Möbius band

immersion via f: the domain of f:D→^3 is a subset D⊂^2 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}] ;

[Graphics:../HTMLFiles/index_447.gif]

The metric tensor g on D⊂^2 is the pullback of the standard metric tensor Id[3] on ^3 via f.

ℊ = ΜPullback[f, 2, Id[3]] ; ℊ//MF

( {{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 x_2=0. In this case, the length is simply 2πr^2, the circumferrence of a circle.

∫_0^(2π) Det[ℊ] x_1

π (2 r^2 + 3 x_2^2)

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

ShowΜ[ℊ]

ShowΜ[ℊ, {0, 0}]

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

christ↓

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

riemann↓

{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}} )

scalar  -(8 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}} )

christ↓

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

 Γ_ (i, j)^2 =    ( {{-r, 0}, {0, 0}} )

riemann↓

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

!!! curvature identities

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

scalar  -2/r^2

For some reason the following line takes forever to execute. In the next line we introduce a new coordinate x_3, which parametrizes the local normal vector field.

φ = ΜInduce[f, 2, Id[3]] ; φ//MF

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}] ;

[Graphics:../HTMLFiles/index_480.gif]

[Graphics:../HTMLFiles/index_481.gif]


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