We consider
Mn,
n ≥ 3, a complete, connected submanifold of a space form $¥tilde{M}^{n+p}(¥tilde{c})$ , whose non vanishing mean curvature vector
H is parallel in the normal bundle. Assuming the second fundamental form
h of
M satisfies the inequality <
h>
2 ≤
n2 |
H|
2/(
n - 1), we show that for $¥tilde{c}$ ≥ 0 the codimension reduces to 1. When
M is a submanifold of the unit sphere, then
Mn is totally umbilic. For the case $¥tilde{c}$ < 0, one imposes an additional condition that is trivially satisfied when $¥tilde{c}$ ≥ 0. When
M is compact and has non-negative Ricci curvature then it is a geodesic hypersphere in the hyperbolic space. An alternative additional condition, when $¥tilde{c}$ < 0, reduces the codimension to 3.
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