  We show that the focal radius of any submanifold $N$ of positive dimension in
a manifold $M$ with sectional curvature greater than or equal to $1$ does not
exceed $\frac{\pi }{2}.$ In the case of equality, we show that $N$ is totally
geodesic in $M$ and the universal cover of $M$ is isometric to a sphere or a
projective space with their standard metrics, provided $N$ is closed.
  Our results also hold for $k^{th}$--intermediate Ricci curvature, provided
the submanifold has dimension $\geq k.$ Thus in a manifold with Ricci curvature
$\geq n-1,$ all hypersurfaces have focal radius $\leq \frac{\pi }{2},$ and
space forms are the only such manifolds where equality can occur, if the
submanifold is closed.
  To prove these results, we develop a new comparison lemma for Jacobi fields
that exploits Wilking's transverse Jacobi equation.
