Some Relatively "Simple" Stuff As To Tangential And Co-Tangential Bundles

Let us initially consider two different sets of cohesive discrete energy quanta, that each operate to perform both of their respective individually taken common tasks.  Let us next consider that these two different sets of cohesive discrete energy, are here to bear two different distinct mappable Lagrangian-based paths, that are here to work to be subtended by an angle "theta." If one were to take the tangential bundle of that path integral, that is here to be directly corresponding to the Lagrangian-based path of one of these inferred orbifold eigensets -- while then taking the Cross-Product of this resultant mappable Lagrangian-based path With the co-tangential bundle of that path integral, that is here to be directly corresponding to the Lagrangian-based path of the second of these two inferred orbifold eigensets, -- the consequential mathematical result, would then be equal to the sine of theta times the product of these two said path integrals.  Furthermore; if one were to initially do the prior stated math, (taking the product of the tangential bundle of one of the two correlative path integrals With the co-tangential bundle of the other of the two correlative path integrals), yet instead, in this second particular case, one were here to take the Dot-Product of the multiplicative result of the coupling of the tangential bundle of one of such orbifold eigenset With the co-tangential bundle of the other of the two said path integrals, -- then, one would consequently get the equivalence of the cosine of theta times the product of these two said path integrals. In both of these two different general types of situations, this often tends to be the case --  in so long as either one is working with two different Bianchi-related paths, or, if both of these two eluded-to Lagrangian-based paths are here to work to involve a different set of directorals.  In so long as both the directly corresponding tangential bundle And the directly corresponding co-tangential bundle are here to be of a cartesian tense of a respective divergence/convergence, Or, in so long as both the directly corresponding tangential bundle And the directly corresponding co-tangential bundle are here to be of a hyperbolic tense of a respective divergence/convergence, -- then, the earlier general tense of a situation would often tend to be viable. Samuel David Roach.