Pursell-Shanks Type Theorem for Orbit Spaces of $G$-Manifolds

Abstract

Pursell and Shanks [8] proved that a Lie algebra isomorphism between Lie algebras of all C°° vector fields with compact support on paracompact connected C°° manifolds M and N yields a diffeomorphism between the manifolds M and N. Similar results hold for some other structures on manifolds. Indeed, Omori [6] proved the corresponding results in the case of volume structures, symplectic structures, contact structures and fibering structures with compact fibers. The case of complex structures was studied by Amemiya [1]. Koriyama [5] proved that in the case of Lie algebras of vector fields with invariant submanifolds. Recently, Fukui [4] studies the case of Lie algebras of G-invariant C°° vector fields with compact support on paracompact free smooth Gmanifolds when G is a compact connected semi-simple Lie group. The corresponding result is no longer true when G is not semi-simple or G does not act freely. In this paper, we consider Pursell-Shanks type theorem for orbit spaces of smooth G-manifolds in the case of G a compact Lie group. For a smooth G-manifold M, the orbit space M/G inherits a smooth structure by defining a function on M/G to be smooth if it pulls back to a smooth function on M, and the Zariski tangent space of M/G can be defined. This smooth structure of the orbit space was studied by Schwarz [9], [11], Bierstone [2], Poenaru [7] and Davis [3], Schwarz [10] defined a Lie algebra 9£ (M/G) of smooth vector fields on the orbit