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

170 Chapter 4. General theory of surfaces
This system is, again, completely integrable, because the integrability condition
is equivalent to the condition
∂2r ∂ui∂u j = ∂2r ∂u j∂ui ∂r ′ i
(4.17.26)
∂u j = ∂r′ j
∂ui (4.17.27)
which is true, as one can convince oneself looking at the first equation (4.17.22), because
of the symmetry of the second matrix (h ji = hi j) and because of the symmetry of the
Christoffel’s coefficients in the lower indices. Therefore, applying again the existence
and uniqueness theorem, it follows that there exists an open neighborhood V ⊂ W ⊂ U
of u0 and a single C3 function r : V → R3 such that r(u0) = p0.
We are not done yet, because we still have to show that gi j and hi j are the first two
fundamental forms of the parameterized surface defined by r. Apparently, we also have
to show that r is regular. But this follows immediately if we prove that g is the first
fundamental form, because
gi j = ∂r ∂r
·
∂ui ∂u j
and then
∂r ∂r
× � 0,
∂ui ∂u j
because the square of the norm of this vector is not zero (as it is equal to the determinant
of the first fundamental form, which is strictly greater then zero, since the form is positively
defined). It is, actually, enough to show that, all over V, the following relations
are fulfilled: ⎧⎪⎨⎪⎩
r ′ i · r′ j = gi j,
r ′ i · n = 0, . (4.17.28)
n · n = 1
To this end, we shall compute the derivatives with respect to the coordinates of the scalar
products, taking into account the Gauss-Weingarten equations and we get the system of
first order partial differential equations:
⎧
∂(r ′ i · r′ j )
⎪⎨
⎪⎩
∂u k
∂(r ′ j
∂ui = 2�
Γ
l=1
l ik (r′ l · r′ 2�
j ) + Γ
l=1
l jk (r′ l · r′ i ) + hik(r ′ j · n) + h jk(r ′ i · n)
· n)
= − 2�
∂( n · n)
∂u i
l,k=1
= −2 2�
hilglk (r ′ k · r′ 2�
j ) + Γ
l=1
l i j (r′ l · n) + hi j( n · n)
hilg
l,k=1
lk (r ′ k
· n)
. (4.17.29)