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

Definition 3.1. f is a Morse function if it has no degenerate critical points on M.
Morse functions are very useful for understanding the geometry and topology of
compact manifolds and the standard reference for their properties is [9]. Here are
some of the statements most relevant for our study:
Theorem 3.2. Morse Lemma. [9] Let p be a nondegenerate critical point of f and
let k be the index of f at p. Then there is a local coordinate system (x1, ..., xn) in a
neighborhood U of p with xi(p) = 0 for all i such that in the entire neighborhood U
we have f = f(p) − x 2 1 − ... − x 2 k + x2 k+1 + ... + x2 n.
Theorem 3.3. [9] If f is a Morse function on M and every M a = f −1 (−∞, a] is
compact, then M has the homotopy type of a CW-complex, with one cell of dimension
k for each critical point of index k.
Theorem 3.4. (Weak) Morse Inequalities. [9] If bk is the k-th Betti number of
M and Ck denotes the set of critical points of index k of a Morse function on M then
bk ≤ #Ck for any k and �
(−1) k #Ck = �
(−1) k bk = χ(M).
k
k
Morse theory also works on a manifold with boundary, if we count nondegenerate
critical points carefully. If M is a compact manifold with boundary ∂M and f :
M → R is a smooth function, consider f|M\∂M and f|∂M separately. A critical point
of f|M\∂M is a point p ∈ M \ ∂M where the induced map from Tp(M \ ∂M) to Tf(p)R
is zero, and degeneracy is defined in the same fashion as before. Since ∂M is an
(n − 1)-manifold, we can simply consider f|∂M : ∂M → R; a critical point is again
a point p ∈ ∂M where the map (f|∂M)∗ : Tp∂M → Tf(p)R is zero. Such a critical
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