Does there exist an equivalent to Seifert surfaces for other Riemannian manifolds than $\mathbb{R}^3$? More precisely:
Let $M$ be a simply-connected Riemannian manifolds and $K \subset M$ a (tame) knot. Does there exist a surface $\Sigma$ embedded into $M$ satisfying $\partial \Sigma=K$?
I am also interested in particular cases if they exist (other than $\mathbb{S}^3$ of course).