EQUATIONS OF ELASTIC HYPERSURFACES

48 SHELLS
is the contravariant metric tensor (the inverse to the covariant metric tensor):
G S G −1
S = I or g jm(p)g mk (p) = g jm (p)g mk (p) = δ jk . (2.199)
Due to the orthogonality of the systems { } n−1
g k and { g k} n−1
we easily establish that
k=1 k=1
the covariant and the contravariant frames are related as follows:
∑n−1
∑n−1
g j = g jk g k , g j = g jk g k , j, k = 1, . . . , n − 1 . (2.200)
k=1
k=1
From (2.199) and (2.200), invoking the symmetry property g kj = g jk , we derive the following
connection between the covariant and the contravariant representation of a tangential
vector field (so called ”raising“ and ”lowering“ the indices):
∑n−1
U = Ũ j g j =
j=1
∑n−1
j,k=1
∑n−1
g kj Ũ j g kj g j = Ũ k g k ,
∑n−1
∑n−1
Ũ k := g kj Ũ j , Ũ k := g kj Ũ j , k = 1, . . . , n − 1 for U ∈ TS .
j=1
j=1
k=1
(2.201)
From (2.195), (2.196) and (2.190) we derive easily that the coefficients b jk in (2.195),
known as the covariant curvature tensor (or the coefficients of the second fundamental
form; cf. (2.232)) and the coefficients b k j in (2.196), known as the mixed curvature tensor,
are related as follows:
b k j =
∑n−1
m=1
g km b mj , b jk =
∑n−1
m=1
g km b m j , j, k = 1, . . . , n − 1 . (2.202)
The coefficients b jk are symmetric b jk = b kj , while b k j in general are not. Thus,
B S (X ) := [ b k j (X ) ] (n−1)×(n−1) = G−1 S (X )B S (X ) ,
B S (X ) := [ b jk (X ) ] (n−1)×(n−1) = B⊤ S (X ) , X ∈ S (2.203)
and the eigenvalues κ 1 (X ), . . . , κ n−1 (X ) of B S (X ) are called the principal curvatures,
while
H S (X ) := Tr B S (X )
n − 1
∑n−1
, Tr B S (X ) :=
j=1
K S (X ) := det B S (X ) = det B S (X )
det G S (X ) = κ j(X ) ,
n−1
b j j (X ) = ∑
j=1
κ j (X )
n − 1 ,
X ∈ S
(2.204)
the mean curvature and the Gaußian curvatures of S .
A point X = Θ(x) ∈ S ⊂ R 3 of two-dimensional hypersurface is called elliptic,
parabolic or hyperbolic if the Gaußian curvature is positive K S (x) = λ 1 (x)λ 2 (x) > 0,