Structuring the graph of |x| so it is not a smooth manifold

[shoescreen]

Hello,

I am learning about smooth manifolds through Lee's text. One thing that I have been pondering is describing manifolds such as |x| which are extremely well behaved but not smooth in an ordinary setting.

It is simple to put a smooth structure on this manifold, however that is unsatisfactory for what I am envisioning. Can you help me think of a reasonable topological manifold which describes this set, but is not smooth?

I suppose what I am really looking for is some sort of canonical differential structure to give to embeddings of R^n that when looking at graphs of real valued functions, the manifold would be smooth if and only if the function is smooth.

Standard and usual definitions apply, e.g. subspace topology, smooth means C-infinity

Thanks!

 

[lavinia]

Hello,

I am learning about smooth manifolds through Lee's text. One thing that I have been pondering is describing manifolds such as |x| which are extremely well behaved but not smooth in an ordinary setting.

It is simple to put a smooth structure on this manifold, however that is unsatisfactory for what I am envisioning. Can you help me think of a reasonable topological manifold which describes this set, but is not smooth?

I suppose what I am really looking for is some sort of canonical differential structure to give to embeddings of R^n that when looking at graphs of real valued functions, the manifold would be smooth if and only if the function is smooth.

Standard and usual definitions apply, e.g. subspace topology, smooth means C-infinity

Thanks!

the embedded manifold will be smooth if at each of its points there is an open neighborhood in the ambient manifold whose intersection with the embedded manifold is diffeomorphic to an open subset of R^n.