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This article delivers the characterization of a pseudo $B$ symmetric spacetimes and we illustrate that a pseudo $B$ symmetric spacetime admitting Codazzi type of $B$-tensor represents a perfect fluid spacetime and if this spacetime admits the time-like convergence criterion, then the pseudo $B$ symmetric spacetime fulfills cosmic strong energy criterion and contains pure matter. Besides, we find in a pseudo $B$ symmetric spacetime with Codazzi type of $B$-tensor the electric part of the Weyl tensor vanishes and has Riemann and Weyl compatible vector fields. Furthermore, it is established that the chosen spacetime with Codazzi type of $B$-tensor is conformally flat and represents a Robertson-Walker spacetime. Also, we calculate the scale factor $\varPsi (t)$ for these spacetimes in a spatially flat Robertson-Walker spacetime. Finally, we study the impact of this spacetime under $f(R)$ gravity scenario and deduce several energy conditions by considering a new model $f\left(\mathcal{R}\right)= e^{(\alpha \mathcal{R})}-ln(\beta \mathcal{R})$ in which $\alpha$ and $\beta$ are positive constants., Comment: 19 pages, 4 figures. arXiv admin note: text overlap with arXiv:2406.16113

This article concerns with the characterization of a spacetime and modified gravity, such as $f\left(R,G\right)$, $f\left(R,T\right)$ and $f\left(R,L_{m}\right)$-gravity equipped with the projective curvature tensor. We establish that a projectively flat perfect fluid spacetime represents dark energy era. Also, we prove that a projectively flat perfect fluid spacetime is either locally isometric to Minkowski spacetime or a de-Sitter spacetime. Furthermore, it is shown that a perfect fluid spacetime permitting harmonic projective curvature tensor becomes a generalized Robertson-Walker spacetime and is of Petrov type $I$, $D$ or $O$. Lastly, we investigate the effect of projectively flat perfect fluid spacetime solutions in $f\left(R,G\right)$, $f\left(R,T\right)$ and $f\left(R,L_{m}\right)$-gravity, respectively. We also investigate the spacetime as a $f\left(R,G\right)$-gravity solution of and use the flat Friedmann-Robertson-Walker metric to establish a relation among jerk, snap, and deceleration parameters. Numerous energy conditions are studied in terms of Ricci scalar with the model $f\left(R,G\right)=\exp(R)+\alpha \left(6G\right)^{\beta}$. For this model, the strong energy condition is violated but the weak, dominant and null energy conditions are fulfilled, which is in excellent accordance with current observational investigations that show the universe is now accelerating.

The almost contact metric structure that we have on a real hypersurface $M$ in the complex quadric $Q^{m}=SO_{m+2}/SO_mSO_2$ allows us to define, for any nonnull real number $k$, the $k$-th generalized Tanaka-Webster connection on $M$, $\hat{\nabla}^{(k)}$. Associated to this connection we have Cho and torsion operators, $F_X^{(k)}$ and $T_X^{(k)}$, respectively, for any vector field $X$ tangent to $M$. From them and for any symmetric operator $B$ on $M$ we can consider two tensor fields of type (1,2) on $M$ that we will denote by $B_F^{(k)}$ and $B_T^{(k)}$, respectively. We will classify real hypersurfaces $M$ in $Q^m$ for which any of those tensors identically vanishes, in the particular case of $B$ being the structure Lie operator $L_{\xi}$ on $M$., Comment: 9 pages

In this paper, we introduce a notion of quadratic Killing structure Jacobi operator (simply, Killing structure Jacobi operator) and its geometric meaning for real hypersurfaces in the complex two-plane Grassmannians. In addition, we give a classification theorem for Hopf real hypersurfaces with quadratic Killing structure Jacobi operator in complex two-plane Grassmannians., Comment: 25 pages

In \cite{S 2017}, Suh gave a non-existence theorem for Hopf real hypersurfaces in the complex quadric with parallel normal Jacobi operator. Motivated by this result, in this paper, we introduce some generalized conditions named $\mathcal C$-parallel or Reeb parallel normal Jacobi operators. By using such weaker parallelisms of normal Jacobi operator, first we can assert a non-existence theorem of Hopf real hypersurfaces with $\mathcal C$-parallel normal Jacobi operator in the complex quadric $Q^{m}$, $m \geq 3$. Next, we prove that a Hopf real hypersurface has Reeb parallel normal Jacobi operator if and only if it has an $\mathfrak A$-isotropic singular normal vector field., Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1907.04661, arXiv:2003.07231, arXiv:1512.03671, arXiv:1605.05316

In this paper, first, we investigate the commuting property between the normal Jacobi operator~${\bar R}_N$ and the structure Jacobi operator~$R_{\xi}$ for Hopf real hypersurfaces in the complex quadric~$Q^m = SO_{m+2}/SO_mSO_2$, $m \geq 3$, which is defined by ${\bar R}_N R_{\xi} = R_{\xi}{\bar R}_N$. Moreover, a new characterization of Hopf real hypersurfaces with $\mathfrak A$-principal singular normal vector field in the complex quadric~$Q^{m}$ is obtained. By virtue of this result, we can give a remarkable classification of Hopf real hypersurfaces in the complex quadric~$Q^{m}$ with commuting Jacobi operators., Comment: 20 pages. arXiv admin note: text overlap with arXiv:1907.04661, arXiv:1605.05316

In this paper, we first introduce the full express of the Riemannian curvature tensor of a real hypersurface $M$ in complex quadric $Q^{m}$ from the equation of Gauss. Next we derive a formula for the structure Jacobi operator $R_{\xi}$ and its derivative under the Levi-Civita connection of $M$. We give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, $\nabla_{\xi}R_{\xi} =0$, in the complex quadric $Q^{m}$, $m \geq 3$., Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:1605.05316, arXiv:1512.03671; text overlap with arXiv:1907.03374; text overlap with arXiv:1710.10627 by other authors

We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator., Comment: 23pages

Kenmotsu geometry is a valuable part of contact geometry with nice applications in other fields such as theoretical physics. In this article, we study the statistical counterpart of a Kenmotsu manifold, that is, Kenmotsu statistical manifold with some related examples. We investigate some statistical curvature properties of Kenmotsu statistical manifolds. We prove that a Kenmotsu statistical manifold is not a Ricci-flat statistical manifold with an example. Finally, we prove a very well-known Chen-Ricci inequality for statistical submanifolds in Kenmotsu statistical manifolds of constant $\phi-$sectional curvature by adopting optimization techniques on submanifolds. This article ends with some concluding remarks., Comment: 15 pages

We give new necessary and sufficient conditions on the Weyl tensor for generalized Robertson-Walker (GRW) space-times to be perfect-fluid space-times. For GRW space-times, we determine the form of the Ricci tensor in all the O(n)-invariant subspaces provided by Gray's decomposition of the gradient of the Ricci tensor. In all but one, the Ricci tensor is Einstein or has the form of perfect fluid. We discuss the corresponding equations of state that result from the Einstein equation in dimension 4, where perfect-fluid GRW space-times are Robertson-Walker., Comment: 10 pages, final version

We give a new proof of the classification of contact real hypersurfaces with constant mean curvature in the complex hyperbolic quadric ${Q^m}^* = SO_{m,2}^o/SO_mSO_2$, where $m\geq 3$. We show that a contact real hypersurface $M$ in ${Q^m}^*$ for $m\geq 3$ is locally congruent to a tube of radius $r{\in}{\mathbb R}^+$ around the complex hyperbolic quadric ${Q^{m-1}}^*$, or to a tube of radius $r\in\mathbb{R}^+$ around the $\mathfrak A$-principal $m$-dimensional real hyperbolic space ${\mathbb R}H^m$ in ${Q^m}^* = SO_{m,2}^o/SO_mSO_2$, or to a horosphere in ${Q^{m-1}}^*$ induced by a class of $\mathfrak A$-principal geodesics in ${Q^m}^*$., Comment: Extensive revision of the first version. The Introduction has been rewritten completely, in particular including a reference to an earlier proof of the classification. Section 2 has been rewritten to replace the incorrect model of the complex hyperbolic quadric from v1 with a correct one. Sections 4 and 5 have also been revised to make the arguments clearer and easier to understand. 24 pages

We investigate the structure of real hypersurfaces with isometric Reeb flow in Kaehler manifolds. As an application we classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type., Comment: 26 pages; extended classification in previous version to all irreducible Hermitian symmetric spaces of compact type

We classify real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadrics ${Q^*}^{m} = SO^{o}_{2,m}/SO_mSO_2$, $m \geq 3$. We show that $m$ is even, say $m = 2k$, and any such hypersurface becomes an open part of a tube around a $k$-dimensional complex hyperbolic space ${\mathbb C}H^k$ which is embedded canonically in ${Q^*}^{2k}$ as a totally geodesic complex submanifold or a horosphere whose center at infinity is $\frak A$-isotropic singular. As a consequence of the result, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics ${Q^*}^{2k+1}$, $k \geq 1$., Comment: arXiv admin note: substantial text overlap with arXiv:1301.0411

First we introduce the notion of parallel structure Jacobi operator for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ . Next we give a complete classification of real hypersurfaces in $Q^m = SO_{m+2}/SO_mSO_2$ with parallel structure Jacobi operator., Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:1512.03671

In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_{\xi}\phi T=TR_{\xi}\phi$, where $T=A$ or $T=S$ for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators, we give a complete classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting condition respectively., Comment: arXiv admin note: text overlap with arXiv:1409.6387, arXiv:1310.5436

We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ . It is shown that the commuting Ricci tensor gives that the unit normal vector field $N$ becomes $\frak A$-principal or $\frak A$-isotropic. Then according to each case, we give a complete classification of real hypersurfaces in $Q^m = SO_{m+2}/SO_mSO_2$ with commuting Ricci tensor.