Einstein, Weyl and Asymmetric Geometries

In a previous paper, we presented new results on non-Riemannian geometry. For an asymmetric connection, we showed that a projective change in the symmetric part generates a vector field that is not arbitrary, but is the gradient of a non-arbitrary scalar field. As a consequence, Weyl's geometry is a conformal differential geometry and is associated with asymmetric geometry by this projective change. In the present paper, important differences between light-like and non-light-like intervals are analysed. We show that integrability condition in Weyl's geometry implies microscopic spacetime oscillations of Weyl's and Einstein's geometries. We show that the integrability condition in Weyl's geometry, together with the condition that null and massive particles interact locally in an Einstein spacetime. We construct an equation for linear transverse waves and make a qualitative application in a hydrogen atom, which, from qualitative viewpoint, agrees with Bohr's hydrogen model and Schrodinger's quantum mechanics.