Surfaces of revolution • from [Kü03] pp.52-54

We walk thru the computations carried out on pages 53-54 in [Kü03] to compare the results.

f = {r[x_1] Cos[x_2], r[x_1] Sin[x_2], h[x_1]}

ShowΜ[ℊ = ΜPullback[f, 2, Id[3]]]

{Cos[x_2] r[x_1], r[x_1] Sin[x_2], h[x_1]}

metric ( {{h^′[x_1]^2 + r^′[x_1]^2, 0}, {0, r[x_1]^2}} )

rieman {Null}

scalar -(2 h^′[x_1] (-r^′[x_1] h^′′[x_1] + h^′[x_1] r^′′[x_1]))/(r[x_1] (h^′[x_1]^2 + r^′[x_1]^2)^2)

We make the same assumptions as [Kü03].

assum = {h^′[x_1]^2 + r^′[x_1]^2>0, r^′[x_1] >0}

φ = ΜInduce[f, 2, Id[3]] ; φ//MF

φ - f/.x_3→1//S

{h^′[x_1]^2 + r^′[x_1]^2>0, r^′[x_1] >0}

All figures below are labelled. Without additional assumptions, the geometrical expressions get quite complicated.

ShowΜ[ℊ = ΜPullback[φ, 3, Id[3]], 2] ;

christ↓

Γ_ (i, j)^1
Γ_ (i, j)^2
Γ_ (i, j)^3

II->Γ

To simplify the discussion, at this point [Kü03] makes the stronger assuptions that the curve (r,h) is parametrised according to arc length.

assum = {h^′[x_1]^2 + r^′[x_1]^2 == 1, r^′[x_1] >0}

ℊ = ΜPullback[φ, 3, Id[3]] ; ℊ//MF

ℊ◦ {x_1, x_2, 0}//S//MF

{h^′[x_1]^2 + r^′[x_1]^2 == 1, r^′[x_1] >0}

( {{1, 0, 0}, {0, r[x_1]^2, 0}, {0, 0, 1}} )

All expressions are labelled:

ShowΜ[ℊ, 2] ;

christ↓

Γ_ (i, j)^1
Γ_ (i, j)^2
Γ_ (i, j)^3

shape= ( {{r^′[x_1] h^′′[x_1] - h^′[x_1] r^′′[x_1], 0, 0}, {0, h^′[x_1]/r[x_1], 0}, {0, 0, 0}} )

   κ_1= h^′[x_1]/r[x_1]   v_1= {0, 1}

   κ_2= r^′[x_1] h^′′[x_1] - h^′[x_1] r^′′[x_1]   v_2= {1, 0}

K=Πκ   (h^′[x_1] (r^′[x_1] h^′′[x_1] - h^′[x_1] r^′′[x_1]))/r[x_1]

H=Σκ/n  1/2 (h^′[x_1]/r[x_1] + r^′[x_1] h^′′[x_1] - h^′[x_1] r^′′[x_1])

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