Geometric and topological rigidity of pinched submanifolds in Riemannian manifolds

We study the rigidity of compact submanifolds of Riemannian manifolds of arbitrary codimension that satisfy a sharp pinching condition involving the norm of the second fundamental form and the mean curvature. Without assuming that the ambient manifold is a space form, we show that this condition imposes strong geometric and topological restrictions on the submanifold. The resulting theorems are sharp and provide extensions of several known results in the literature, particularly sphere theorems, without requiring additional assumptions.