76 research outputs found
Separation of Variables and the Geometry of Jacobians
This survey examines separation of variables for algebraically integrable
Hamiltonian systems whose tori are Jacobians of Riemann surfaces. For these
cases there is a natural class of systems which admit separations in a nice
geometric sense. This class includes many of the well-known cases.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Real projective structures on a real curve
Given a compact connected Riemann surface equipped with an
antiholomorphic involution , we consider the projective structures on
satisfying a compatibility condition with respect to . For a projective
structure on , there are holomorphic connections and holomorphic
differential operators on that are constructed using . When the
projective structure is compatible with , the relationships between
and the holomorphic connections, or the differential operators,
associated to are investigated. The moduli space of projective structures
on a compact oriented surface of genus has a natural
holomorphic symplectic structure. It is known that this holomorphic symplectic
manifold is isomorphic to the holomorphic symplectic manifold defined by the
total space of the holomorphic cotangent bundle of the Teichm\"uller space
equipped with the Liouville symplectic form. We show that
there is an isomorphism between these two holomorphic symplectic manifolds that
is compatible with .Comment: Indagationes Math. (to appear
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