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\begin{document}

\title{Legendrian warped product immersions of Sasakian space forms for
characterizing spheres via differential equations}



\author[1]{Akram Ali}%
\affil[1]{Universidade Federal do Amazonas}%


\vspace{-1em}



  
  \date{May 08, 2024}


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\begin{abstract}
The goal of this paper is to investigate the geometry of the warping
function on a \$n\$-dimensional compact Legenderian warped product
submanifold \$M\^{}n\$ of Sasakian space form with free boundary. We
establish sharp estimates to the squared norm of the second fundamental
form and the Laplacian of the warping function. Besides, we provide some
triviality results for \$M\^{}n\$ by using the Ricci curvature along the
gradient of the warping function. Taking the clue from the Bochner
formula and second-order ordinary differential equation, we find the
characterization for the base of \$M\^{}n\$ via the first non-zero
eigenvalue of the warping function and proved that it is isometric to
Euclidean space \$\textbackslash{}mathbb\{R\}\^{}p\$ or Euclidean sphere
\$\textbackslash{}mathbb\{S\}\^{}p\$ under some extrinsic conditions.%
\end{abstract}%



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