The foundations of Wesson’s induced matter theory are analyzed. It is shown that the empty—without matter—5-dimensional bulk must be regarded as a Weylian space rather than as a Riemannian one. Revising the geometry of the bulk, we have assumed that a Weylian connection vector and a gauge function exist in addition to the metric tensor. The framework of a Weyl–Dirac version of Wesson’s theory is elaborated and discussed. In the 4-dimensional hypersurface (brane), one obtains equations describing both fields, the gravitational (...) and the electromagnetic. The result is a geometrically based unified theory of gravitation and electromagnetism with mass and current induced by the bulk. In special cases on obtains on the brane the equations of Einstein–Maxwell, or these of the original induced matter theory. (shrink)

The problem of measurement in theories based on geometry with nonmetricity and contorsion is analyzed. In order to enable the use of atoms as measuring standards, one has to remove the nonintegrability of length in the interior of atoms. Geometrical descriptions appropriate fo this purpose are found in the general case and in the case of two-covariant theories.