Let $M^m \subseteq N^n$ be a submanifold, with $(N, g)$ a Riemannian manifold and the Levi-civita connection denoted by $\nabla$. Consider the second fundamental form generalized to arbitrary codimension
$$B: TM \times TM \to N(M)$$ $$v,w \in TM, \qquad B(v,w) = (\nabla_v w)^{\perp}$$
can anything be said about the surjectivity of this operator based on $m$, $n$, and $g$? I would think $$ m \geq n - m $$ is a necessary condition, otherwise something like a curved line in $\mathbb{R}^3$. Similarly, if the connection is trivial (maybe flat is the right word?), then the map is probably not surjective.