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This paper studies the characteristics of quarter-symmetric metric connections. Some invariants with respect to the projective transformation are obtained.

We introduce a generalized quarter-symmetric metric recurrent connection and study its geometrical properties. We also derive the Schur's theorem for the generalized quarter-symmetric metric recurrent connection.

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In this paper, we study submanifolds in a Riemannian manifold with a semi-symmetric non-metric connection. We prove that the induced connection on a submanifold is also semi-symmetric non-metric connection. We consider the total geodesicness and minimality of a submanifold with respect to the semi-symmetric non-metric connection. We obtain the Gauss, Cadazzi, and Ricci equations for submanifolds with respect to the semi-symmetric non-metric connection.