The authors prove a characterization theorem for compact, oriented and connected \(n\)-submanifolds in the unit sphere \(S^{n+p}(1)\). Let \(M\) be an \(n\)-dimensional \((n\geq 3)\) compact, oriented and connected submanifold in the unit sphere \(S^{n+p}(1)\) with scalar curvature \(n(n-1)r\). It is known in [H. Li, Math. Ann. 305, 665–672 (1996; Zbl 0864.53040)] that if \(p=1\) and \(r\) is constant with \(r \geq 1\), then \(M\) is isometric to either the totally umbilical sphere \(S^n(r)\) or the Riemannian product \(S^1(\sqrt{1-c^2})\times S^n(c)\) with \(c^2 = \frac{n-2}{nr}\). Also the second case happens if
\[ S \leq (n-1)\frac{n(r-1)+2}{n-2} + \frac{n-2}{n(r-1)+2}, \]
where \(S\) denotes the square norm of the second fundamental form of \(M\). The authors generalize this result to the sphere with higher codimension \(p\) with some conditions using the Lawson-Simons formula for the nonexistence of stable currents in [H. B. Lawson jun. and J. Simons, Ann. Math. (2) 98, 427–450 (1973; Zbl 0283.53049)]. Precisely, let
\[ \alpha(n, r) = \begin{cases}(n-1)\frac{n(r-1)+2}{n-2} + \frac{n-2}{n(r-1)+2} &\text{for } p \leq 2,\\ n(r-1)+x_1(r) &\text{for } p \geq 3, \end{cases} \]
where \(x_1(r)\) satisfies an equation containing only \(n\) and \(r\). The authors prove that if \(M\) has nowhere-zero mean curvature and
\[ r \geq \frac{n-2}{n-1}\quad\text{and}\quad S \leq \alpha(n,r), \]
then either the fundamental group of \(M\) is finite, and \(M\) is diffeomorphic to a spherical space form if \(n=3\), and \(M\) is homeomorphic to a sphere if \(n\geq 4\); or \(M\) is isometric to the Riemannian product \(S^1(\sqrt{1-c^2})\times S^{n-1}(c)\) with \(c^2 = \frac{n-2}{nr}\). That the scalar curvature of \(M\) is constant is not assumed in the theorem and the condition \(H \neq 0\) on \(M\) is necessary for proving the theorem.