arXiv2026
We prove a semidirect-product formula for the Hofer-like norm on the identity component \(G_ω(M)\) of the symplectic diffeomorphism group, expressing it as an infimum over the harmonic locus plus a Hamiltonian residue. Specialising to zero flux yields the geometric criterion, for the inclusion of the Hamiltonian group, endowed with the Hofer norm, into \(G_ω(M)\), endowed with the Hofer-like norm, to be an isometric embedding. The criterion is verified in two principal cases: on closed symplectic manifolds admitting a compatible metric of non-negative Ricci curvature for which the harmonic \(1\)-forms are integral and the harmonic flows have period \(1\), covering flat tori and Calabi-Yau manifolds; and on a non-vacuous family of harmonic diffeomorphisms on \(T^2\) equipped with a non-flat compatible metric. The two norms induce the same topology on the Hamiltonian group, a result due to Buss-Leclercq for which we give a self-contained proof; under an explicit boundedness hypothesis (H\(_K\)) on the two-parameter vector-field family attached to zero-flux paths, the inclusion is moreover bi-Lipschitz with distortion at most \(1+3L_0K\). We give self-contained proofs of the non-degeneracy, the positivity of the displacement energy, and the \(C^0\)-stability of limits, three fundamental results of Hofer-like geometry.