{x_2→0, x_3→t} }/.R→3], {x_1, 0, 2π}, {t, -1, 1}, PlotPoints→ {30, 9}] ;$

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Created by Mathematica (December 22, 2006)