More About Possible Equations For Vector Bundles

 Let us now consider, in this particular case scenario, -- a Ward-Cauchy-related situation, in which one is to define the Motion of a given arbitrary cohesive set of discrete energy quanta -- to Act as being the "Lagrangian-Based Operator." Here and once again; this is to where one is to be considering, the actual phenomenology or "thing" of the said cohesive set of discrete energy quanta, to act as the prior referred to respective Hamiltonian Operator. This would consequently be working, to define the actual Motion of the mentioned Hamiltonian Operator (a.k.a the actual Motion of the mentioned entity, that is here to be of the phenomenology of the said cohesive set of discrete energy quanta, to be the Lagrangian of the herein considered set of energy quanta), to behave of as displaying the general physical characteristic, of exhibiting the nature of the said attribute of a Lagrangian-Based Operator. Think of the consequential directional-related metric-based phenomenology, that is here to be directly associated, with such a directly corroborative stated Lagrangian-Based Operator (a.k.a.; Think of the consequential directional-related metric-based phenomenology, that is here to be of such a directly corroborative stated Motion of such a mentioned Hamiltonian Operator), to act, as displaying the physical characteristic, of being exhibited, as a "vector bundle." In such a given arbitrary respective case; one is here to minimize the focus, upon any invariably latent Nijenhuis tensors. Consequently; one may be able to work to determine, certain equations, -- that may be derived, in so as to help to mathematically describe the general Nature, of both the non abelian tense of such an implied Lagrangian-Based Vector Bundle, as well as deriving the mathematical description of the general Nature, of the abelian tense of such an implied Lagrangian-Based Vector Bundle. The following two equations, is what I happened to come up with, in regards to this.:

The General Equation, For The Lagrangian-Based Reductional Non Abelian Vector Bundle:

[[[The Real Riemann-Related Component Of The Force Of The Lagrangian-Based Operator] - 

[The Imaginary Component Of The Force Of The Lagrangian-Based Operator] - 

[(i*The Directly Corresponding Lagrangian-Based Operator)/(2*PI*D)]](as taken, via a correlative set of directorals)].

&; 

The General Equation, Of The Lagrangian-Based Reductional Abelian Vector Bundle:

[[[The Real Riemann-Related Component Of The Force Of The Lagrangian-Based Operator] -

[The Imaginary Component Of The Force Of The Lagrangian-Based Operator] - 

[(The Directly Corresponding Lagrangian-Based Operator)/(2*PI*D)]](as taken, via a correlative set of directorals)].

Please Let Me Know If These Two Basic Equations Are Here To Make Adequate Sense To You!

To Be Continued! Sincerely, SAMUEL DAVID ROACH.