For a complete Riemannian manifold M with (1,1)-elliptic Codazzi self-adjoint tensor field A, we use the divergence type operator $${L_A}(u): = div(A\nabla u)$$ and an extension of Ricci to extend some major comparison theorems in geometry. In fact such as mean curvature theorem, Bishop–Gromov volume Cheeger–Gromoll splitting theorem their famous topological consequences. Also get upper bound f...

In this paper, we extend the work of Cao-Chen [9] on Bach-flat gradient Ricci solitons to classify $n$-dimensional ($n\ge 5$) complete $D$-flat steady solitons. More precisely, prove that any noncompact soliton with vanishing $D$-tensor is either Ricci-flat, or isometric Bryant soliton. Furthermore, proof extends shrinking case and expanding as well.

This paper aims to study some semi-symmetric and curvature tensor conditions on α-Kenmotsu pseudo-metric manifolds. Some of semi-symmetric, locally symmetric, the Ricci are considered such Also, relationships between M-projective conformal tensor, concircularly conharmonic investigated. Finally, an example structure is given.

We provide necessary and sufficient conditions for some particular couples ( g , ? ) of pseudo-Riemannian metrics affine connections to be statistical structures if we have gradient almost Einstein, Ricci, Yamabe solitons, or a more general type solitons on the manifold. In cases, establish formula volume manifold give lower an upper bound norm Ricci curvature tensor field.

In this article, pseudoparallel submanifolds for almost Kenmotsu $\left( \kappa,\mu,\nu\right) -$space are investigated. The is considered on the concircular curvature tensor. Submanifolds of these manifolds with properties such as pseudoparallel, $2-$pseudoparallel, Ricci generalized and $2-$Ricci has been characterized. Necessary sufficient conditions given invariant to be total geodesic acco...