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Eigenvalue pinching theorems on compact symmetric spaces

Authors:	Yuuichi Suzuki Hajime Urakawa

Affiliation:	Mathematics Laboratories, Graduate School of Information Sciences, Tohoku University, Katahira, Sendai, 980-8577, Japan Hajime Urakawa ; Mathematics Laboratories, Graduate School of Information Sciences, Tohoku University, Katahira, Sendai, 980-8577, Japan

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 with a bi-invariant Riemannian metric on with respect to $-frac{1}{2} ,B$ , being the Killing form of the Lie algebra , the first eigenvalue $lambda _{1}(h)$ would satisfy $begin{equation}sum _{j=1}^{2}sum _{ell =3}^{n} \|[v_{j},v_{ell }]\|^{2}>n(2lambda _{1}(h)-1),end{equation}$ for all orthonormal bases ${v_{j}}_{j=1}^{n}$ of tangent spaces of (cf. Corollary 3). This problem arose in an attempt to give a spectral geometric proof that $pi _{2}(G)=0$ for a Lie group .

Keywords:	First eigenvalue pinching theorems symmetric spaces

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