Automorphisms of Submanifolds

Abstract

The paper deals with local symmetries of the infinite-order jet space of C ∞ -smooth n-dimensional submanifolds in R m n . Transformations under consideration are the most general possible. They need not preserve the distinction between dependent, and independent variables, the order of derivatives and the hierarchy of finite-order jet spaces.

On the internal approach to differential equations 3. Infinitesimal symmetries

“…We recall that even the structure of all higher-order symmetries of the trivial (empty) systems of differential equations (that is, of the infinite-order jet spaces without any differential constraits) is unknown [1,2,3]. The same can be said for the "linearized theory" of the higher-order infinitesimal transformations treated in this article.…”

Report on the Absolute Differential Equations I

“…where X i and W j are given functions of a finite number of variables (1). They are interpreted as the transformation formulae: the functions ).…”

“…The structure of the totality of all symmetries unexpectedly manifests as an unheard-of mystery [1], [2]. Just the symmetries are important since they produce the higher-order equivalences of differential equations.…”

“…. , f K are given functions of variables (1). Together with the "primary" equations f 1 = · · · = f K = 0, also all consequences (9) are taken into account.…”

Automorphisms of Ordinary Differential Equations

“…We will later see that they are insufficient to analyze the seemingly easier symmetry problem of one first-order equation with three unknown functions (alternatively, two Pfaffian equations with five variables) in full generality since the order of derivatives need not be preserved in this case and the finite-order jet spaces may be destroyed. Recall that even the higher-order symmetries (automorphisms) of empty systems of differential equations (i.e., of the infinite order jet spaces without any additional differential constraints) are nontrivial [2][3][4] and cannot be included into the classical Lie-Cartan theory of transformation groups. Such symmetries need not preserve any finite-dimensional space and therefore the invariant differential forms (the Maurer-Cartan forms, the moving coframes) need not exist.…”