Physical Science International Journal

Two classes of three-dimensional metric spaces are identified. They are the conventional three-dimensional metric space and a new ‘three-dimensional’ absolute intrinsic metric space. Whereas an initial flat conventional proper metric space IE^′3 can transform into a curved three-dimensionalRiemannian metric space IM^′3 without any of its dimension spanning the time dimension (or in the absence of the time dimension), in conventional Riemann geometry, an initial flat ‘three-dimensional’ absolute intrinsic metric space ∅IˆE³ (as a flat hyper-surface) along the horizontal, evolves into a curved ‘three-dimensional’ absolute intrinsic metric space ∅IˆM³, which is curved (as a curved hyper-surface) toward the absolute intrinsic metric time ‘dimension’ along the vertical, and it is identified as ‘three-dimensional’ absolute intrinsic Riemannian metric space. It invariantly projects a flat ‘three-dimensional’ absolute proper intrinsic metric space ∅IE^′3_ab along the horizontal, which is made manifested outwardly in flat ‘three-dimensional’ absolute proper metric space IE^′3_ab, overlying it, both as flat hyper-surfaces along the horizontal. The flat conventional three-dimensional relative proper metric space IE^′3 and its underlying flat three-dimensional relative proper intrinsic metric space ∅IE′3 remain unchanged. The observers are located in IE^′3. The projective ∅IE^′3_ab is imperceptibly embedded in ∅IE^′3 and IE^′3_ab in IE^′3. The corresponding absolute intrinsic metric time ‘dimension’ is not curved from its vertical position simultaneously with ‘three-dimensional’ absolute intrinsic metric space. The development of absolute intrinsic Riemannian geometry is commenced and the conclusion that the resulting geometry is more all-encompassing then the conventional Riemannian geometry on curved conventional metric space IM^′3 only is reached.