In this paper we develope the fundamentals of the generalized symplectic geometry on the bundle of linear frames LM of an n-dimensional manifold M that follows upon taking the R n -valued soldering 1-form θ on LM as a generalized symplectic potential. The development is centered around generalizations of the basic structure equation df = −X f ω of standard symplectic geometry to LM when the symplectic 2-form ω is replaced by the closed and non-degenerate R n -valued 2-form β = dθ = dθ i r i . The fact that dθ is R n -valued necessitates generalizing from R-valued observables to vector-valued observables on LM , and there is a corresponding increase in the number of Hamiltonian vector fields assigned to each observable. We show that the algebras of symmetric and anti-symmetric contravariant tensor fields on the base manifold have natural interpretations in terms of symplectic geometry on LM . For the analysis we consider in place of each rank p contravariant tensor field on the base manifold the uniquely related algebra under a generalized super bracket. The naturally defined brackets of the tensorial functions on LM give the Schouten differential concomitants when reinterpreted on the base manifold. Generalized symplectic geometry on the frame bundle of a manifold thus unifies and clarifies the many different approaches to the differential concomitants of Schouten. Two applications of the geometry to physics are presented. First the dynamics of free inertial observers in spacetime is shown to follow upon taking the metric tensor as the Hamiltonian for free observers. We then show that the Dirac equation arises in a natural way as an eigenvalue equation for a naive prequantization operator assigned to the spacetime metric tensor Hamiltonian.ii

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We show that covariant field theory for sections of π : E → M lifts in a natural way to the bundle of vertically adapted linear frames L π E. Our analysis is based on the fact that L π E is a principal fiber bundle over the bundle of 1-jets J 1 π. On L π E the canonical soldering 1-forms play the role of the contact structure of J 1 π. A lifted Lagrangian L:L π E → R is used to construct modified soldering 1-forms, which we refer to as the Cartan-Hamilton-Poincaré 1-forms. These 1-forms on L π E pass to the quotient to define the standard Cartan-Hamilton-Poincaré m-form on J 1 π. We derive generalized Hamilton-Jacobi and Hamilton equations on L π E, and show that the Hamilton-Jacobi and canonical equations of Carathéodory-Rund and de DonderWeyl are obtained as special cases. The manifold L π E emerges as a natural arena for a unified theory that contains, in addition to the sector for sections of π, dynamical sectors for a geometry for M and a geometry for the fibers of E.

The Poisson and graded Poisson Schouten-Nijenhuis algebras of symmetric and anti-symmetric contravariant tensor fields, respectively, on an n-dimensional manifold M are shown to be n-symplectic. This is accomplished by showing that both brackets may be defined in a unified way using the n-symplectic structure on the bundle of linear frames LM of M . New results in n-symplectic geometry are presented and then used to give globally defined representations of the Hamiltonian operators defined by the Schouten-Nijenhuis brackets.

Abstract. A general framework for constructing isospectral flows in the space gl(n) of n by n matrices is proposed. Depending upon how gl(n) is split, this framework gives rise to different types of abstract matrix factorizations. When sampled at integer times, these flows naturally define special iterative processes, and each flow is associated with the sequence generated by the corresponding abstract factorization. The proposed theory unifies as special cases the well-known matrix decomposition techniques used in numerical linear algebra and is likely to offer a broader approach to the general matrix factorization problem.

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n-symplectic geometry on the adapted frame bundle λ : LπE → E of an n = (m + k)-dimensional fiber bundle π : E → M is used to set up an algebra of observables for covariant Lagrangian field theories. Using the principle bundle ρ : LπE → J 1 π we lift a Lagrangian L :and then use L to define a "modified n-symplectic potential"θL on LπE, the Cartan-Hamilton-Poincaré (CHP) R n -valued 1-form. If the lifted Lagrangian is non-zero then (LπE, dθL)is an n-symplectic manifold. To characterize the observables we define a lifted Legendre transformation φL from LπE into LE. The image QL := φL(LπE) is a submanifold of LE, and (QL, d(θ|Q L )) is shown to be an n-symplectic manifold. We prove the theorem thatθL = φ * L (θ|Q L ), and pull back the reduced canonical n-symplectic geometry on QL to LπE to define the algebras of observables on the n-symplectic manifold (LπE, dθL). To find the reduced n-symplectic algebra on QL we set up the equations of nsymplectic reduction, and apply the general theory to the model of a k-tuple of massless scalar fields on Minkowski spacetime. The formalism set forth in this paper lays the ground work for a geometric quantization theory of fields.

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