  Michael Farber introduced the Lusternik-Schnirelmann category cat$(M,\xi)$
for the pair of finite CW complex $M$ and first-order cohomology $\xi$. It is
inspired by the Morse-Novikov theory, which is a closed 1-form version of the
Morse theory. An important result of this theory is that if the number of zeros
of a closed 1-form $\omega$ on a closed manifold is less than
cat$(M,[\omega])$, then any gradient flows of $\omega$ has at least one
homoclinic cycle. This paper begins with an explanation of
Lusternik-Schnirelmann theory of closed 1-form, and extends Farber's results to
general non-compact manifolds. We will also explain that Farber's results hold
equally well on non compact manifolds, and explain the new phenomena related
gradient flows of $\omega$ that occurs on non compact manifolds.
