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