Almost complex structures on cotangent bundles and generalized geometry

We study a class of complex structures on the generalized tangent bundle of a smooth manifold M$M$ endowed with a torsion free linear connection, ∇$\nabla$. We introduce the concept of ∇$\nabla$-integrability and we study integrability conditions. In the case of the generalized complex structures introduced by Hitchin (2003) in [2], we compare the two concepts of integrability. Moreover, as an application, we describe almost complex structures on the cotangent bundle of M$M$ induced by complex structures on the generalized tangent bundle of M$M$.     

A frame bundle generalization of multisymplectic momentum mappings

We construct momentum mappings for covariant Hamiltonian field theories using a generalization of symplectic geometry to the bundle L_VY of vertically adapted linear frames over the bundle of field configurations Y. Field momentum observables are vector-valued momentum mappings generated from automorphisms of Y, using the (n + k)-symplectic geometry of L_VY. These momentum observables on L_VY generalize those in covariant multisymplectic geometry and produce conserved field quantities along flows. Three examples illustrate the utility of these momentum mappings: orthogonal symmetry of a Kaluza-Klein theory generates the conservation of field angular momentum, affine reparametrization symmetry in time-evolution mechanics produces a version of the parallel axis theorem of rotational dynamics, and time reparametrization symmetry in time-evolution mechanics gives us an improvement upon a parallel transport law.     

On symplectic submanifolds of cotangent bundles

Necessary and sufficient conditions are given for a symplectic submanifold of a cotangent bundle to itself be a cotangent bundle.Partially supported by NSF grant DMS-9222241.     

Differential geometry of the Fermat quartic and theta functions

The universal curve over a finite cover of the moduli space of elliptic curves with level four structure is embedded in CP³ as the Fermat quartic and is parametrized via the four Jacobi theta functions. Constructions from completely integrable systems have shown the importance of looking at the curvature of certain spaces and here we compute sectional curvatures. For our computations, we choose the ambient Fubini-Study metric of CP³. We also derive several theta identities which arise from the quartic’s holomorphic two-form.     

On the geometry of isomonodromic deformations

This note examines the geometry behind the Hamiltonian structure of isomonodromy deformations of connections on vector bundles over Riemann surfaces. The main point is that one should think of an open set of the moduli of pairs (V,∇)$(V,\nabla )$ of vector bundles and connections as being obtained by “twists” supported over points of a fixed vector bundle _V0$V_0$ with a fixed connection _∇0${\nabla }_0$; this gives two deformations, one, isomonodromic, of (V,∇)$(V,\nabla )$, and another induced from the isomonodromic deformation of (_V0,_∇0)$(V_0,{\nabla }_0)$. The difference between the two will be Hamiltonian.     

Reduction of Lagrangian mechanics on Lie algebroids

We prove that if a surjective submersion which is a homomorphism of Lie algebroids is given, then there exists another homomorphism between the corresponding prolonged Lie algebroids and a relation between the dynamics on these Lie algebroid prolongations is established. We also propose a geometric reduction method for dynamics on Lie algebroids defined by a Lagrangian and the method is applied to regular Lagrangian systems with nonholonomic constraints.     

Invariant characterization of Liouville metrics and polynomial integrals

In this paper a criterion for a metric on a surface to be Liouville is established, and it is given in terms of differential invariants of the metric. Moreover, here we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals the geodesic flow of a given metric possesses? The method is also applied to recognition of higher degree polynomial integrals of geodesic flows.     

Grading of spinor bundles and gravitating matter in noncommutative geometry

The gravitating matter is studied within the framework of noncommutative geometry. The noncommutative Einstein-Hilbert action on the product of a four-dimensional manifold with discrete space gives models of matter fields coupled to the standard Einstein gravity. The matter multiplet is encoded in the Dirac operator which yields a representation of the algebra of universal forms. The general form of the Dirac operator depends on a choice of the grading of the corresponding spinor bundle. A choice is given, which leads to the nonlinear vectorσ-model coupled to the Einstein gravity.     

Differential Galois obstructions for non-commutative integrability

We show that if a holomorphic Hamiltonian system is holomorphically integrable in the non-commutative sense in a neighbourhood of a non-equilibrium phase curve which is located at a regular level of the first integrals, then the identity component of the differential Galois group of the variational equations along the phase curve is Abelian. Thus necessary conditions for the commutative and non-commutative integrability given by the differential Galois approach are the same.     

Geometric discrete analogues of tangent bundles and constrained Lagrangian systems

Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vector. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete tangent bundles as phase spaces for discretizations of the variational principles of Lagrangian systems, up to the generality of nonholonomic mechanical systems with nonlinear constraints. We obtain a self-contained and transparent development, where regularity, equations of motion, symmetry and momentum, and structure preservation, all have natural expressions.     

Strong Bogomolov inequality for stable vector bundles

Inspired by the recent conjectures concerning the existence of stable bundles on Calabi–Yau threefolds arising from string theory, we consider the possibility of strengthening the classical Bogomolov inequality. We show the existence of stable bundles violating such inequality on many complete intersections.     

On the conformal geometry of transverse Riemann–Lorentz manifolds

Physical reasons suggested in [J.B. Hartle, S.W. Hawking, Wave function of the universe, Phys. Rev. D41 (1990) 1815–1834] for the Quantum Gravity Problem lead us to study type-changing metrics on a manifold. The most interesting cases are Transverse Riemann–Lorentz Manifolds. Here we study the conformal geometry of such manifolds.     

A Class of Calogero Type Reductions of Free Motion on a Simple Lie Group

The reductions of the free geodesic motion on a non-compact simple Lie group G based on the G ₊ × G ₊ symmetry given by left- and right-multiplications for a maximal compact subgroup are investigated. At generic values of the momentum map this leads to (new) spin Calogero type models. At some special values the ‘spin’ degrees of freedom are absent and we obtain the standard BC _n Sutherland model with three independent coupling constants from SU(n + 1,n) and from SU(n,n). This generalization of the Olshanetsky-Perelomov derivation of the BC _n model with two independent coupling constants from the geodesics on G/G ₊ with G = SU(n + 1,n) relies on fixing the right-handed momentum to a non-zero character of G ₊. The reductions considered permit further generalizations and work at the quantized level, too, for non-compact as well as for compact G.      

Induction for weak symplectic Banach manifolds

The symplectic induction procedure is extended to the case of weak symplectic Banach manifolds. Using this procedure, one constructs hierarchies of integrable Hamiltonian systems related to the Banach Lie–Poisson spaces of k$k$-diagonal trace class operators.     

Two-dimensional superintegrable metrics with one linear and one cubic integral

We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta.     

On the reduced phase space of a cotangent bundle

The geometric prequantization of a reduced phase space of a cotangent bundle is described and its relation with the geometric prequantization of the cotangent bundle is pointed out.     

Remarks on Symplectic Connections

This note contains a short survey on some recent work on symplectic connections: properties and models for symplectic connections whose curvature is determined by the Ricci tensor, and a procedure to build examples of Ricci-flat connections. For a more extensive survey, see Bieliavsky et al. [Int. J. Geom. Methods Mod. Phys. 3, 375–420 2006]. This note also includes a moment map for the action of the group of symplectomorphisms on the space of symplectic connections, an algebraic construction of a large class of Ricci-flat symmetric symplectic spaces, and an example of global reduction in a non-symmetric case.     

On the universal covering group of the real symplectic group

A model for the universal covering group of the symplectic group as a Lie group, and some calculations based on the model, as well as defining a similar model for the Lagrangian Grassmannian and relating our construction to the Maslov Index.     

Generalized holomorphic bundles and the B-field action

On a generalized complex manifold, there is an associated definition of a generalized holomorphic bundle, introduced by Gualtieri. In the case of an ordinary complex structure, this notion yields an object which we call a co-Higgs bundle, and we consider the B-field action of a closed form of type (1,1)$(1,1)$, both local and global. The effect makes contact with both Nahm’s equations and holomorphic gerbes.