  A rigorous foundation for the contact homology of Legendrian submanifolds in
a contact manifold of the form $P\times \R$ where $P$ is an exact symplectic
manifold is established. The class of such contact manifolds include 1-jet
spaces of smooth manifolds. As an application, contact homology is used to
provide (smooth) isotopy invariants of submanifolds of $\R^n$ and, more
generally, invariants of self transverse immersions into $\R^n$ up to
restricted regular homotopies. When $n=3$, this application is the first step
in extending and providing a contact geometric underpinning for the new knot
invariants of Ng
