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Since connections in general do not require that we have a Riemannian manifold, but only a smooth manifold, I find it kind of weird that the only examples of connections that I find in the internet are those which use the Levi-Civita connection.

So, I wanted to know of any examples of non-Levi-Civita connections.

If somebody could also give an example of an easily-visualizable parallel transport (like on S^2) using that connection it would be great.

Thanks in advance!