Immersions of positively curved manifolds into manifolds with curvature bounded above

Abstract:

Let $M$ be a compact, connected, orientable Riemannian manifold of dimension $n - 1 \geqslant 2$, and let $x$ be an isometric immersion of $M$ into an $n$-dimensional Riemannian manifold $N$. Let $K$ denote sectional curvature and $i$ denote the injectivity radius. Assume, for some constant positive constant $c$, that $K(N) \leqslant 1/(4{c^2}),\quad 1/{c^2} \leqslant K(M)$, and $\pi c \leqslant i(N)$. Then the radius of the smallest $N$-ball containing $x(M)$ is less than $\tfrac {1} {2}\pi c$ and $x$ is in fact an imbedding of $M$ into $N$, whose image bounds a convex body.