In basic models concerning two-level systems, we deal with manifolds such as the Bloch sphere and torus. I believe that the Chern number is what dominates the theory in terms of ties to differential geometry, but what how could one look at the Gaussian curvature of a surface in terms of associated physical factors? Is it somehow related to some aspect of quantum system.

The closest I got was the Gaussian curvature being made analogous to topological charges (source: https://surface.syr.edu/cgi/viewcontent.cgi?article=1125&context=phy_etd), but I do not understand what that means in the context of a two-level system, for example.

I've read several other resources about the development of topological insulators, but most of those expositions dealt with Gaussian curvature only until its involvement with the Gauss-Bonnet theorem.

Some material I read: http://wwwphy.princeton.edu/~haldane/talks/pccmsummerschool2012_haldane.pdf https://books.google.com/books?hl=en&lr=&id=1LZlSZ7ORrQC&oi=fnd&pg=PP1&dq=gaussian+curvature+in+condensed+matter+physics&ots=0hREtMuoTD&sig=g7jvs3J0lYsnzZNJPTcy4QQvjRU#v=onepage&q=gaussian%20curvature%20in%20condensed%20matter%20physics&f=false

I would appreciate any insight!

In basic models concerning two-level systems, we deal with manifolds such as the Bloch sphere and torus. I believe that the Chern number is what dominates the theory in terms of ties to differential geometry, but what how could one look at the Gaussian curvature of a surface in terms of associated physical factors? Is it somehow related to some aspect of quantum system.

The closest I got was the Gaussian curvature being made analogous to topological charges (source), but I do not understand what that means in the context of a two-level system, for example.

I've read several other resources about the development of topological insulators, but most of those expositions dealt with Gaussian curvature only until its involvement with the Gauss-Bonnet theorem.

Some material I read are Duncan and Haldane's notes and The Language of Shape: The Role of Curvature in Condensed Matter: Physics Chemistry and Biology by By S. Hyde et all.

I would appreciate any insight!