EQUATIONS OF ELASTIC HYPERSURFACES

2. OUTLINE OF DIFFERENTIAL GEOMETRY 57
Also, if B : B → B is another linear operator, then
n∑
Tr (AB ⊤ ) = 〈h j , Ah k 〉〈h j , Bh k 〉 (2.254)
j,k=1
is independent of a particular frame { h j
} n
j=1 .
Proof: For the representation ÂD := J −1
V
AJ D = [ b jk ] n×n of A in another frame we get, due
to (2.248) and (2.250),
n∑
b jj =
j=1
n∑
m=1 j=1
n∑
g jm a mk g kj =
n∑
δ mk a mk =
and the independence from the frame follows.
The formula with the eigenvalues (2.252) follows if we select for H the frame of eigenvectors
of A (of the matrix representation [ a jk ] n×n of A): Ah j = λ j h j and 〈h j , Ah j 〉 =
λ j 〈h j , h j 〉 = λ j for all j = 1, . . . , n.
Since
n∑
(AB ⊤ ) jj = 〈h j , Ah k 〉〈h j , Bh k 〉 .
formula (2.254) follows from (2.252).
k=1
Consider an example: The tensor of type (0, 2) ∇ S U is defined for arbitrary tangential
vector field U ∈ TS by
The trace
m=1
(∇ S U)(V, W ) := 〈∂ W U, V 〉 = 〈∂ S W U, V 〉 ∀ V, W ∈ TS . (2.255)
Tr(∇ S U) =
n∑
m=1
a kk
n∑
〈∂ hj U, h j 〉 = Div S U (2.256)
j=1
is the divergence and is independent of a frame { h j
} n
j=1 .
Proof of Lemma 2.54: Let us consider the system ̂D := { } n
d j , generating the Gunter’s
j=1
derivatives and described in (2.235). The system ̂D is full in the tangential space TS ⊂
R n , which is also orthogonal to the normal vector field ν. Writing the repreentation of the
Weingarten operator in (2.234) in the full system ̂D we get the following:
〈W S U, V 〉 = II(U, V ) = 〈−∂ S U ν, V 〉 = −
n∑ ( )
Uk D k ν j Vj = 〈−(Dν)U, V 〉 . (2.257)
Since a vector field V ∈ TS is arbitrary, the equality (2.257) implies (2.238).
Due to (2.252) W S is the representation of the Weingarten operator in the canonical
frame { }
e j n
= e j of the Euclidean space j=1 Rn .
j,k=1