EQUATIONS OF ELASTIC HYPERSURFACES

3. CALCULUS OF DIFFERENTIAL OPERATORS ON HYPERSURFACES 69
and therefore we get
⎛
⎜
⎝
=
⎞
∂ 1 N 1 · · · ∂ 1 N n
.
. ..
⎟ . ⎠ =
∂ n N 1 · · · ∂ n N n
1 ( )
∂j ∂ k Ψ S − N j ∂ N ∂ k Ψ S
|∇Ψ S |
1
|∇Ψ S |
Then (3.16) and (3.17) yields (3.15).
Let θ(x) ∈ T x S then
⎛
⎜
⎝
⎞
∂1Ψ 2 S · · · ∂ 1 ∂ n Ψ S
.
. ..
⎟ . ⎠ (I n −N N ⊤ ). (3.17)
∂ n ∂ 1 Ψ S · · · ∂nΨ 2 S
W S (x)θ(x) = { N (x)N ⊤ (x) − I n
}
WS (x)θ(x).
Thus the eigenvalues of the matrices W S (x) and
A S (x) := { N (x)N ⊤ (x) − I n
}
WS (x) (3.18)
coincide and denote them by {κ j (X )} 1≤j≤n , setting the last one zero: κ n = 0.
From this representation we immediately have that the eigenvalues {κ j (x)} 1≤j≤n are the
solutions of the following equation
and the mean curvature is
det(A S (x) − κ I) = 0
H S (x) =
∑n−1
j=1
κ j (x)
n − 1 =
n∑
j=1
TrA S (x)
n − 1
for x ∈ S .
Let us now write (−1) n det(A S (x) − κ I) as a polynomial with respect to κ. Then the
Gauß’s principal curvature at x ∈ S equals to the coefficient at κ.
Example 3.9 Let us demonstrate obtained results for the ellipsoid Er,0 n−1 defined in (2.69).
We have
[ ]
W E
n−1 (x) = δ jk
= diag ( α 1 (x), . . . , α n (x) ) ,
r,0
where
α j (x) =
R r,0 (x)rk
2 n×n
1
rj 2R r,0(x) , N j(x) = x j α j (x) =
R r,0 (x) :=
[ n∑
j=1
(
xj
r 2 j
) 2
] 1/2
.
x j
r 2 j R r,0(x) ,