  In this paper, we present some new results on non-Riemannian geometry, more
specifically, asymmetric connections and Weyl's geometry. For asymmetric
connections, we show that a projective change in the symmetric part generates a
vector field that its not arbitrary, as usually presented, but rather, the
gradient of a non-arbitrary scalar function. We use normal coordinates for the
symmetric part of asymmetric connections as well as for the Weyl's geometry.
This has a direct impact on asymmetric conections, although normal frames are
usual in antisymmetic connections, unlike normal coordinates. In this symmetric
part of asymmetric connections, the vector fields obeys a well-known partial
differential equantion, whereas in Weyl's geometry, gauge vector fields obey an
equation that we believe is presented for the first time in this paper. We
deduce the exact solution of each of these vector fields as the gradient of a
scalar function. For both asymmetric and Weyl's symmetric connections, the
respective scalar functions obey respective scalar partial differential
equations. As a consequence, Weyl's geometry is a conformal differential
geometry and is associated with asymmetric geometry by a projective change. We
also show that a metric tensor naturally appears in asymmetric geometry and is
not introduced via a postulate, as is usually done. In Weyl's geometry, the
eletromagnetic gauge is the gradient of a non-arbitrary scalar function and
eletromagnetic fields are null. Despide the origin in Weyl's differential
geometry, the use of the eletromagnetic gauge is correct in Lagrangean and
Hamiltonian formulations of field theories.
