Search Results (6)

For a compact Riemannian m-manifold $(M^m,g),m>1,$ endowed with a nontrivial conformal vector field $\zeta$ with a conformal factor $\sigma$, there is an associated skew-symmetric tensor $\phi$ called the associated tensor, and also, $\zeta$ admits the Hodge decomposition $\zeta =\overline{\zeta }+\nabla \rho$, where $\overline{\zeta }$ satisfies $\mathrm{div}\overline{\zeta }=0$, which is called the Hodge vector, and $\rho$ is the Hodge potential of $\zeta$. The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on $M^m$. The first result of this article states that a compact Riemannian m-manifold $M^m$ is an m-sphere $S^m\left(c\right)$ if and only if (1) for a nonzero constant c, the function $-\sigma /c$ is a solution of the Poisson equation $\Delta \rho =m\sigma$, and (2) the Ricci curvature satisfies $Ric\left(\overline{\zeta },\overline{\zeta }\right)\ge {∥\phi ∥}^2$. The second result states that if $M^m$ has constant scalar curvature $\tau =m(m-1)c>0$, then it is an $S^m\left(c\right)$ if and only if the Ricci curvature satisfies $Ric\left(\overline{\zeta },\overline{\zeta }\right)\ge {∥\phi ∥}^2$ and the Hodge potential $\rho$ satisfies a certain static perfect fluid equation. The third result provides another new characterization of $S^m\left(c\right)$ using the affinity tensor of the Hodge vector $\overline{\zeta }$ of a conformal vector field $\zeta$ on a compact Riemannian manifold $M^m$ with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold $M^m$, $m>2$, is a Euclidean m-space if and only if it admits a nontrivial conformal vector field $\zeta$ whose affinity tensor vanishes identically and $\zeta$ annihilates its associated tensor $\phi$. Full article

We introduce a special vector field $\omega$ on a Riemannian manifold $(N^m,g)$, such that the Lie derivative of the metric g with respect to $\omega$ is equal to $\rho Ric$, where $Ric$ is the Ricci tensor of $(N^m,g)$ and $\rho$ is a smooth function on $N^m$. We call this vector field a $\rho$-Ricci vector field. We use the $\rho$-Ricci vector field on a Riemannian manifold $(N^m,g)$ and find two characterizations of the m-sphere $S^m\left(\alpha \right)$. In the first result, we show that an m-dimensional compact and connected Riemannian manifold $(N^m,g)$ with nonzero scalar curvature admits a $\rho$-Ricci vector field $\omega$ such that $\rho$ is a nonconstant function and the integral of $Ric\left(\omega ,\omega \right)$ has a suitable lower bound that is necessary and sufficient for $(N^m,g)$ to be isometric to m-sphere $S^m\left(\alpha \right)$. In the second result, we show that an m-dimensional complete and simply connected Riemannian manifold $(N^m,g)$ of positive scalar curvature admits a $\rho$-Ricci vector field $\omega$ such that $\rho$ is a nontrivial solution of the Fischer–Marsden equation and the squared length of the covariant derivative of $\omega$ has an appropriate upper bound, if and only if $(N^m,g)$ is isometric to m-sphere $S^m\left(\alpha \right)$. Full article

The target of the current research article is to investigate the solitonic attributes of relativistic magneto-fluid spacetime (MFST) if its metrics are Ricci–Yamabe soliton (RY-soliton) and gradient Ricci–Yamabe soliton (GRY-soliton). We exhibit that a magneto-fluid spacetime filled with a magneto-fluid density $\rho$, magnetic field strength H, and magnetic permeability $\mu$ obeys the Einstein field equation without the cosmic constant being a generalized quasi-Einstein spacetime manifold $\left(GQE\right)$. In such a spacetime, we obtain an EoS with a constant scalar curvature $\mathcal{R}$ in terms of the magnetic field strength H and magnetic permeability $\mu$. Next, we achieve some cauterization of the magneto-fluid spacetime in terms of Ricci–Yamabe solitons with a time-like torse-forming vector field $\xi$ and a $\phi \left(\mathcal{R}ic\right)$ vector field. We establish the existence of a black hole in the relativistic magneto-fluid spacetime by demonstrating that it admits a shrinking Ricci–Yamabe soliton and satisfies the time-like energy convergence criteria. In addition, we examine the magneto-fluid spacetime with a gradient Ricci–Yamabe soliton and deduce some conditions for an equation of state (EoS) $\omega =-\frac{1}{5}$ with a Killing vector field. Furthermore, we demonstrate that the EoS $\omega =-\frac{1}{5}$ of the magneto-fluid spacetime under some constraints represents a star model and a static, spherically symmetric perfect fluid spacetime. Finally, we prove that a gradient Ricci–Yamabe soliton with the conditions $\mu =0$ or $H=2$; $\mu \ne 0$, $H>2$ and obeying the equation of state $\omega =-\frac{1}{5}$ is conceded in a magneto-fluid spacetime, and a naked singularity with a Cauchy horizon subsequently emerges, respectively. Full article

$f(\mathcal{R},T)$-gravity is a generalization of Einstein’s field equations $\left({EFE}_s\right)$ and $f\left(\mathcal{R}\right)$-gravity. In this research article, we demonstrate the virtues of the $f(\mathcal{R},T)$-gravity model with Einstein solitons $\left(ES\right)$ and gradient Einstein solitons $\left(GES\right)$. We acquire the equation of state of $f(\mathcal{R},T)$-gravity, provided the matter of $f(\mathcal{R},T)$-gravity is perfect fluid. In this series, we give a clue to determine pressure and density in radiation and phantom barrier era, respectively. It is proved that if a $f(\mathcal{R},T)$-gravity filled with perfect fluid admits an Einstein soliton $(g,\rho ,\lambda )$ and the Einstein soliton vector field $\rho$ of $(g,\rho ,\lambda )$ is Killing, then the scalar curvature is constant and the Ricci tensor is proportional to the metric tensor. We also establish the Liouville’s equation in the $f(\mathcal{R},T)$-gravity model. Next, we prove that if a $f(\mathcal{R},T)$-gravity filled with perfect fluid admits a gradient Einstein soliton, then the potential function of gradient Einstein soliton satisfies Poisson equation. We also establish some physical properties of the $f(\mathcal{R},T)$-gravity model together with gradient Einstein soliton. Full article

Minimal compact hypersurface in the unit sphere ${\mathbb{S}}^{n+1}$ having squared length of shape operator ${∥A∥}^2<n$ are totally geodesic and with ${∥A∥}^2=n$ are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in ${\mathbb{S}}^{n+1}$. One finds a naturally induced vector field w called the associated vector field and a smooth function $\rho$ called support function on the hypersurface M of ${\mathbb{S}}^{n+1}$. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in ${\mathbb{S}}^5$ to be totally geodesic is that the support function $\rho$ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in ${\mathbb{S}}^{n+1}$, ($n>2$), provided the scalar curvature $\tau$ is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in ${\mathbb{S}}^{n+1}$ is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field $Aw$. Full article

We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field $\xi$ of the Sasakian manifold induces a vector field ${\xi }^T$ on the hypersurface, namely the tangential component of $\xi$ to hypersurface, and it also gives a smooth function $\rho$ on the hypersurface, which is the projection of the Reeb vector field on the unit normal. First, we find volume estimates for a compact orientable hypersurface and then we use them to find an upper bound of the first nonzero eigenvalue of the Laplace operator on the hypersurface, showing that if the equality holds then the hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector field $\nabla \rho$ on a compact orientable hypersurface in a Sasakian manifold in order to find another geometric condition (in terms of mean curvature and integral curves of ${\xi }^T$ ) under which the hypersurface is isometric to a sphere. Finally, we study compact orientable hypersurfaces with constant mean curvature in a Sasakian manifold and find a sharp upper bound on the first nonzero eigenvalue of the Laplace operator on the hypersurface. In particular, we show that this upper bound is attained if and only if the hypersurface is isometric to a sphere, provided that the Ricci curvature of the hypersurface along $\nabla \rho$ has a certain lower bound. Full article