I wonder if the following statement is true: If $M^n$ is a smooth complete submanifold of $\mathbb R^{n+p}$ with $\nabla A=0,$ where $A$ is the second fundamental form of $M,$ then $M=S^k\times\mathbb R^{n-k}$ for some $k.$
By Lawson's theorem in $\textit{Local Rigidity Theorems for Minimal Hypersurfaces},$ we know that the statement is true when $M$ is a hypersurface (i.e., $p=1$), but I am not sure if it is true in the higher codimension case.
Any ideas or comments are appreciated!