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 | References | Similar Articles | Additional Information

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 , being the Killing form of the Lie algebra , the first eigenvalue would satisfy

for all orthonormal bases of tangent spaces of (cf. Corollary 3). This problem arose in an attempt to give a spectral geometric proof that for a Lie group .

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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