J˜-Tangent Affine Hypersurfaces with an Induced Almost Paracontact Structure

The subjects of our study are affine hypersurfaces f:M→R2n+2 considered with a transversal vector field C, which is J˜-tangent. By J˜ we understand the canonical paracomplex structure on R2n+2. The vector field C induces on the hypersurface f an almost paracontact structure (φ,ξ,η). We obtain a complete classification of hypersurfaces admitting a metric induced almost paracontact structure with respect to the second fundamental form. We show that, in this case, the J˜-tangent transversal vector field is restricted to centroaffine and the hypersurface must be a piece of hyperquadric. It is demonstrated that these hyperquadrics have a very specific form. A three-dimensional example is also given. Moreover, we establish an equivalence relation between almost paracontact metric structures, para α-contact metric structures, and para α-Sasakian structures. Methods of affine differential geometry, as well as paracomplex/paracontact geometry, are used.