MORSE THEORY AND THE GAUSS-BONNET FORMULA Alina ...

4 An Extrinsic Proof
4.1 A proof of the Gauss-Bonnet theorem
Let M n be a compact manifold embedded isometrically in R N for some N > n with
unit normal bundle ν 1 M.
Theorem 4.1. Extrinsic Gauss-Bonnet Theorem
�
ν 1 M
�
ν 1 M
det AudV ν 1 M(u) = χ(M) Vol(S N−1 )
Notice that if n is odd then det A−u = − det Au. By Fubini’s theorem,
� �
�
det AudVν1M(u) = det AudVν1 pMdp = 0dp = 0, and χ(M) = 0.
M
ν 1 pM
The proof is a beautiful application of Morse theory and integral geometry and
we will describe it in the remainder of this section.
Definition 4.2. For a unit vector v ∈ S N−1 , define the height function hv : M → R
by hv(p) = 〈p, v〉. In this context, p represents a point in M as well as the vector
connecting the origin of R N with that point.
Proposition 4.3. S0 = {v ∈ S N−1 : hv is a Morse function } has full measure in
S N−1 .
From the definition of height functions we can deduce that p ∈ M is a critical point
for the height function hv if and only if v is orthogonal to TpM. This is equivalent
to v being parallel to some normal vector u ∈ ν 1 M based at p. Moreover, there is a
close connection between height functions and the second fundamental form:
M
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