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