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
G
with a bi-invariant Riemannian metric
h
h
on
G
G
with respect to
−
1
2
B
-\frac {1}{2} \,B
,
B
B
being the Killing form of the Lie algebra
g
\mathfrak {g}
, the first eigenvalue
λ
1
(
h
)
\lambda _{1}(h)
would satisfy
∑
j
=
1
2
∑
ℓ
=
3
n
|
[
v
j
,
v
ℓ
]
|
2
>
n
(
2
λ
1
(
h
)
−
1
)
,
\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
\{v_{j}\}_{j=1}^{n}
of tangent spaces of
G
G
(cf. Corollary 3). This problem arose in an attempt to give a spectral geometric proof that
π
2
(
G
)
=
0
\pi _{2}(G)=0
for a Lie group
G
G
.