  In this paper we first apply the chain level Floer theory to the study of
Hofer's geometry of Hamiltonian diffeomorphism group in the cases without
quantum contribution: we prove that any quasi-autonomous Hamiltonian path on
weakly exact symplectic manifolds or any autonomous Hamiltonian path on
arbitrary symplectic manifolds is length minimizing in its homotopy class with
fixed ends, as long as it has a fixed maximum and a fixed minimum which are not
over-twisted and all of its contractible periodic orbits of period less than
one are sufficiently $C^1$-small. Next we give a construction of new invariant
norm of Viterbo's type on the Hamiltonian diffeomorphism group of arbitrary
compact symplectic manifolds.
