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I was reading Sharpe's text on Cartan geometry, and I started to wonder:

  1. Does the theory change in any significant way if the base manifold for the Cartan principal bundle is allowed to be a smooth manifold with boundary?

  2. Is there any significant interest in general Cartan geometries for manifolds with boundary?

I have only worked with smooth manifolds with empty boundary in the past, so I am not very familiar with the nuances of manifolds with boundary. As far as I can tell, there should be no problem extending the theory to the case where the manifold has a non-empty boundary. But perhaps there is something I am missing?

With regards to the second question, I know manifolds with boundary are of interest in the case of semi-Riemannian geometry. But I am unfamiliar with other sorts of Cartan geometries (CR, conformal, etc.), so I am not sure whether geometers working on other classes of Cartan geometries are interested in the case of non-empty boundary.

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    I know that hyperbolic surfaces with boundary are very interesting. You might also check out what the people in the GEAR network are interested in. It's a group of folks who study geometric structures and representation varieties, which includes Cartan geometries. – Neal May 27 '15 at 18:42

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