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.

The paper deals with the study of Z-symmetric manifolds (ZS)n admitting certain cases Schouten tensor (specifically: Ricci-recurrent, cyclic parallel, Codazzi type and covariantly constant), investigate some geometric physical properties manifold. Moreover, we also (ZS)4 spacetimes tensor. Finally, construct an example to verify our result.

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...

We present the nonrelativistic limit of stellar structure equations Ricci-based gravities, a family metric-affine theories whose Lagrangian is built via contractions metric with Ricci tensor an priori independent connection. find that this characterized by four parameters arise in expansion several geometric quantities powers stress-energy matter fields. discuss relevance result for phenomenolo...

In this paper, we study Kropina metrics with isotropic scalar curvature. First, obtain the expressions of Ricci curvature tensor and Then, characterize on by analysis.

In this paper, we consider $\left(LCS\right)_{n}$ manifold admitting almost $\eta-$Ricci solitons by means of curvature tensors. Ricci pseudosymmetry concepts soliton have introduced according to the choice some special tensors such as pseudo-projective, $W_{1}$, $W_{1}^{\ast}$ and $W_{2}.$ Then, again tensor, necessary conditions are searched for be semisymmetric. Then characterizations obtain...

• Euclid synthetic geometry 300 BC • Descartes analytic geometry 1637 • Gauss – complex algebra 1798 • Hamilton – quaternions 1843 • Grassmann – Grasmann Algebra 1844 • Cayley – Matrix Algebra 1854 • Clifford – Clifford algebra 1878 • Gibbs – vector calculus 1881 – used today • Sylvester – determinants 1878 • Ricci – tensor calculus 1890 • Cartan – differential forms 1908 • Dirac, Pauli – spin ...