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Summary: THE GAUSS-BONNET 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 outward-pointing 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 unit-outward-pointing 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 Gauss-Bonnet formula
is this:
(1) 2 -
b
a
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