  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.
