A rigorous geometric proof of the Lie's Theorem on nonlinear superposition rules for solutions of non-autonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an alternative but equivalent definition of a superposition rule: it is considered as a foliation with some suitable properties. The problem of uniqueness of the superposition function is solved, the key point being the codimension of the foliation constructed from the given Lie algebra of vector fields. Finally, as a more convincing argument supporting the use of this alternative definition of superposition rule, it is shown that this definition allows an immediate generalization of Lie's Theorem for the case of systems of partial differential equations.

A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F, or, equivalently, from an Euler vector field. This is used in showing that double (or higher) vector bundles present in the literature can be equivalently defined as manifolds with a family of commuting Euler vector fields. Higher vector bundles can be therefore defined as manifolds admitting certain $\N^n$-grading in the structure sheaf. Consequently, multi-graded (super)manifolds are canonically associated with higher vector bundles that is an equivalence of categories. Of particular interest are symplectic multi-graded manifolds which are proven to be associated with cotangent bundles. Duality for higher vector bundles is then explained by means of the cotangent bundles as they contain the collection of all possible duals. This gives, moreover, higher generalizations of the known `universal Legendre transformation' T*E->T*E*, identifying the cotangent bundles of all higher vector bundles in duality. The symplectic multi-graded manifolds, equipped with certain homological Hamiltonian vector fields, lead to an alternative to Roytenberg's picture generalization of Lie bialgebroids, Courant brackets, Drinfeld doubles and can be viewed as geometrical base for higher BRST and Batalin-Vilkovisky formalisms. This is also a natural framework for studying n-fold Lie algebroids and related structures.Comment: 27 pages, minor corrections, to appear in J. Geom. Phy

Various problems concerning the geometry of the space u * (H) of Hermitian operators on a Hilbert space H are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into Kähler submanifolds. It is also shown that the space D(H) of density states on an n-dimensional Hilbert space H is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space D k (H) of rank-k states, k = 1, . . . , n, is a smooth manifold of (real) dimension 2nk − k 2 − 1 and this stratification is maximal in the sense that every smooth curve in D(H), viewed as a subset of the dual u * (H) to the Lie algebra of the unitary group U (H), at every point must be tangent to the strata D k (H) it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition H = H 1 ⊗ H 2 , an abstract criterion of entanglement is proved.

We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T n Q playing a fundamental role in higher order Lagrangian formalisms. Graded bundles are graded manifolds in the sense that we can choose an atlas whose local coordinates are homogeneous functions of degrees 0, 1, . . . , n. We prove that graded bundles have a convenient equivalent description as homogeneity structures, i.e. manifolds with a smooth action of the multiplicative monoid (R ≥0 , ·) of non-negative reals. The main result states that each homogeneity structure admits an atlas whose local coordinates are homogeneous. Considering a natural compatibility condition of homogeneity structures we formulate, in turn, the concept of double (r-tuple, in general) graded bundle -a broad generalization of the concept of double (r-tuple) vector bundle. Double graded bundles are proven to be locally trivial in the sense that we can find local coordinates which are simultaneously homogeneous with respect to both homogeneity structures.MSC 2010: 53C15 (Primary); 53C10, 55R10, 58A32, 58A50, 18D05 (Secondary).