Geodesic
First nonzero eigenvalue of a~pseudo-umbilical hypersurface in the unit sphere
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2014), pp. 69-78

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S. Deshmukh has obtained interesting results for first nonzero eigenvalue of a minimal hypersurface in the unit sphere. In the present article, we generalize these results to pseudo-umbilical hypersurface and prove: what conditions are satisfied by the first nonzero eigenvalue $\lambda_1$ of the Laplacian operator on a compact immersed pseudo-umbilical hypersurface $M$ in the unit sphere $S^{n+1}$. We also show that a compact immersed pseudo-umbilical hypersurface of the unit sphere $S^{n+1}$ with $\lambda_1=n$ is either isometric to the sphere $S^n$ or for this hypersurface an inequality is fulfilled in which sectional curvatures of the hypersuface $M$ participate.
Mots-clés : pseudoumbilical hypersurface
Keywords: eigenvalue of Laplacian operator. 
@article{IVM_2014_8_a6,
     author = {Majid Ali Choudhary},
     title = {First nonzero eigenvalue of a~pseudo-umbilical hypersurface in the unit sphere},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {69--78},
     publisher = {mathdoc},
     number = {8},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2014_8_a6/}
}
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Majid Ali Choudhary. First nonzero eigenvalue of a~pseudo-umbilical hypersurface in the unit sphere. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2014), pp. 69-78. http://geodesic.mathdoc.fr/item/IVM_2014_8_a6/