Isomorphisms between Diffeomorphism Groups
In his “Erlanger Programme” (1872) , Klein defined a geometry of a set X as the study of thoses properties of “figures” (subsets of X) which remain invariant under a group G(X) of transformations of X. Here let us quote Greenberg’s beautiful book  on “Euclidean and non Euclidean Geometries,” p. 213 Invariance and groups are the unifying concepts in Klein’s Erlanger Programme. Groups of transformations had been used in geometry for many years, but Klein’s originality consisted in reversing the roles, in making the group the primary object of interest and letting it operate on variuous geometries, looking for invariants. Klein proposed to translate geometric problems in projective geometry into algebraic problems in invariant theory.
KeywordsVector Field Symplectic Form Smooth Manifold Symplectic Manifold Symplectic Case
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