An introduction to the Lie-Santilli theory (Q1383860)

One of the most visible differences in the transition from special relativity with the Minkowskian line element \[ ds^2= dx^\mu \eta_{\mu\nu} dx^\nu, \quad \eta= \text{diag} (1, 1, 1,-1), \quad x=(x, c_0t), \] to general relativity for the exterior gravitational problem in vacuum with the Riemannian line element \(ds^2= dx^\mu g_{\mu\nu} (x)dx^\nu\) is that the former is reducible to the primitive Poincaré symmetry P(3.1), while the latter is not believed to be reducible to a single symmetry. The problem has been solved by R. Santilli using isostructure theory. This paper presents the methodology for the solution of this problem. The section headings are the following: (i) symmetries of exterior gravitation, (ii) symmetries of interior gravitation, (iii) bibliographical notes, (iv) isotopies and isodualities of the unit, (v) isotopies and isodualities of fields, (vi) isospaces and their isoduals, (vii) isotopies and isodualities of the transformation theory, (iix) isotopies and isodualities of functional analysis, (ix) isotopies and isodualities for universal enveloping associative algebras, (x) isotopies and isodualities of Lie algebras, (xi) isotopies and isodualities of Lie groups, (xii) the fundamental theorem on isosymmetries, (xiii) isotopies and isodualities of the rotational symmetry, (xv) isotopies and isodualities of the Lorentz and Poincaré symmetries, (xiv) mathematical and physical applications.