We prove that the first nonzero eigenvalue of the Laplace-Beltrami operator of equator-like minimal submanifold embedded in the sphere $S^{n+1}$ is equal to $n$. The proof uses the spectral properties of the heat kernel operator corresponding to the submanifold.

In this article, we find the different sufficient conditions for a compact minimal hypersurface $M$ of the unit sphere $S^{n+1},n\in {\mathbb{Z}}^{+}$ to be the Clifford hypersurface $S^{\ell }(\sqrt{\frac{\ell }{n}})\times S^m(\sqrt{\frac{m}{n}}),$ where $\ell ,m\in {\mathbb{Z}}^{+},\ell +m=n$ or the sphere $S^n$. This classification is achieved by applying constraints to the tangent and normal components of the immersion.

Given an $m$-dimensional submanifold $\left(N^m,g\right)$ of the Euclidean space $\left(R^{m+k},\overline{g}\right)$, we explored the existence of a vector field $\xi$ and a constant $\lambda$ so that we got the Yamabe soliton $\left(N^m,g,\xi ,\lambda \right)$. In order to find the vector field $\xi$, we used a constant unit vector $\overrightarrow{a}$ on the Euclidean space $\left(R^{m+k},\overline{g}\right)$, took $\xi$ as the tangential component of $\overrightarrow{a},$ called the Nash vector and took $\mathrm{\Gamma }$ as the normal component of $\overrightarrow{a}$ called the Nash normal. Also, we used a function $\phi$ on the submanifold $\left(N^m,g\right)$ given by $\phi =\overline{g}\left(\mathrm{\Gamma },H\right)$ called the Nash function, where $H$ is the mean curvature vector of the submanifold. Also, we had an operator defined by $K=A_{\mathrm{\Gamma }}$ the shape operator in the direction of the normal $\mathrm{\Gamma }$ called the Nash operator. We found the condition under which $\left(N^m,g,\xi ,\lambda \right)$ is a Yamabe soliton. The submanifold $\left(N^m,g,\xi ,\lambda \right)$ of the Euclidean space $\left(R^{m+k},\overline{g}\right)$, which becomes a Yamabe soliton under certain conditions, mentioned above, is called the Nash Yamabe soliton. First, we studied properties of the Nash Yamabe soliton and then found several conditions under which the non-compact Nash Yamabe soliton $\left(N^m,g,\xi ,\lambda \right)$'s metric has constant scalar curvature, that is, a Yamabe metric. In the first result, we assumed that the mean curvature $H$ of the Nash Yamabe soliton $\left(N^m,g,\xi ,\lambda \right)$ was parallel in the normal bundle and that the Nash function $\phi$ was the solution of the Fischer-Marsden equation, which necessarily implies that $g$ is a Yamabe metric. In a second result, we assumed that the Nash vector $\xi$ of the complete noncompact Nash Yamabe soliton $\left(N^m,g,\xi ,\lambda \right)$ had the following properties: (ⅰ) div $\xi$ does not change sign and (ⅱ) $\Vert \xi \Vert$ is Lebesgue integrable on $N^m,$ and proved that in this case $g$ was a Yamabe metric. Finally, we assumed that the mean curvature $H$ of the Nash Yamabe soliton $\left(N^m,g,\xi ,\lambda \right)$ was parallel in the normal bundle and that the Ricci operator $Q$ was invariant under the Nash vector $\xi$ to prove that the metric $g$ was a Yamabe metric.

In this research paper, we discussed some geometric axioms of a relativistic string cloud spacetime attached with strange quark matter. We determined the conformal $\eta$-Ricci soliton on a relativistic string cloud spacetime attached with strange quark matter with a $\phi (\mathcal{R}ic)$-vector field. In addition, we illustrated some physical significance of conformal pressure $P$ in terms of conformal $\eta$-Ricci soliton with the same vector field. Besides this, we deduced a generalized Liouville equation from the conformal $\eta$-Ricci soliton. Furthermore, we examine the harmonic relevance of conformal $\eta$-Ricci soliton on string cloud spacetime attached with strange quark matter with a potential function $\psi$. Finally, we turned up a necessary and sufficient condition for the 1-form $\eta$, which is the $g$-dual of the vector field $\gamma$ on a string cloud spacetime attached with strange quark matter to be a solution for the Schrödinger-Ricci equation.