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440 PHILLIP A. GRIFFITHS
do- A A n
P,(h, t)dM A l
where P,(h, t) is given by (1.11). Consequently
d" cte It I,,(RM)dM.
In case the codimension m is odd, we may use
x(H) x(M) x(S m- 1)
2 x(M)
together with (1.15) to conclude that
Im (1.17) Cte I,,(RM)dM x(M).
In case m is even we simply consider M C IRN / by adding trivially an extra
coordinate to IRN, and then the argument still applies.
Setting In(RM)dM KdM, formula (1.17) is the famous Gauss-Bonnet theorem.
As mentioned above, we have more or less retraced its original derivation,
where it remains to show that the formula is valid for any Riemannian metric.
One way, of course, is by quoting the Nash embedding theorem. For pedagogical
purposes it is probably better to show directly that for a 1-parameter
family of Riemannian metrics the variation of the integral
by differentiation under the integral sign and using the Chern-Weil formalism to
write
Ot
(I,,(R,,M)dM)
and then applying Stokes’ theorem. (7
A concluding remark is that, like many of the most beautiful formulas of
geometry, the Gauss-Bonnet is an intrinsic relation which was however discovered
and first proved by extrinsic methods. Chern’s subsequence intrinsic
proof (s was based on the tangent sphere bundle rather than the extrinsic normal
sphere bundle.
(c) Gauss mapping and the Gauss-Bonnet theorem. This last observation
suggests that we consider directly the tangential Gauss mapping on M, which
will be done following some observations on Grassmannians.
We denote by Gn(n, N) the Grassmann manifold of oriented n-planes through
the origin of IRN. Note that Gn(1, N) is the sphere S N and that Gn(N n, N)