Then [Laplace] is naturally a sub line bundle of the

vector bundle [member of] = [O.sup.[cross product]2.sub.D] corresponding to [C.sup.2].

The Uncertainty Principle plays the important role of generating a

vector bundle, out of the total uncertainty space E of trivial displacement 4-vectors from which a closed compact manifold X is formed, i.e., ([pi] : E [right arrow] X).

A Hom-Lie algebroid is a quintuple (A, [phi], [x, x], [[rho].sub.A], [[alpha].sub.A]), where A is a

vector bundle over a manifold M, [phi]: M [right arrow] M is a smooth map, [x, x] : [GAMMA](A) [cross product] [GAMMA](A) [right arrow] [GAMMA](A) is a bilinear map, called bracket, [[rho].sub.A] : [GAMMA](A) [right arrow] [phi]'TM is a

vector bundle morphism, called anchor, and [[alpha].sub.A] : [GAMMA](A) [right arrow] [GAMMA](A) is a linear endomorphism of [GAMMA](A), for X, Y [member of] [GAMMA](A), f [member of] [C.sup.[infinity]](M) such that

Given a real

vector bundle [xi] over a space X and a subspace U of X, we shall use the superscript notation U[xi] for the Thom space of the restriction [xi] | U of the

vector bundle [xi] to the subspace U.

Radu Miron presented his main discoveries: generalized Finsler metrics, Lagrange spaces, generalized Lagrange spaces, Hamilton spaces as well as a geometry of the total space of a

vector bundle based on the use of a nonlinear connection.

generalized the bundle shift [12] to the case of Bergman spaces, constructed a

vector bundle model for analytic Toeplitz operator [T.sub.[phi]] on the Bergman space [L.sup.2.sub.a](D), and tried to build

vector bundle models for restrictions of [T.sub.[phi]] to its minimal reducing subspaces, but it is not completed.

A key point is that derivations of the structure ring of graded functions on an N-graded manifold, unlike the [Z.sup.2]-graded one, are represented by sections of the smooth

vector bundle [V.sub.E] (102) over its body manifold Z (Theorem 50).

Denote by [[nabla].sup.[phi]] the connection of the

vector bundle [[phi].sup.*] TN induced from the Levi-Civita connection [[nabla].sup.h] of (N,h) .

Let us consider a

vector bundle [pi] : [M.sub.1] [right arrow] M with n-dimensional base manifold M and r-dimensional fibres.

We therefore propose to represent them as a

vector bundle < with an added information loop: O.

More precisely, a Lie algebroid structure on a real

vector bundle A on a manifold M is defined by a

vector bundle map [[rho].sub.A]: A [right arrow] TM, the anchor of A, and an R-Lie algebra bracket on [GAMMA](A), [[,].sub.A] satisfying the Leibnitz rule