Page 348 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 28: Local Intrinsic Geometry of Surfaces


Two Riemannian surface patches (U, g) and U, g  are isometric if there is a diffeomorphism Notes


 : U  U such that
l
g  g   m j , ...(1)

lm
ij
i
l
l
i
where    u / u . In fact, Equation (1) reads:

i

d * g g, 

where d* g is the pull-back of g by the Jacobian of  at u.  We then say that  is an isometry
between (U, g) and  U, g . As before, we denote by g the inverse of the matrix g .

ij
 
ij
We also denote the Riemannian metric:
ds = g du du, j
2
i
ij
and at times refer to it as a line element. The arc length of a curve :[a,b]  U is then given by:
b
i
 
L  a g    j dt.

ij
 , 
Note that the arc length is simply the integral of g   .
Example: Let U   be open, and let   be the identity matrix, then (U, ) is a
2
let 
ij
Riemannian surface. The Riemannian metric d will be called the Euclidean metric.
Example: Let X : U   be a parametric surface, and let g be the coordinate
3
representation of its first fundamental form, then (U, g) is a Riemannian surface patch. We say
3


that the metric g is induced by the parametric surface X. If X  X  : U   is a reparametrization


of X and g  the coordinate representation of its first fundamental form, then U, g  is isometric

to (U, g).
2
2
2
Example: (The Poincaré Disk). Let D  (u,v): u  v   1 be the unit disk in  , and let
4
g  2  ij
ij
1 r 2 
2
2
where r  u  v is the Euclidean distance to the origin. We can write this line element also as
2
2
ds  4 du  dv 2 2 . ...(2)
1 u 2  v 2 
The Riemannian surface (D, g) is called the Poincar´e Disk. Let U = {(x, y) : y > 0} be the upper half-
plane, and let
1
h  y 2  ij .
ij

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