A LOWER BOUND FOR THE RADIUS OF THE SMALLEST BALL
CONTAINING A MANIFOLD WITH CURVATURE BOUNDED FROM ABOVE

Fernando Gim\'enez

1991 Math. Subject Classification: 53C40, 53C42, 53C21.

Key words: Isometric immersion, sectional curvature bounded from above,
geodesic ball, extrinsic radius.

From the Introduction.

Let ${\cal F}$ be  a
certain family of triples $(\varphi,N,M)$ where $N$ and $M$ are riemannian
manifods and $\varphi :N\rightarrow M$ is
an isometric immersion. We look for bounds for the
function
   $\rho:{\cal F}\rightarrow {\Bbb R}^{+}$
which assigns to each  $(\varphi,N,M)$ the  radius of the smallest
$M$-ball containing $\varphi (N)$ (extrinsic radius).

      Spruck ([Sp]) proved that if ${\cal F}$ is the family of triples
$(\varphi,N,{\Bbb R}^{n})$ such that  $N$ is a compact,
connected riemannian manifold
of dimension $n-1$ with sectional curvature $K\geq \beta >0$, then
the supremum of the function $\rho$ is $\frac{\pi}{2 \sqrt{\beta}}$.

Menninga in [Me] has generalized Spruck's result by considering a family of
triples
$(\varphi,N,M)$ where $N$ is a compact,
connected riemannian manifold
of dimension $s\geq 2$ with sectional curvature $K\geq \beta >0$
and $M$ is a riemannian manifold of dimension $n$ with
injectivity radius $i(M)\geq \frac{\pi}{ \sqrt{\beta}}$
and  sectional curvature $\overline{K}\leq \frac{\beta}{4}$.

For the infimum of the function $\rho$ Jorge and Koutroufiotis
have obtained in  [JK] a result which implies that if
${\cal F}$ is the family of triples
$(\varphi,N,M)$ where $N$ is a complete manifold of dimension
$s$ with scalar curvature  bounded
from below and   sectional
curvature $K\leq \beta$, and $M$ is
a complete connected riemannian manifold of dimension $n>2s$
with sectional curvature $\lambda \geq \overline{K}\geq \alpha $
and the diameter of $\varphi (N)$ is less than
$\frac{\pi}{2\sqrt{\lambda}}$ if $\lambda >0$, then $\beta >\alpha$ and
a lower bound  for the
function $\rho$ is
\[ \left[ \frac{C_{\lambda}}{S_{\lambda}}\right]^{-1}
\left(\sqrt{\beta-\alpha}\right) \]
where
\[ S_{\lambda}(t) = \left\{ \begin{array}{cc}
\frac{\sin(\sqrt{\lambda}t)}{\sqrt{\lambda}} & {\rm if} \ \ \ \lambda>0\\
t & {\rm if} \ \ \  \lambda=0\\
\frac{\sinh(\sqrt{\left|\lambda\right|}t)}
{\sqrt{\left|\lambda\right|}}& {\rm if} \ \ \  \lambda<0
\end{array} \right. \]
and $C_{\lambda}(t)=S_{\lambda}'(t)$.

This bound is a minimum if $\lambda = \alpha $. In fact, it is attained
at
$(\varphi,N,M) \equiv (i,S_{R}^{s},{\Bbb K}^{n}(\lambda))$, where
${\Bbb
K}^{n}(\lambda) $ is the simply connected space of constant sectional
curvature $\lambda$,
$S_{R}^{s}$ is
the geodesic sphere of radius $R=\left[
\frac{C_{\lambda}}{S_{\lambda}}\right]^{-1}
\left(\sqrt{\beta-\lambda}\right)$ in
${\Bbb K}^{s+1}(\lambda)$
(where ${\Bbb K}^{s+1}(\lambda)$  is considered as a totally geodesic
submanifold
of ${\Bbb K}^{n}(\lambda)$) and  $i:S_{R}^{s}\rightarrow
{\Bbb K}^{n}(\lambda)$ is the canonical immersion. One has
\[ \rho (i,S_{R}^{s},{\Bbb K}^{n}(\lambda))= R= \left[
\frac{C_{\lambda}}{S_{\lambda}}\right]^{-1}
\left(\sqrt{\beta-\lambda}\right) \]

The aim of this note is to obtain lower bounds for the function $\rho$
defined on some families of triples $(\varphi,N,{\Bbb C}K^{n}(\lambda))$
where ${\Bbb C}K^{n}(\lambda)$ is the simply connected K\"ahler
manifold with constant holomorphic sectional curvature $4\lambda$ and
real dimension $2n$.