EIGENVALUE PINCHING THEOREMS ON COMPACT SYMMETRICAL SPACES

We prove two first eigenvalue pinching theorems for Riemannian symmetr ic spaces (Theorems 1 and 2). As their application, we answer negative ly a question raised by Elworthy and Rosenberg, who proposed to show t hat for every compact simple Lie group G with a bi-invariant Riemannia n metric h on G with respect to -1/2 B, B being the Killing form of th e Lie algebra g, the first eigenvalue lambda(1)(h) would satisfy [GRAP HICS] for all orthonormal bases {nu(j)}(j=1)(n) of tangent spaces of G (cf. Corollary 3). This problem arose in an attempt to give a spectra l geometric proof that pi(2)(G) = 0 for a Lie group G.