Page 303 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY
P. 303
Complex Analysis and Differential Geometry
Notes a tangent vector. Note that covariant and contravariant indices have different transformation
laws, cf. (1) and (2).
3
Proposition 1. Let X : U be a parametric surface, and let X be a reparametrization
X
j X ,
i
of X. Then T X T X. X, and Z z X z j then:
Furthermore, if Z T
u u u i
i
z z j u i j , ...(1)
u
where d u / u . i j
Proof. By the chain rule, we have:
u i
X X . ...(2)
i
j
u j
Thus T X T (u) X, and since we can interchange the roles of X and X, we conclude that
u
j
T X T (u) X. Substituting (2) in z X , we find:
j
u
i
z X z j u i j X ,
i
i
du
and (1) follows.
3
,
Definition 4. A vector field along a parametric surface X :U is a smooth mapping
. A vector field Y is tangent to X if Y (u) T X for all u U. A vector field Y is normal
Y : U 3 2 u
to X if Y(u) T X for all u U.
u
Example: The vector fields X and X are tangent to the surface. The vector field X × X 2
2
1
1
is normal to the surface.
We call the unit vector field
X X
N 1 2
X X 2
1
the unit normal. Note that the triple (X , X , N), although not necessarily orthonormal, is positively
2
1
oriented. In particular, we can see that the choice of an orientation on X, e.g., X X , fixes a unit
2
1
normal, and vice-versa, the choice of a unit normal fixes the orientation. Here we chose to use
the orientation inherited from the orientation u u on U.
2
1
2
Definition 5. We call the map N : U the Gauss map.
The Gauss map is invariant under orientation-preserving reparametrization.
Proposition 2. Let X : U be a parametric surface, and let N : U be its Gauss map. Let
2
3
X X be an orientation-preserving reparametrization of X. Then the Gauss map of X is N .
2 We often visualize Y(u) as being attached at X(u), i.e. belonging to the tangent space of at X(u); cf.
3
see footnote 1.
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