150 D. SALAMON, FLOER HOMOLOGY
Definition 1.6. Let ( M, w) be a compact symplectic manifold. Then the minimal
Chern number of (M,w) is the integer
N = inf { k > 0 13 v : 82 ~ M, fs
2
v* c1 = k}.
If fs 2 v*c1 = 0 for every v : 82 ~ M we call N = oo the minimal Chern number.
If N =I-oo then (c1,7r2(M)) =NZ.
In the following we shall assume ( 4) and, in the case T =I- 0, normalize the
symplectic form such that fs 2 v*w E 'lJ., for every smooth map v : 82 ~ M.
1.3. The Morse-Smale-Witten complex
Let M be a compact smooth Riemannian manifold and f : M ~ IR be a Morse
function. Denote by Crit(J) = {x EM : df(x) = O} the set of critical points off.
The Morse condition asserts that the critical points are all nondegenerate. Thus
the Hessian d^2 f(x) : Tx M x TxM ~ IR is nondegenerate for every x E Crit(J).
In lo cal coordinates d^2 f(x) is given by the matrix of second partial derivatives and
the nondegeneracy condition asserts that this matrix is nonsingular.
Exercise 1. 7. Let V denote the Levi-Ci vita connection of the Riemannian metric.
Prove that the linear operator V^2 f(x) : T xM ~ T xM defined by V^2 f(x)~ =
VE. V f(x) for~ E T xM is symmetric with respect to the given Riemannian metric.
If df(x) = 0 prove that
(VE.\lf(x),TJ) = d^2 f(x)(~,TJ)
for all~' rJ E T xM. D
Consider the (negative) gradient flow
(5) u=-Vf(u)
and denote by cp^8 : M ~ M the flow of (5). The Morse condition implies that the
critical points off are hyperbolic fixed points of (5). It follows that the stable and
unstable manifolds
W^8 (x ; f) = { z EM : lim cp^8 (z) = x},
S->00
Wu(x; f) = { z EM : s~-oo Jim cp^8 (z) = x}
are smooth submanifolds of M for every critical point x of f. The Morse index of a
critical point is the number of negative eigenvalues of the Hessian (when regarded
as a linear operator V^2 f ( x)) and it agrees with the dimension of the unstable
manifold. It is denoted by
ind1(x) = v -(d^2 f(x)) =dim wu(x; f).
The gradient flow (5) is called a Morse-Smale system if, for any pair of critical
points x, y off, the stable and unstable manifolds intersect transversally. In this
case the set