Eigenvalues of the laplacian on a compact manifold without boundary Let $ M $ be a compact manifold WITHOUT boundary. It is clear that the first eigenvalue of the Laplace operator $ -\Delta $ is $ \lambda_0=0 $. Now we suppose that M has constant sectional curvature c. My question: is it true that $ \lambda_1= nc $ ($ \lambda_1 $ is the second eigenvalue of the laplacian)?
Thanks

If $c>0$, then $\lambda_1=nc$ implies that $M$ is isometric to the standard round sphere of radius ${1}/\sqrt{c}$. This is known as Lichnerowicz-Obata's theorem. See Theroem 5.1 in Peter Li's "Lecture notes on geometric analysis".
