  We present a new optimal systolic inequality for a closed Riemannian manifold
X, which generalizes a number of earlier inequalities, including that of C.
Loewner. We characterize the boundary case of equality in terms of the geometry
of the Abel-Jacobi map, A_X, of X. For an extremal metric, the map A_X turns
out to be a Riemannian submersion with minimal fibers, onto a flat torus. We
characterize the base of J_X in terms of an extremal problem for Euclidean
lattices, studied by A.-M. Berg\'e and J. Martinet. Given a closed manifold X
that admits a submersion F to its Jacobi torus T^{b_1(X)}, we construct all
metrics on X that realize equality in our inequality. While one can choose
arbitrary metrics of fixed volume on the fibers of F, the horizontal space is
chosen using a multi-parameter version of J. Moser's method of constructing
volume-preserving flows.
