  In this paper we study submanifold with nonpositive extrinsic curvature in a
positively curved manifold. Among other things we prove that, if $K\subset
(S^n, g)$ is a totally geodesic submanifold in a Riemannian sphere with
positive sectional curvature where $n\ge 5$, then $K$ is homeomorphic to
$S^{n-2}$ and the fundamental group of the knot complement $\pi_1(S^n-K)\cong
\Bbb Z$.
