EQUATIONS OF ELASTIC HYPERSURFACES

3. CALCULUS OF DIFFERENTIAL OPERATORS ON HYPERSURFACES 99
B. If a Killing’s vector field vanishes at n points U(X k ) = 0, X 1 , · · · , X n ∈ S and
the differences { x j − x 1} n
are linearly independent, then U vanishes identically
j=2
U(X ) ≡ 0 on S .
In particular, if a Killing’s vector field U vanishes on the non-empty part of the surface
U(X ) ≡ 0 on S 0 ⊂ S , mes S 0 > 0, then U vanishes identically U(X ) ≡ 0 on S .
C. The space of Killing’s vector fields R(S ) is finite dimensional
dim R(Ω) ≤ dim R(R n ) =
n(n + 1)
2
. (3.151)
Proof: Due to (2.225) and to (2.226)
0 = D jk (U) = (Def S U) jk = 1 2 (U j;k + U k;j ) ,
U k;j := ∂ j U k −
∑n−1
m=1
Γ m kjU m , ∀ j, k = 1, . . . , n − 1 .
(3.152)
∑n−1
If the tangential vector field V = V j g j is also written in cartesian coordinates
j=1
∑n−1
V = V j g j =
j=1
n∑
Ṽ j e j (3.153)
j=1
the coefficients are related as follows
D jk (V )(x) = ( D jm (V )g j mg m k
)
(X ) for all X = Θ(x) , x ∈ Ω (3.154)
(cf. [Ci3, Theorem 1.3.1]), where
D jk (V )(x) := 1 2[
∂k Ṽ j (x) + ∂ j Ṽ k (x) ] , x ∈ Ω , j, k = 1, 2, . . . , n − 1 . (3.155)
Then, from (3.152) and from (3.154), follows
D jk (V ) ≡ 0 ∀ x ∈ Ω , ∀j, k = 1, 2, . . . , n − 1 . (3.156)
By differentiating (3.156) and recalling that ∂ k ∂ l Ṽ j = ∂ l ∂ k Ṽ j , we get
∂ j ∂ k Ṽ m = ∂ j D km (V ) + ∂ k D jm (V ) − ∂ m D jk (V ) = 0 for all j, k, m = 1, 2, . . . , n − 1 .
Therefore,
or
Ṽ j (x) = a j + b j1 x 1 + · · · + b jn x n
j = 1, 2, . . . , n
V (x) = a + B · x with B = [b jk ] n×n . (3.157)