Blaga P. Lectures on the differential geometry of - tiera.ru

5.2. Minimal surfaces 209
asymptotic line from the second family has an infinity of Bertrand mates. But then, as
we know, they have to be cylindrical helices. �
We shall give now, following Barbosa and Colares, a more direct prove of the Catalan’s
theorem, using a local parameterization of the ruled surface.
Another proof of the Catalan’s theorem. Let S ⊂ R 3 be a ruled surface. Then from the
definition, S has local parameterizations of the form
r(u, v) = α(u) + vγ.
We may choose the directrix α of the ruled surface to be an orthogonal trajectory of the
family of rulings, and γ, the directing vector of the rulings, to be a unit vector, at each
point of the directrix. We shall assume, moreover, that u is the natural parameter of the
directrix. A straightforward computation provides, for the coefficients of the first two
fundamental forms of the surface, the expressions:
E = 1 + 2vα ′ · γ ′ + v 2 γ ′2 ,
F = 0,
G = 1,
1
D = �
1 + 2vα ′ · γ ′ + v2γ ′2 (α′ + vγ ′ , γ, α ′′ + vγ ′′ ),
D ′ =
D ′′ = 0.
Thus, the minimality condition reads
or, which is the same,
1
� 1 + 2vα ′ · γ ′ + v 2 γ ′2 (α′ + vγ ′ , γ, γ ′ ),
(α ′ + vγ ′ , γ, α ′′ + vγ ′′ ) = 0,
(α ′ × γ) · α ′′ + v � (α ′ × γ) · γ ′′ + (γ ′ × γ) · α ′′� + v 2 (γ ′ × γ) · γ ′′ = 0.
As the left hand side of the previous equation is a polynomial function in v, the equality
holds for any v iff the coefficients of the polynomial function vanish simultaneously:
(α ′ × γ) · α ′′ = 0 (a)
(α ′ × γ) · γ ′′ + (γ ′ × γ) · α ′′ = 0 (b)
(γ ′ × γ) · γ ′′ = 0. (c)