6.5 Proof of local Gauss-Bonnet
Theorem.
By loke (http:/blog.nus.edu.sg/ma4271/author/loke/).
Finally we can give a proof of the local Gauss Bonnet Theorem!
First we recall the local Gauss-Bonnet Theorem.
The notations are explained in Section 6.2
(http:/

At a point
.
6.2 Local Gauss-Bonnet theorem
on one of the curves
Hence
At the point when
By loke (http:/blog.nus.edu.sg/ma4271/author/loke/).
, we could compute its geodesic curvature
is a function on the curve
and
.
meets, there is a sudden change of dir

5.5 Geodesic curvature
is the called the geodesic curvature of
By loke (http:/blog.nus.edu.sg/ma4271/author/loke/).
Since
We will review geodesic curvature which we have seen in the tutorial.
Let
be a curve parametrized by arc length on a regular surface

5.7 Differential equations
satisfied by geodesics
By loke (http:/blog.nus.edu.sg/ma4271/author/loke/).
2. Suppose
is perpendicular to the tangent space.
Then it is perpendicular to
and
, i.e.
We state an existence result about geodesics.
Proposition.
Let

For (i)
5.6 Geodesic
By loke (http:/blog.nus.edu.sg/ma4271/author/loke/).
(ii),
is the orthogonal projection of
It is zero if and only if
(ii)
(i)
(iv). By definition
It is zero if and only
Given a curve
parametrized by arc length on a regular surface , t

6.1 Line integrals and surface
integrals review
Review surface integrals.
Let
Let
be a regular surface.
be a coordinate function.
Let
By loke (http:/blog.nus.edu.sg/ma4271/author/loke/).
be a function on the surface.
In this chapter, we will discuss a bea

6.3 Global Gauss-Bonnet
Theorem with boundaries
By loke (http:/blog.nus.edu.sg/ma4271/author/loke/).
(https:/blog.nus.edu.sg/ma4271/files/2016/03/Torus-triang-21o8ct6.png)
We now state the global Gauss-Bonnet theorem.
A result says that every nice region

Theorem. (Gauss-Bonnet Theorem for compact surfaces without boundary)
Let be a compact regular surface. Then
6.4 Global Gauss Bonnet Theorem
By loke (http:/blog.nus.edu.sg/ma4271/author/loke/).
Proof.
Since there is no boundary, only the middle term on su

(https:/blog.nus.edu.sg/ma4271/files/2016/03/walphat-vergtg.jpg)
5.4 Differentiating vector fields
and Christoffel symbols
By loke (http:/blog.nus.edu.sg/ma4271/author/loke/).
In order to compute
Reference. Page 239 in [DoCarmo].
, we discard the
componen

where
5.3 Differentiating vector fields
along a curve
By loke (http:/blog.nus.edu.sg/ma4271/author/loke/).
is the unit normal vector.
The end result is a vector on the tangent space at
and we denote this vector by
Definition.
The above vector
is called th