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

the second fundamental form of M \ ∂M if u is based at some x ∈ M \ ∂M or the
second fundamental form of ∂M if u is based on the boundary.
Theorem 5.1. Gauss-Bonnet for manifolds with boundary
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det AudVν1M\∂M(u) +
u∈ν 1 (M\∂M)
u∈ν1 + ∂M
det AudVν1 + ∂M(u) = χ(M) Vol(S N−1 )
Proof. For every v ∈ S N−1 we consider the height function hv : M → R,
hv(p) = 〈v, p〉. The result follows via essentially the same proof as for theorem 4.1.�
Dillen and Kühnel offer an alternative proof of the theorem via the tube method,
using the relation χ(NM) = (1 + (−1) N )χ(M); the details can be found in [5].
5.2 Submanifolds of the sphere
Let M n be a compact manifold without boundary isometrically embedded in a unit
sphere S N . Applying Morse theory to distance functions on M will allow us to
obtain yet another version of the Gauss-Bonnet formula, with the advantage that the
integrand has a very interesting geometric interpretation.
Consider the tube of radius r around M as defined in section 2.1. If r is small
enough, the tube is an embedded submanifold of S N but for r = π the tube has self-
intersections. However, its signed volume contains important topological information
about M. More precisely, consider the normal disk bundle:
ν π M = {(x, w) : x ∈ M, w ∈ TxS N , w ⊥ Tx and �w� ≤ π}
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