Page 399 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes is also a geodesic, and

 


 (t, v) = ( t, v).

  
From lemma 2, for v  0, if (1,v) is defined, then


  v  
  v ,     (1,v).
 v 
 
This leads to the definition of the exponential map.
Definition 2: Given a surface X :   E and a point p = X(u, v) on X, the exponential map expp
3
is the map
exp : U  X()
p
defined such that


   v  
exp (v)    v ,     (1,v),
p
 v 
 
 
     v
2
where (0,v)  p and U is the open subset of  (= T (X)) such that for every v  0,   v ,   is
p
  v  

defined. We let exp (0) = p. It is immediately seen that U is star-like. One should realize that in
p
general, U is a proper subset of .
Example: In the case of a sphere, the exponential map is defined everywhere. However,

given a point p on a sphere, if we remove its antipodal point –p, then exp (v) is undefined for
p
points on the circle of radius .
Nevertheless, expp is always well-defined in a small open disk.

Lemma 3: Given a surface X :   E , for every point p = X(u, v) on X, there is some  > 0, some
3
open disk B of center (0, 0), and some open subset V of X() with p  V, such that the exponential

map exp : B  V is well defined and is a diffeomorphism.
p
A neighborhood of p on X of the form exp (B) is called a normal neighborhood of p.
p
The exponential map can be used to define special local coordinate systems on normal
neighborhoods, by picking special coordinates systems on the tangent plane.
In particular, we can use polar coordinates (, ) on  . In this case, 0 <  < 2. Thus, the closed
2
half-line corresponding to  = 0 is omitted, and so is its image under exp . It is easily seen that in
p
such a coordinate system, E = 1 and F = 0, and the ds is of the form
2
ds = dr + G d .
2
2
2
The image under exp of a line through the origin in  is called a geodesic line, and the image
2
p
of a circle centered in the origin is called a geodesic circle. Since F = 0, these lines are orthogonal.

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