<p>For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha (n,k,H,c)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>H</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of mean curvature, we prove that the submanifold is either isometric to the Einstein Clifford torus, or a topological sphere for the maximal bound <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha (n,\left[ \frac{n}{2}\right] ,H,c)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mfenced close="]" open="["> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> <mo>,</mo> <mi>H</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, or has up to <i>k</i>-th homology groups vanishing. This gives an almost complete (except for the differentiable sphere theorem) characterization of compact submanifolds with pinched Ricci curvature, generalizing celebrated rigidity results obtained by Ejiri, Xu–Tian, Xu–Gu, Xu–Leng-Gu, Vlachos, Dajczer–Vlachos.</p>