On almost paracontact Riemannian manifolds of type \((n,n)\) (Q1601377)

A \((2n+1)\)-dimensional differentiable manifold \(M\) is said to have an almost paracontact structure \((\varphi,\xi,\eta)\), if it admits a \((1,1)\)-tensor \(\varphi\), a vector field \(\xi\) and a 1-form \(\eta\) satisfying the following conditions: \[ \eta(\xi)=1, \quad \varphi^2= I_{2n+1}-\eta \otimes\xi, \quad tr\varphi =0. \] A positive definite Riemannian metric \(g\) is said to be compatible with the almost paracontact structure if it satisfies the following conditions: \[ g(\cdot,\xi) =\eta(\cdot),\;g(\varphi \cdot, \varphi\cdot) =g-\eta \otimes\eta. \] Then, \((\varphi,\xi,\eta,g)\) is called an almost paracontact Riemannian structure of type \((n,n)\). In this paper the authors give a classification with eleven basic classes of almost paracontact Riemannian manifolds of type \((n,n)\) with respect to the covariant derivative of the \((1,1)\)-tensor of the almost paracontact structure.