Let M be an n-dimensional
(n ≥ 3) compact, oriented and connected submanifold in the unit sphere 𝕊n+p(1),
with scalar curvature n(n− 1)r and nowhere-zero mean curvature. Let S denote the
squared norm of the second fundamental form of M and let α(n,r) denote a certain
specific function of n and r. Using the Lawson–Simons formula for the nonexistence
of stable k-currents, we obtain that, if r ≥ (n − 2)∕(n − 1) and S ≤ α(n,r), then
either M is isometric to the Riemannian product 𝕊1× 𝕊n−1(c) with
c2= (n − 2)∕(nr), or the fundamental group of M is finite. In the latter case, M is
diffeomorphic to a spherical space form if n = 3, or homeomorphic to a sphere if
n ≥ 4.
Keywords
unit sphere, submanifold, curvature structure, topology
