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S^3 can be seen as a U(1)-bundle over S^2 (Hopf fibration). It has first Chern number 1 or -1. Denoting with z_1,z_2 coordinates on C^2 and restricting to S^3 the natural connection form is \omega_1=\bar z_1 dz_1+\bar z_2 dz_2

Consider now the U(1) bundles over S^2 with Chern number n>1. They have total space S^3/Z_n where the Z_n action on C^2 is generated by (z_1,z_2)->(z_1\exp(i2\pi/n),z_2\exp(i2\pi/n)). I know that the appropriate connection form which generalises the one of the Hopf bundle is \omega_n=n(\bar z_1 dz_1+\bar z_2 dz_2) I do not understand where the factor of n comes from. In other terms how do I see, from the defining properties of a connection form (or otherwise), that a factor of n is needed and/or that any other factor would give a one-form which is not a connection form?

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If you work out how the circle group acts on the fibres then the connection form has to pull-back to the Maurer-Cartan form on the circle group. That will fix the multiple.

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