Page 312 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 24: Two Fundamental Form

i

where we have used the notation f  u i f   f / u . According to the definition, we are Notes
i
requiring that this matrix has rank 2, or equivalently that the vectors

1
2
3
1
2
x , x , x and X  x , x , x 3 2  are linearly independent. Another equivalent requirement
2
1
2
2
1
2
2   is injective.
3
is that dX : 
3
 

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.

If Y  T X, then it can be expressed as a linear combination in X and X :
 u 1 2
2
1
2 
2
i
Y  y X  y X  y X ,
i
1
i 1

i
where y  are the components of the vector Y in the basis X , X of T X. We will use the
2
u
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.
3 , 3 2
 A vector field along a parametric surface X :U   is a smooth mapping Y : U   . A
vector field Y is tangent to X if Y (u)  T X for all u  U. A vector field Y is normal to X if
u
Y(u)  T X for all u  U.
u
A symmetric bilinear form on a vector space V is function B : V V   satisfying:

B(aX + bY,Z) = aB(X,Z) + bB(Y,Z), for all X, Y  V and a, b  R.

B(X, Y) = B(Y, X), for all X, Y  V.

The symmetric bilinear form B is positive definite if B(X, X)  0, with equality if and only
if X = 0.
With any symmetric bilinear form B on a vector space, there is associated a quadratic form
Q(X) = B(X, X). Let V and W be vector spaces and let T : V  W be a linear map. If B is a
symmetric bilinear form on W, we can define a symmetric bilinear form T* Q on V by
T* Q(X, Y) = Q(TX, TY). We call T* Q the pull-back of Q by T. The map T is then an isometry
between the inner-product spaces (V, T* Q) and (W,Q).
24.6 Keywords
2

Diffeomorphism: 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 sa that  is an
orientation-preserving diffeomorphism.
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.
Gauss map: The Gauss map is invariant under orientation-preserving reparametrization.

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