Eigenvalue pinching theorems on compact symmetric spaces

Abstract:

We prove two first eigenvalue pinching theorems for Riemannian symmetric spaces (Theorems 1 and 2). As their application, we answer negatively a question raised by Elworthy and Rosenberg, who proposed to show that for every compact simple Lie group $G$ with a bi-invariant Riemannian metric $h$ on $G$ with respect to $-\frac {1}{2} B$, $B$ being the Killing form of the Lie algebra $\mathfrak {g}$, the first eigenvalue $\lambda _{1}(h)$ would satisfy \begin{equation*}\sum _{j=1}^{2}\sum _{\ell =3}^{n} |[v_{j},v_{\ell }]|^{2}>n(2\lambda _{1}(h)-1),\end{equation*} for all orthonormal bases $\{v_{j}\}_{j=1}^{n}$ of tangent spaces of $G$ (cf. Corollary 3). This problem arose in an attempt to give a spectral geometric proof that $\pi _{2}(G)=0$ for a Lie group $G$.