Closedness of the Tangent Spaces to the Orbits of Proper Actions
Copyright policy )Closedness of the Tangent Spaces to the Orbits of Proper Actions
In this note we show that for any proper action of a Banach--Lie group $G$ on a Banach manifold $M$, the corresponding tangent maps $\g \to T_x(M)$ have closed range for each $x \in M$, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $M/G$ carries a natural manifold structure.
Mathematics - Differential Geometry, Group structures and generalizations on infinite-dimensional manifolds, Infinite-dimensional Lie groups and their Lie algebras: general properties, proper action, 58B25, Dynamical Systems (math.DS), 57E20, Banach-Lie group, Differential Geometry (math.DG), Topology of infinite-dimensional manifolds, Banach manifold, FOS: Mathematics, Mathematics - Dynamical Systems, 22E65, 22E65; 58B25; 57E20
Mathematics - Differential Geometry, Group structures and generalizations on infinite-dimensional manifolds, Infinite-dimensional Lie groups and their Lie algebras: general properties, proper action, 58B25, Dynamical Systems (math.DS), 57E20, Banach-Lie group, Differential Geometry (math.DG), Topology of infinite-dimensional manifolds, Banach manifold, FOS: Mathematics, Mathematics - Dynamical Systems, 22E65, 22E65; 58B25; 57E20
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