We determine a new class of paracontact paracomplex Riemannian manifolds derived from certain cone construction, called para-Sasaki-like Riemannian manifolds, and give explicit examples. We define a hyperbolic extension of a paraholomorphic paracomplex Riemannian manifold, which is a local product of two Riemannian spaces of equal dimension, and show that it is a para-Sasaki-like Riemannian manifold. If the original paraholomorphic paracomplex Riemannian manifold is a complete Einstein space of negative scalar curvature, then its hyperbolic extension is a complete Einstein para-Sasaki-like Riemannian manifold of negative scalar curvature. Thus, we present new examples of complete Einstein Riemannian manifolds of negative scalar curvature.

我们确定了从某些圆锥结构派生的一类新型的对接超复黎曼流形，称为对萨萨基式黎曼流形，并给出了明确的例子。我们定义了一个副全同形副复杂黎曼流形的双曲扩展，它是两个等维黎曼空间的局部乘积，并表明它是一个类似萨萨基的黎曼流形。如果原始的拟全同形副复黎曼流形是一个负标量曲率的完整爱因斯坦空间，那么它的双曲扩展是一个负标量曲率的完整爱因斯坦拟Sasaki样黎曼流形。因此，我们提出了负标量曲率的完整爱因斯坦黎曼流形的新例子。