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Curvature bounds for the spectrum of a compact einstein-like
manifold

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*Sharief Deshmukh*

**E-mail:** shariefd@ksu.edu.sa
**Abstract.** Let $(M,g)$ be a connected compact $n$-dimensional,
Riemannian manifold which is of class ${\cal {A}}$ according to Gray [3,
p.261]. If the sectional curvatures of $M$ are bounded below by a constant
$k_0$ and the square of the length of curvature tensor field $\Vert R_X\Vert
^2$ in the direction of a vector field $X$ satisfies $\Vert R_X\Vert ^2\leq
2(n-1)k_0^2L\Vert X\Vert ^2$, where $L$ is a constant (for the unit $n$%
-sphere $S^n$ it satisfies $\Vert R_X\Vert ^2=2(n-1)\Vert X\Vert ^2)$,
then it is shown that either $M$ is isometric to a sphere or else each
nonzero eigenvalue $\lambda $ of the Laplacian satisfies one of the following
(i) $% \lambda >\frac{2(3n-2)}3k_0$, \ (ii) $\lambda$

**AMSclassification.** Primary 53C20, 58G25; Secondary 53B25, 53C20,
53C40

**Keywords.** Einstein-like manifolds, eigenvalues of Laplacian,
isometric to a sphere