Gradient Flow & Morse Theory

Morse function: f: M → ℝ with non-degenerate critical points (Hessian invertible).
Gradient flow: γ'(t) = -∇f(γ(t)) — curves flowing down to minima.
Index k = number of negative eigenvalues of Hessian at critical point.

Morse inequalities: βₖ ≤ Cₖ (Betti numbers ≤ critical point counts)
Euler characteristic: χ = Σ(-1)ᵏCₖ = Σ(-1)ᵏβₖ

Handle decomposition: each index-k critical point contributes a k-handle. Together they reconstruct the topology of M.