Page 337 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Interchanging n and i and adding, we get (17). Now, differentiate (21), and taking into account
bj
ij
ia
that g   g g ab,l g , substitute (17) to get:
,l
bm
jm
N   k g X  k g ja  n al g   n bl g na g X m
ij
ij,l
m
nb
i,l
jm
 k g jm  a ml X  k m l g jm N   k  ij,l  k  n jl  g X  k k m l g jm N.
ij
ij
m
in
a
Note that the last term is symmetric in i and l so that interchanging i and l, and subtracting, we
get:
jm
N  N   k ij,l  k il,j   n ij k   n il k jn  g X m
i,l
ln
l,i
which vanishes by (19). Thus, it follows that (23) is satisfied. We conclude that given values for
X , X , N at a point u  U there is a unique solution of (20)–(21) in U. We can choose the initial
0
1
2
values to that X  X = g , N  X = 0, and N  N = 1 at u . Using (20) and (21), it is straightforward
j
i
0
i
ij
to check that the functions h = X  X, p = N  X and q = N  N, satisfy the differential equations:
i
ij
i
i
j
h ij,l   n il h   n jl h  k p  k p ,
ni
il
j
jl
nj
i
p i,j   k g lm h mi   m p  k q,
jl
il
ij
m
jm
q   2k g p .
m
ij
i
However, the functions h  g ,p  0 and q = 1 also satisfy these equations, as well as the same
ij
ij
i

initial conditions as h  X X ,p  N X and q  N N at u . Thus, by the uniqueness statement


i
0
ij
i
i
j
mentioned above, it follows that X N  g , N N  0, and N  N = 1. Clearly, in view of (20) we


j
i
ij
i
have X = X , hence there is a function X : U   whose partial derivatives are X . Since   is
g
3
ij
i,j
i
j,i
positive definite we have that X , X are linearly independent, hence X is a parametric surface
1
2
g
with first fundamental form   . Furthermore, it is easy to see that the unit normal of X is N,
ij
and X X   N X   k , hence, the second fundamental form of X is k . This completes the


i
j
ij
ij
ij
proof of the existence statement.
Assume now that X is another surface with the same first and second fundamental forms. Since

X and X have the same first fundamental form, it follows that there is a rigid motion R(x) = Qx


 . Let
R


  , QX u 
  
  , QN u 
+ y with Q SO(n; )  such that X u 0 X u 0 i   X u 0   N u 0
0
i
0
ˆ
ˆ

ˆ

ˆ
X R X. Since the two triples  X , X , N and  X , X , N both satisfy the same partial





1
2
1
2
differential equations (20) and (21), it follows that they are equal everywhere, and consequently
ˆ


X  X R X.

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