Different definitions for submanifolds

Solution 1:

This is not an answer but a quick observation. Consider $M:=\mathbb{R}^2$ and $M':=\{0\}\times [0,\infty) \cup [0,\infty)\times \{0\}$. Let $\tilde F: (x,y)\mapsto x-y$. Then $F:=\tilde F|_{M'}$ is a bijective mapping $M'\to \mathbb{R}$. This is not a counterexample because (ii) in (II) is not satisfied, but it shows that care has to be taken and (ii) is crucial.

Solution 2:

Just going to leave this here as a note to posterity: not all definitions of submanifold are equivalent. So be careful!

For example, Guillemin and Pollack use submanifold to mean "embedded submanifold," whereas Warner uses submanifold to mean "immersed submanifold." These are not equivalent meanings.

Solution 3:

From a categorical point of view, the notion of immersed submanifold is probably the best overall definition of a submanifold. This is because it is a kind of 2-subobject, in the sense both the tangent map and the base map are subobjects in the classical categorical sense, that is they are monics and hence injective.

Nevertheless, the other definitions are also useful. One other definition you might not be aware of, is that of initial submanifold, an account is given in Michor, Kolar & Slovak's Natural Operations in Differential Geometry. This definition characterises all initial submanifolds, in the categorical sense, that are also immersed submanifolds.