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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!

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I'm not sure how specifically you're asking about Gaussian curvature, since Gauss only studied scalar curvature of the tangent bundle of surfaces embedded in three dimensions. I'll assume by Gaussian curvature you're just asking about geometric curvature more generally and about its relation to Berry phase.

Although they're very interesting, it's not just the global invariants of the Berry connection that are important. Even when all these indices are trivial, the Berry connection can be nontrivial, and can cause interesting things to happen.

One of the earliest "local in the Brillouin zone" properties that was worked out in the presence of Berry curvature is the "anomalous velocity". Consider an electron (or more precisely a Bloch quasiparticle state) of Bloch momentum $k$ in a constant electric field $E$ and let $\epsilon(k)$ describe the energy of the band where the electron lives. Hamilton's equations for the wavepacket read

$$\dot x = d\epsilon(k)/dk$$ $$\dot k = eE,$$ where $e$ is the charge of the electron. Note that unlike Hamilton's equations for a particle, which would have $\dot x = k$, what appears in $\dot x$ is the group velocity, which is affected by the changing slope of the band. When there is Berry connection, the motion of this wavepacket is also distorted by the Berry curvature $\Omega$, leading to an extra term, which schematically is

$$\dot x = d\epsilon(k)/dk + \dot k \times \Omega = d\epsilon(k)/dk + e E \times \Omega.$$

So you see that the Berry connection always works to deflect the applied force. In this way, it's very similar to the Coriolis effect for forces applied on curved spaces like the surface of the Earth, where the Gaussian curvature appears instead of the Berry curvature.

For much much more on electronic properties of Berry curvature, see this paper: https://arxiv.org/abs/0907.2021 .

I would also recommend "Gauge Theory of the Falling Cat", by Montgomery, and "Gauge Theories of Swimming" by Wilczek, to see some fun physics problems about "condensed" matter (like cats and amoebas) where gauge theory, geometric curvature, and Berry phase nicely intermingle.

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