 
Summary: THE GAUSSBONNET THEOREM ON THE UNIT SPHERE
GREG W. ANDERSON
1. Statement of the formula
Let S be the unit sphere
x2
+ y2
+ z2
= 1
oriented by choosing the outwardpointing unit normal vector at each point of S.
Let : [a, b] S be a closed curve on the sphere bordering some region D S on
the sphere. Let N(t) denote the unitoutwardpointing normal vector to S at the
point (t). (This is just a fancy way of saying N(t) = (t) but it is meant to instill
the right attitude toward the geometry.) Let T(t) denote the unit tangent vector
to the curve at the point (t). (This is just a fancy way of saying T(t) = (t)
 (t)
but as before it is meant to instill the right attitude.) The GaussBonnet formula
is this:
(1) 2 
b
a
