If one has a Riemannian manifold $M^3$ which is embedded in a $4$-manifold $N^4$ such that $M^3$ has metric $g$ and second fundamental form $k$, does having scalar curvature $R_g$ always larger than or equal to zero necessarily imply that $k$ is zero?
If not, does it at least imply that the trace of the second fundamental form $k_i^i = 0$?