  The paper will study a new quarter-symmetric non-metric connection on a
generalized Riemannian manifold. It will determine the relations that the
torsion tensor satisfies. The exterior derivative of the skew-symmetric part
$F$ of generalized metric $G$ with respect to the Levi-Civita connection
coincides with that of skew-symmetric part $F$ with respect to
quarter-symmetric non-metric connection, which implies that the
even-dimensional manifold endowed with $F$ is symplectic manifold if and only
if it is closed with respect to quarter-symmetric non-metric connection. The
linearly independent curvature tensors of this connection and its dual
(conjugate) connection are determined and the properties of these tensors are
discussed. Finally, the condition is given that the connection should be
conjugate symmetric.

Differential Geometry