Eigenvalue pinching theorems on compact symmetric spaces
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- by Yuuichi Suzuki and Hajime Urakawa PDF
- Proc. Amer. Math. Soc. 126 (1998), 3065-3069 Request permission
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 $G$.References
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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
- Received by editor(s): November 21, 1996
- Received by editor(s) in revised form: February 10, 1997
- Communicated by: Christopher Croke
- © Copyright 1998 American Mathematical Society
- 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