Page 353 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 28.4 Summary

2

Let U   be open. A Riemannian metric on U is a smooth function g : U   2 2 . A
 
Riemannian surface patch is an open set U equipped with a Riemannian metric.
The tangent space of U at u  U is  . The Riemannian metric g defines an inner-product on
2
each tangent space by:
g(Y,Z) = g y z,
i j
ij
where y and z are the components of Y and Z with respect to the standard basis of  . We
i
2
j
2
will write Y  g(Y,Y), and omit the subscript g when it is not ambiguous.
g
Let U   be open, and let   be the identity matrix, then (U, ) is a Riemannian
2
let 
 ij
surface. The Riemannian metric d will be called the Euclidean metric.
Let (U, g) be a Riemannian surface. The Christoffel symbols of the second kind of g are

defined by:
1
m
  2 g mn  g ni,j  g nj,i  g ij,n  .
ij
The Gauss curvature of g is defined by:
1
m
n
K  g ij   m    m    m  .
2 ij,m ij nm im nj
3
,
If (U, g) is induced by the parametric surface X :U   then these definitions agree.



Let Y and Z be vector fields on U, and let : U   U be a diffeomorphism, then

  , d Z . 

d        d Y     
 Y,Z 



Let Y be a compactly supported vector field on the Riemannian surface (U, g). Then, we

have:
 U div Y dA  0.
28.5 Keywords
Riemannian metric: A Riemannian metric on U is a smooth function g : U   2 2 . A Riemannian


surface patch is an open set U equipped with a Riemannian metric.
Euclidean metric: 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.
28.6 Self Assessment
1. A ................. on U is a smooth function g : U   2 2 . A Riemannian surface patch is an open


set U equipped with a Riemannian metric.
2. Let U   be open, and let   be the identity matrix, then (U, ) is a Riemannian
2
let 
ij
surface. The Riemannian metric d will be called the .................

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