Let $U^{m} \subset \mathbb{R}^{m}$ be an open set. Suppose $\varphi$ is an immersion of $U^{m}$ into $\mathbb{R}^{m+n}$ satisfying the following condition:

For each point $p \in \varphi(U^{m})$, the nullity of the second fundamental form of $\varphi(U^{m})$ is equal to $m-1$.

Then, it is well-known that

- $\varphi(U^{m})$ is foliated by $(m-1)$-planes, along which the tangent space of $\varphi(U^{m})$ is constant (i.e., isomorphic to the same linear subspace of $\mathbb{R}^{m}$);
- With the induced metric, $\varphi(U^{m})$ is flat.

Recall that a submanifold $M$ of a Riemannian manifold $\overline{M}$ is said to be a *full submanifold* if it is not contained in any totally geodesic submanifold $N$ of $\overline{M}$ with $\dim N < \dim \overline{M}$.

In general, can $\varphi(U^{m})$ be a full submanifold?