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For the curved n-body problem in S^3, we show that a regular polygonalconfiguration for n masses on a geodesic is an equilibrium configuration if andonly if n is odd and the masses are equal. The equilibrium configuration isassociated with a one-parameter family (depending on the angular velocity) ofrelative equilibria, which take place on S^1 embedded in S^2. We then study thestability of the associated relative equilibria on two invariant manifolds,T^*((\S^1)^n\D) and T^*((\S^2)^n\D). We show that they are Lyapunov stable onS^1, they are Lyapunov stable on S^2 if the absolute value of angular velocityis larger than a certain value, and that they are linearly unstable on S^2 ifthe absolute value of angular velocity is smaller than that certain value.

Regular polygonal equilibrium configurations on S^1 and stability of the associated relative equilibria