We prove that if a Sasakian metric is a ∗-Ricci Soliton, then it is either positive Sasakian, or null-Sasakian. Next, we prove that if a complete Sasakian metric is an almost gradient ∗-Ricci Soliton, then it is positive-Sasakian and isometric to a unit sphere [Formula: see text]. Finally, we classify nontrivial ∗-Ricci Solitons on non-Sasakian [Formula: see text]-contact manifolds.

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In the present, we first obtain Chen–Ricci inequality for C-totally real warped product submanifolds in cosymplectic space forms. Then, we focus on characterizing spheres and Euclidean spaces, by using the Bochner formula and a second-order ordinary differential equation with geometric inequalities. We derive the characterization for the base of the warped product via the first eigenvalue of the warping function. Also, it proves that there is an isometry between the base $\mathbb{N}_{1}$ N 1 and the Euclidean sphere $\mathbb{S}^{m_{1}}$ S m 1 under some different extrinsic conditions.

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In this paper, we consider the CPE conjecture in the frame-work of K-contact and (κ, µ)-contact manifolds. First, we prove that if a complete K-contact metric satisfies the CPE is Einstein and is isometric to a unit sphere S 2n+1 . Next, we prove that if a non-Sasakian (κ, µ)-contact metric satisfies the CPE, then M 3 is flat and for n > 1, M 2n+1 is locally isometric to E n+1 × S n (4).Mathematics Subject Classification 2010: 53C25, 53C20, 53C15