Page 331 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes We also have that detB = detD. Thus taking into the account that (7) holds for D:





logdetB  logdetD   tr D D  tr B B . 
1
1


In order to prove the general case, it is more convenient to look at the equivalent identity:



1

detB  tr   detB B B .   ...(8)
Note that by Kramer’s rule, the matrix (detB)B is the matrix of co-factors of B, hence its
–1
components being determinants of minors of B, are multivariate polynomials in the components
of B. Thus, both sides of the identity (8) are linear polynomials
n n
 p B ;B    pij(B)b ,  ij  q B ;B    qij(B)b ,  ij
i,j 1 i,j 1


in the components b of B, whose coefficients p (B) and q (B) are themselves multivariate
ij
ij
ij
polynomials in the components b of B. Since the set of matrices with distinct eigenvalues is an
ij



open set U  S n n , we have already proved that  p B ;B   q B ;B holds for all values of B, and






all B  U. For each such B  U the equality  p B ;B   q B ;B for all B implies that p (B) = q (B)

ij
ij
for i, j = 1, , n. Since this holds for all B in an open set, we conclude that p = q , and hence
ij
ij
p = q.
We remark that the more general identity (8) in fact holds, as easily shown, for all square
matrices B. An immediate consequence of the proposition is that:

1
 detB   1 tr B B  detB, ...(9)

2
for any continuously differentiable family of symmetric positive definite matrices B. We are
now ready to prove the proposition.
Proof of Proposition 6. Differentiating the area (4) under the integral sign, and using (9), we get:
dA (V)  1 g ij dg ij det g 1 2 1 g ij dg ij dA.
F
dt 2  V dt   du du  2  V dt
ij
Since Y is smooth, we have at t = 0 that dF /dt = (dF/dt) = Y , and thus
i
i
i
dg
g ij ij  g ij  Y X  X X  2g X Y .
ij


 j
dt i j i i j
This completes the proof of the proposition.
Since the variation of the area dA (V)/dt is a linear functional in the generator dF/dt of the
F
variation, it is possible to decompose any variation into tangential and normal components. We
begin by showing that the area doesn’t change under a tangential variation. This is simply the
infinitesimal version of Proposition (5).
Proposition 7. Let X : U   be a parametric surface, and let F(u; t) be a compactly supported
3
tangential variation. If V  U is open with V compact in U, and the support of F contained in V,
then dA (V)/dt = 0.
F
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