Jun 03 2024

## Isometry and Homotopy

In Chemistry, “This phenomenon of two or more compounds having the same molecular formula but different structures is called **isomerism**.” ([1], pg. 405) In a positive charge groupoid, traveling through an open gamma ray field, coming from a given nucleus, there is an isometry. Though the homotopy comes across “*an m-dimensionable manifold M of class C ^{r}*” ([2], pg. 646), the gravitons have different phases as the groupoid travels on, in an outer tangent space, and “

*the homotopy problem is equivalent to an extension problem.*” ([3], pg. 175)

“16.2. LEMMA. *If (E,S) is a cell and its boundary, then (E *X* 0)* ꓴ *(S *X* I) is a retract of E *X* I*.” ([3], pg. 84)

Thus, “π is an infinite cyclic group,” ([3], pg. 199). Steenrod calls S “A system S of coordinates” ([3], pg. 22), or “a bundle of coefficients.” ([3], pg. 190) In this case, with “*E *X* I*“, *I* is an isotopy, and the zero space is the tangent space. There is “no left distributive law” ([3], pg. 122), because orbital electrons absorb gravitons at various rates.

It may be that the tangent space is all that the charged particle puts out, and that Lie groups form in the open gamma ray field.

[1] Hein, Morris, “Foundations of College Chemistry, Fourth Edition”, DICKENSON PUBLISHING COMPANY, INC., c. 1977

[2] Whitney, Hassler, “Differential Manifolds”, The Annals of Mathematics, Second Series, Vol. 37. No. 3 (Jul., 1936) pp. 645-680

[3] Steenrod, Norman, “The Topolgy of Fibre Bundles”, Princeton University Press, c. 1951

on 07 Jun 2024 at 10:16 amHere is an isotopy definition from Springer:

https://encyclopediaofmath.org/wiki/Isotopy_(in_topology)

X is the electric field, and Y is the magnetic field, of an EM wave.

on 11 Jun 2024 at 2:28 pmAs a piece of the shell breaks off from a charged particle, it becomes a tangent space in the open gamma ray field and experiences an exponential growth of the Lie group for a short while.

“Until now we have considered Exp only locally, restricted to a neighborhood of the zero vector at each point of the manifold.” … “which gives conditions that the domain D of Exp be the entire tangent bundle T (M).” … “in all cases the domain D is an open set.” ([4], pg. 332)

[4] Boothby, William M., “An Introduction to Differentiable Manifolds and Riemannian Geometry”, Academic Press, 2003

on 11 Jun 2024 at 3:13 pm“Most writers call Gxx the isotropy group at x. This usage can be distracting and I have retained the older term vertex group. However I have replaced the other use of the word isotropy with stabilizer.” ([5], pg. 49)

[5] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

on 12 Jun 2024 at 11:08 am“Let L’ be the isotropy subgroup of L at a point 0 of M” ([6], pg. 484) and “L’ is self normalizing in L”. Then “L’ is a maximal parabolic subalgebra of L.” ([6], pg. 489)

As a piece of the shell grows a Lie group, it becomes a parabolic subalgebra.

[6] Yamaguchi. Keizo, “Differential Systems Associated with Simple Graded Lie Algebras”, c. 1991, Advanced Studies in Pure Mathematics 22, 1993, Progress in Differential Geometry, pp. 413-494