Let $M$ be an $n$-dimensional complete connected Riemannian
manifold with sectional curvature $K_M\geq 1$ and radius
$\operatorname{rad}(M)>\pi /2$.
For any $x\in M$, denote by $\operatorname{rad} (x)$ and $\rho
(x)$ the radius and conjugate radius of $M$ at $x$, respectively. In
this paper we show that
if
$\operatorname{rad}
(x)\leq \rho (x)$ for all $x\in M$, then $M$ is isometric to a
Euclidean $n$-sphere. We also show that
the radius of any
connected nontrivial (i.e., not reduced to a point)
closed totally geodesic submanifold of $M$ is greater than or equal to
that of $M$.