Lichnerowicz–Obata eigenvalue theorem

Remarks

The proof of the inequality applies the Bochner formula to an eigenfunction $f$ with $\Delta f = \lambda_1 f$. With $\nabla^2 f$ the Hessian, the formula reads $-\frac12 \Delta \abs{\nabla f}^2 = \abs{\nabla^2 f}^2 - \ip{\nabla f}{\nabla \Delta f} + \operatorname{Ric}(\nabla f, \nabla f)$. Integrating over $M$ and using $\abs{\nabla^2 f}^2 \ge (\Delta f)^2 / n$ gives $\lambda_1 \ge nk$. In the equality case the Hessian of $f$ equals $-k f g$, and Obata showed that a complete manifold carrying a nonconstant solution of this equation is a round sphere. On the unit sphere $S^n$ one has $\operatorname{Ric} = (n-1) g$ and $\lambda_1 = n$, attained by the restrictions of the coordinate functions, so the bound is sharp. The hypothesis $k > 0$ is needed: flat tori have $\operatorname{Ric} = 0$, and the flat torus $\R^n / L\Z^n$ has $\lambda_1 = 4\pi^2 / L^2$, which tends to $0$ as $L$ grows.

Going deeper

For a student

This is the model comparison theorem of spectral geometry: a curvature bound gives an eigenvalue bound, with rigidity in the extreme case. It sits beside the Bonnet–Myers theorem, which under the same hypothesis bounds the diameter by $\pi/\sqrt{k}$. The Bochner technique used in the proof is the same one used for vanishing theorems for harmonic forms. For a lower bound on $\lambda_1$ without curvature assumptions, see Cheeger's inequality.