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

202 Chapter 5. Special classes of surfaces
surface. Eisenhart claims that Meusnier actually showed that for minimal surfaces the
mean curvature vanishes. This is not quite correct: Meusnier was not aware of the
true meaning of the the relation he found. In fact, the notion of mean curvature was
introduced only fifty years later, by Sophie Germain.
An alternative proof of the fact that the minimal surfaces are the solution of the
variational problem was found by Darboux. A particularly clear exposition of this proof
can be found in the book of Hsiung.
For further convenience, we shall need the following definition:
Definition 5.2.2. A local parameterization (U, r) of a surface S is called isothermic if,
with respect to this parameterization, the coordinate lines are orthogonal (i.e. F = 0)
and E = G. This means, actually, that the matrix of the first fundamental form is just the
unit matrix multiplied by a positive function.
On any surface which is at least C 2 there exist isothermic parameterizations. The
proof of this claim is based on the theory of partial differential equations and it is not
interesting for our exposition.
Proposition 5.2.1. A surface is minimal iff its asymptotic directions are orthogonal.
Proof. A surface is minimal iff the coefficients of the first two fundamental forms verify
the equation
ED ′′ − 2FD ′ + GD = 0.
Let us choose, for the surface, an isothermic parameterization, with respect to which we
have
E = G = λ 2 (u, v), F = 0.
Then the minimality condition for the surface becomes
E(D ′′ + D) = 0
and, as E � 0, the surface being regular, we have
D ′ + D ′′ = 0.
On the other hand, the equation for the slopes of the asymptotic directions is
Dt 2 + 2D ′ t + D ′′ = 0
and one can see immediately that the asymptotic directions are orthogonal iff the roots
of this equation verify t1t2 = −1, which is equivalent to D = −D ′′ . �