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  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