  This paper constructs and studies the Gromov-Witten invariants and their
properties for noncompact geometrically bounded symplectic manifolds. Two
localization formulas for GW-invariants are also proposed and proved. As
applications we get solutions of the generalized string equation and dilation
equation and their variants. The more solutions of WDVV equation and quantum
products on cohomology groups are also obtained for the symplectic manifolds
with finitely dimensional cohomology groups. To realize these purposes we
further develop the language introduced by Liu-Tian to describe the virtual
moduli cycle (defined by Liu-Tian, Fukaya-Ono, Li-Tian, Ruan and Siebert).
