Fifteen years of contact circles and contact spheres (Q356541)

The paper under review is a nice and detailed survey of the theory of contact circles and contact spheres. A contact sphere on a three-dimensional manifold \(M^3\) ``is a triple of one-forms \((\alpha_1, \alpha_2,\alpha_3)\) such that \(\alpha_c= c_1\alpha_1+c_2\alpha_2+c_3\alpha_3\) is a contact form for each non-zero constant vector \(\mathbf c=(c_1,c_2,c_3)\)'' and is a quaternionic counterpart of a contact structure. ``A contact circle on \(M\) is a pair of one-forms \((\alpha_1, \alpha_2 )\) such that \(\alpha_c= c_1\alpha_1+c_2\alpha_2 \) is a contact form for each non-zero constant vector \(\mathbf c=(c_1,c_2)\).'' The authors review the main results in this topic and provide two new proofs.