On the equivalence problem for toric contact structures on \(S^3\)-bundles over \(S^2\) (Q2510044)

The contact equivalence problem for toric contact structures on \(S^3\)-bundles over \(S^2\) is considered. The main results are the following two theorems. Theorem 1. The two toric contact structures \(D_{p_1,p_2,\ell,\ell}\) and \(D_{p_1',p_2',\ell',\ell'}\) on \(S^2\times S^3\) or \(X_\infty\) are inequivalent contact structures if \[ p_1'+ p_2'\neq p_1+ p_2. \] Theorem 2. The two contact structures \(D_{p_1,p_2,\ell,\ell}\) satisfying \(p_1'+ p_2'= p_1+ p_2\) are equivalent if \[ \text{gcd}(\ell,p_2- p)= \text{gcd}(\ell,p_2'- p_1'). \] Some results concerning extremal Sasakian structures are also established. For example, it is proved that for each contact structure \(D_p\) of special type there are \(\phi(p)\) compatible Sasakian-Einstein metrics that are inequivalent as Riemannian metrics, where \(\phi\) is the Euler \(\phi\)-function.