cianfa72
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- About the definition of different differential structures over a given topological manifold
Given a topological manifold, this may or may not admit a ##C^1## atlas (i.e. starting from its maximal atlas it is or it isn't possible to rip charts from it to get an atlas of ##C^1## compatible charts).
A theorem due to Whitney states that from such a topological manifold ##C^1##-atlas (if any) is always feasible to rip charts to get a ##C^{\infty}## atlas.
My question is: starting from such a ##C^1## atlas could there be the case that one can "extract" two different ##C^{\infty}## atlases whose charts are not compatible each other (even when the differential structures they define turn out to be diffeomorphic) ?
A theorem due to Whitney states that from such a topological manifold ##C^1##-atlas (if any) is always feasible to rip charts to get a ##C^{\infty}## atlas.
My question is: starting from such a ##C^1## atlas could there be the case that one can "extract" two different ##C^{\infty}## atlases whose charts are not compatible each other (even when the differential structures they define turn out to be diffeomorphic) ?