Static spherically symmetric constant density relativistic and Newtonian stars in the Lobachevskyan geometry

The present paper has the purpose to illustrate the importance of the ideas and constructions of the Non-Euclidean (Lobachevsky) Geometry, which can be applied even today for solving some conceptually important problems. We study the static and spherically symmetric solutions to the Einstein field equations under the assumption that the space-time may possess an arbitrary number of spatial dimensions. A new exact solution of a perfect fluid sphere of constant (homogeneous) energy-density which agrees with interior Lobachevsky geometry for 3D and 4D spaces are found. We discuss the property of temporal scalar field arise in lower-dimensional theories as the reduction of extra dimension.