We define curved five-dimensional (5D) space–time from the embedding of 5D surfaces in a 6D flat space. Demanding that the 6D coordinates satisfy a separation of variables form and that the 5D metric is diagonal, we obtain that each curved 5D surface contains 4D hyperboloid de-Sitter subspaces with maximally symmetry SO (4,1). Therefore, we define a very special form for the curved 5D surface where the extra-dimension is perpendicular to the 4D hyperboloid de-Sitter spaces. By relating to a minimally coupled scalar field with a potential which depends on the extra-dimension only, the curved 5D surface's form is satisfied. A mechanism by means of which the extra-dimension can be of a finite size, is found. The borders of the finite extra-dimension are obtained when the scalar field potential goes to infinity for certain finite values of the scalar field. The geodesic lines' equations show that a particle cannot cross such borders.

Two different intraocular lens implants are suggested for age macula degeneration (AMD) eye disease that causes the loss of central vision. An intraocular Lens implant that includes two mirrors with a continuity change in their curvature radius and an intraocular hyperboloid surface lens implant closed by two lenses also with a continuity change in their curvature radiuses and includes an additional lens between the two. Thus, both eyes can see an increased image of a near central object projected on the retina, removed from the degenerated macula and also see the peripheral objects images decreasing continuity according to the continuity change in the curvature radiuses of the mirrors and lenses.

Multidimensional space-times are represented as curved surfaces embedded in higher dimensional flat spaces. The embedding of each surface is based on geometrical principles. According to these geometrical principles, we use variable separated coordinates so that the coordinates parameters become an orthogonal curved coordinates system for each space-time surface. In this way, we obtain that the universe expands and that the expansion is accelerated. By using co-moving coordinates and assuming that there is at least one geodesic which represent a straight line in the curved multidimensional space-time surface (this is a kind of "equivalence principle" of a new type), we obtain the curved multidimensional space-time surface's equation, its metric and accelerated expanded three-sphere surface's particles that also explains the accelerated expansion of the universe.

Our aim in this paper is to interest retinal eye specialists in preventing dry macula degeneration by a special flurry vector field through open or closed curved surfaces. The flux of vector fields through surfaces is based on vector element area and volume element. Therefore, we explain a few geometrical derivations of area and volume elements in curved orthogonal coordinate systems. We hope that by derivation of a spatial vector field flurry against drusen through open or closed surfaces due to the Gauss theorem might select drusen under eye retina cells without destroying the cells and prevent macula degeneration. A changed flurry of a magnetic or electric vector field through a closed line causes an electric or magnetic vector field on the surface closed by the line. We also hope that derivation by Stokes' and Greens' theorems, with the help of iron, might help eye cells to get in life.

Curved multi-dimensional space-times (5D and higher) are constructed by embedding them in one higher-dimensional flat space. The condition that the embedding coordinates have a separable form, plus the demand of an orthogonal resulting space-time, implies that the curved multi-dimensional space-time has 4D de-Sitter subspaces (for constant extra-dimensions) in which the 3D subspace has an accelerated expansion. A complete determination of the curved multi-dimensional spacetime geometry is obtained provided we impose a new type of "equivalence principle", meaning that there is a geodesic which from the embedding space has a rectliniar motion. According to this new equivalence principle, we can find the extra-dimensions metric components, each curved multi-dimensional spacetime surface's equation, the energy-momentum tensors and the extra-dimensions as functions of a scalar field. The generic geodesic in each 5D spacetime are studied: they include solutions where particle's motion along the extra-dimension is periodic and the 3D expansion factor is inflationary (accelerated expansion). Thus, the 3D subspace has an accelerated expansion.