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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Eigenvalue pinching theorems
on compact symmetric spaces


Authors: Yuuichi Suzuki and Hajime Urakawa
Journal: Proc. Amer. Math. Soc. 126 (1998), 3065-3069
MSC (1991): Primary 53C20
DOI: https://doi.org/10.1090/S0002-9939-98-04360-3
MathSciNet review: 1451829
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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 $G$.


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Additional Information

Yuuichi Suzuki
Affiliation: Mathematics Laboratories, Graduate School of Information Sciences, Tohoku University, Katahira, Sendai, 980-8577, Japan

Hajime Urakawa
Affiliation: Mathematics Laboratories, Graduate School of Information Sciences, Tohoku University, Katahira, Sendai, 980-8577, Japan
Email: urakawa@math.is.tohoku.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-98-04360-3
Keywords: First eigenvalue, pinching theorems, symmetric spaces
Received by editor(s): November 21, 1996
Received by editor(s) in revised form: February 10, 1997
Communicated by: Christopher Croke
Article copyright: © Copyright 1998 American Mathematical Society