In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161. It was discovered by Hugo Gieseking . The Gieseking manifold has a double cover homeomorphic to the figure-eight knot complement. The underlying compact manifold has a Klein bottle boundary, and the first homology group of the Gieseking manifold is the integers.