A semi-symmetric metric recurrent connection has already been studied. In
this paper we newly discovered geometrical properties and conjugate
symmetric condition for the mutual connection of a semi-symmetric metric
recurrent connection in a Riemannian manifold.
Two new types of connections, Ricci quarter-symmetric metric recurrent
connection and projective Ricci quarter-symmetric metric recurrent
connection, were introduced and some interesting geometrical and physical
characteristics were achieved.
We introduce a quarter-symmetric projective conformal non-metric connection
family and study its geometrical properties. Further we investigate the
geometries of a symmetric-type quarter-symmetric projective conformal
non-metric connection satisfying the Schur?s theorem.
The authors discuss mainly that the Riemannian manifold Mn admitting a unit preserving circle field ? in the present paper. A sufficient and necessary condition is given that Riemannian manifold Mn is an Einstein manifold by imposing some conditions on W2 curvature tensor. Further, this paper obtains the algebra representation of curvature tensors of a W2-recurrent Riemannian manifold Mn given by R???? = 1/d2 [d?d?R?? ? d?d?R?? + d?d?R?? ? d?d?R??].
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