It is shown that the presence of a non-zero concurrent vector field on a Riemannian manifold poses an obstruction to its topology as well as certain aspects of its geometry. It is shown that on a compact Riemannian manifold, there does not exist a non-zero concurrent vector field. Also, it is shown that a Riemannian manifold of non-zero constant scalar curvature does not admit a non-zero concurrent vector field. It is also shown that a non-zero concurrent vector field annihilates de-Rham Laplace operator. Finally, we find a characterization of a Euclidean space using a non-zero concurrent vector field on a complete and connected Riemannian manifold.

We recall classical themes such as "on hearing the shape of a drum" or "can one hear the shape of a drum?", and the discovery of Milnor, who constructed two flat tori which are isospectral but not isometric. In this article, we consider the question of finding conditions under which an $n$-dimensional closed Riemannian manifold $\left(M^n,g\right)$ having a non-zero eigenvalue $nc$ for a positive constant $c$ (that is, has same non-zero eigenvalue as first non-zero eigenvalue of the sphere $S^n(c)$), is isometric to $S^n(c)$. We address this issue in two situations. First, we consider the compact $\left(M^n,g\right)$ as the hypersurface of the Euclidean space $\left(R^{n+1},⟨,⟩\right)$ with isometric immersion $f:\left(M^n,g\right)\to$ $\left(R^{n+1},⟨,⟩\right)$ and a constant unit vector $\overrightarrow{a}$ such that the function $\rho =⟨f,\overrightarrow{a}⟩$ satisfying $\mathrm{\Delta }\rho =-nc\rho$ for a positive constant $c$ is isometric to $S^n(c)$ if and only if $\left(M^n,g\right)$ is isometric to $S^n(c)$ provided the integral of Ricci curvature $Ric\left(\mathrm{\nabla }\rho ,\mathrm{\nabla }\rho \right)$ has an appropriate lower bound. In the second situation, we consider that the compact $\left(M^n,g\right)$ admits a non-trivial concircular vector field $\xi$ with potential function $\sigma$ satisfying $\mathrm{\Delta }\sigma =-nc\sigma$ for a positive constant $c$ and a specific function $f$ related to $\xi$ (called circular function) is constant along the integral curves of $\xi$ if and only if $\left(M^n,g\right)$ is isometric to $S^n(c)$.

Chen (1993) developed the theory of $\delta$-invariants to establish novel necessary conditions for a Riemannian manifold to allow a minimal isometric immersion into Euclidean space. Later, Siddiqui et al. (2024) derived optimal inequalities involving the CR $\delta$-invariant for a generic statistical submanifold in a holomorphic statistical manifold of constant holomorphic sectional curvature. In this work, we extend the study of such optimal inequality to the contact CR $\delta$-invariant on contact CR-submanifolds in Sasakian statistical manifolds of constant $\varphi$-sectional curvature. This paper concludes with a summary and final remarks.