EQUATIONS OF ELASTIC HYPERSURFACES

3. CALCULUS OF DIFFERENTIAL OPERATORS ON HYPERSURFACES 81
for k = 1, . . . , n; also we recall that
n∑
ν j D j = 0 and M jk = ν j D k − ν k D j (cf. Lemma
j=1
3.11.vi, 3.11.v)) and proceed as follows:
1
2
n∑
j,k=1
M 2
jkϕ= 1 2
n∑
[ν j D k − ν k D j ] 2 ϕ = 1 2
j,k=1
+ ν k D j ν k D j ϕ − ν k D j ν j D k ϕ] =
=
n∑
Dkϕ 2 −
k=1
n∑
[ν j D k ν j D k ϕ − ν j D k ν k D j ϕ
j,k=1
n∑
[ν j D k ν j D k ϕ − ν j D k ν k D j ϕ]
j,k=1
n∑ [
νj ν k D k D j ϕ + ( )
D k ν k νj D j ϕ ] =
j,k=1
n∑
Dkϕ 2 = ∆ S ϕ .
k=1
We remind that the surface gradient ∇ S maps scalar functions to the tangential vector
fields
∇ S : C 1 (S ) → TV (S ) := C(S , TS ) (3.66)
and the scalar product with the normal vector vanishes everywhere on the surface S :
ν(X ) · ∇ S ϕ(X ) ≡ 0 for all ϕ ∈ C 1 (S ) . (3.67)
The directing vectors { } n
d j of Günter’s derivatives { } n
D
j=1 j are tangential and represent
j=1
a full system (cf. (3.25)-(3.27)). But the derivative D j V is not an automorphisms of the
tangential vector space D j : TV (S ) ↛ TV (S ), i.e., did not represent covariant derivative
of vector fields (cf. § 2.7). To improve this we just eliminate, as customary, the normal
component of the vector by applying the canonical orthogonal projection π S onto TS (cf.
(1.11)) and obtain
D S j
V := π S D j V = D j V − 〈ν, D j V 〉ν = D j V + (W S V ) j ν
= D j V + (D V ν j )ν , (3.68)
where W S := [ n∑
D j ν k
]n×n , D V ϕ := V k D k ϕ .
k=1
because
n∑
n∑ [ ]
〈ν, D j V 〉= ν m D j V m = Dj (ν m V m ) − V m D j ν m
m=1
=−
m=1
n∑
V m D j ν m = −(W S V ) j = −
m=1
m=1
n∑
V m D m ν j = −D V ν j .
It is easy to check that the operators D S j
are automorphisms of the tangential vector space
D S j : TV (S ) → TV (S ) . (3.69)