In this paper, we study warped product manifolds admitting $\tau$-quasi Ricci-harmonic(RH) metrics. We prove that the metric of fibre is harmonic Einstein when RH metric. also provide some conditions for $M$ to be a manifold. Finally, necessary and sufficient $g$ by using differential equation system.

Roth (Bull. Aust. Math. Soc. 95, 495-499 (2017)) gave a result for DDVV inequality submanifolds of warped product manifolds. Then Murathan et al. (J. Geom. (2018) 109:30) obtained the Wintgen-like statistical Recently, Gorunus (Inter. Elec. J. 12(1), 43-56 (2019)) established generalized Wintgen Legendrian in almost Kenmotsu Thus, present manuscript, we find submanifold standard manifolds (?-Ke...

* Correspondence: meraj79@gmail. com Department of Mathematics, University of Tabuk, Tabuk, Kingdom of Saudi Arabia Full list of author information is available at the end of the article Abstract In this article, we have obtained necessary and sufficient conditions in terms of canonical structure F on a semi-invariant submanifold of an almost contact manifold under which the submanifold reduced...