Gauss-Bonnet's Formula and Closed Frenet Frames

Closed space curves with
non-vanishing curvature
defines via the Frenet
formulas some closed curves
on the unit 2-sphere, called
the spherical indicatrices.
By using Gauss-Bonnet's
Formula after cutting a
spherical curve into simple
closed sub-curves an index on
the unit 2-sphere is found.
This spherical index may be
seen as a spherical analogy
to the winding number of
closed plane curves. The
spherical index has the
property that the integral
over the unit 2-sphere
of this index, equals the
integrated geodesic
curvature of the spherical
curve. Using this result on
the spherical indicatrices of
a space curve, we obtain
almost similar proofs of some
(generalizations of)
classical theorems. The
spherical index gives both
upper and lower bounds on the
total curvature and
total torsion of space curves.