TY - JOUR
T1 - An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm
AU - Bangert, Victor
AU - Katz, Mikhail
PY - 2004/7
Y1 - 2004/7
N2 - 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, AX, of X. For an extremal metric, the map AX turns out to be a Riemannian submersion with minimal fibers, onto a flat torus. We characterize the base of Ax in terms of an extremal problem for Euclidean lattices, studied by A.-M. Bergé and J. Martinet. Given a closed manifold X that admits a submersion F to its Jacobi torus Tb1(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.
AB - 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, AX, of X. For an extremal metric, the map AX turns out to be a Riemannian submersion with minimal fibers, onto a flat torus. We characterize the base of Ax in terms of an extremal problem for Euclidean lattices, studied by A.-M. Bergé and J. Martinet. Given a closed manifold X that admits a submersion F to its Jacobi torus Tb1(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.
UR - http://www.scopus.com/inward/record.url?scp=4544220180&partnerID=8YFLogxK
U2 - 10.4310/CAG.2004.v12.n3.a8
DO - 10.4310/CAG.2004.v12.n3.a8
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AN - SCOPUS:4544220180
SN - 1019-8385
VL - 12
SP - 703
EP - 732
JO - Communications in Analysis and Geometry
JF - Communications in Analysis and Geometry
IS - 3
ER -