Page 302 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 24: Two Fundamental Form

Write X = (x , x , x ), and each x = x (u , u ), then the Jacobian has the matrix representation: Notes
1
2
1
i
2
i
3
1
 x 1 1 x 
2
 
dX   x 2 1 x 2 2 
 x 3 x 3 
 1 2 
i
where we have used the notation f  u i f   f / u . According to the definition, we are requiring

i
1 

2
3
2
1
1
that this matrix has rank 2, or equivalently that the vectors X = x , x , x and X   x , x , x 3 2 
1
2
2
2
1
2
3
are linearly independent. Another equivalent requirement is that dX :  2   is injective.
Example: Let U   be open, and suppose that f : U   is smooth. Define the graph
2
of f as the parametric surface X(u , u ) = (u , u , f(u , u )). To verify that X is indeed a parametric
1
2
1
2
2
1
surface, note that:
 1 0 
dX    0 1  
 f f 
 1 2 
so that clearly X is non-singular.
2
A diffeomorphism between open sets U, V   is a map  : U  V which is smooth, one-to-
one, and whose inverse is also smooth. If det(d) > 0, then we say that  is an orientation-
preserving diffeomorphism.
3
 

Definition 2. Let X : U   3 , and X : U   be parametric surfaces. We say that X is



reparametrization of X if X  X  , where : U  U is a diffeomorphism. If  is an orientation-

preserving diffeomorphism, then X is an orientation-preserving reparametrization.

Clearly, the inverse of a diffeomorphism is a diffeomorphism. Thus, if X is a reparametrization

of X, then X is a reparametrization of X.

Definition 3. The tangent space T | X of the parametric surface X : U   at u  U is the
3
u
2-dimensional linear subspace of  spanned by the two vectors X and X . 1
3
1
2
If Y  T X, then it can be expressed as a linear combination in X and X :
u
1
2
2
2
1
2 
i
Y  y X  y X  y X ,
1
i
i 1

i
where y  are the components of the vector Y in the basis X , X of T X. We will use the
u
2
1
Einstein Summation Convention: every index which appears twice in any product, once as a subscript
(covariant) and once as a superscript (contravariant), is summed over its range. For example, the
above equation will be written Y = y X . The next proposition show that the tangent space is
i
i
invariant under reparametrization, and gives the law of transformation for the components of
1 Note that the tangent plane to the surface X(U) at u is actually the affine subspace X(u) + T X.
However, it will be very convenient to have the tangent space as a linear subspace of  . u
3
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