# Are initial submanifolds a thing?

## Question

While studying Lie groups, I came across the notion of an "initial submanifold" mentioned in Definition 2.14. here. Seeing a technical definition like this (see details below) I consulted the great wisdom of the internet (looking at you, Wikipedia and nLab); futilely. In fact, I couldn't even find any other independent source defining the notion of an initial manifold.
On the other hand, a quick google scholar search revealed that this seems to be a well-known concept in the sense that people use it, but no one bothers to define it. Some of these papers are even older than the book I got the definition from (e.g. this one).

So I started to wonder (excuse me if that's not a well-defined question):

How established is the concept of an initial submanifold?
Why are there so few sources containing an actual definition of it?

And a question that rather addresses math community as a whole:

Can a concept be seen as folklore even though it is so hard to find an actual source defining it?

## Background

Recall the "usual" definition of a submanifold:

Definition (embedded submanifold): Let $$N$$ be an $$n$$-dimensional manifold.
A subset $$M \subset N$$ is called an embedded submanifold if for each $$p \in M$$ there exists a chart $$(U,u)$$ around $$p$$ such that $$u(U \cap M)) = u(U) \cap \left(\mathbb{R}^m \times \mathbf{0} \right) \subset \mathbb{n}$$.

We can weaken this definition by requiring the essential condition only on path-connected components:

Definition (initial submanifold): Let $$N$$ be a an $$n$$-dimensional manifold.
1) For any point $$p \in N$$ and any subset $$A \subset N$$ we write $$C_p(A)$$ for the path-connected component of $$p$$ in $$A$$.
2) A subset $$M \subset N$$ is called an initial submanifold if for all $$p \in M$$ there exists a chart $$(U,u)$$ around $$p$$ such that $$u(C_p(U \cap M))) = u(U) \cap \left(\mathbb{R}^m \times \mathbf{0} \right) \subset \mathbb{R}^n$$.

The name initial submanifold already hints towards a certain universal property:

Definition (universal property): Let $$i:M \rightarrow N$$ be an injective immersion. We say that $$i$$ satisfies the universal property if for any manifold $$Z$$ a mapping $$f: Z \rightarrow M$$ is smooth if and only if $$i \circ f: Z \rightarrow N$$ is smooth.

Indeed we have

Lemmas:
1) Let $$f: M \rightarrow N$$ be an injective immersion between manifolds satisfying the universal property. Then $$f(M)$$ is an initial submanifold of $$N$$.
2) Let $$M$$ be an initial submanifold of a manifold $$N$$. Then there is a unique $$C^{\infty}$$-manifold structure on $$M$$ such that the injection $$i: M \rightarrow N$$ is an injective immersion.
3) Any initial submanifold $$M$$ of a manifold $$N$$ with injective immersion $$i: M \rightarrow N$$ has the universal property.

And hence (the authors don't mention that one explicitly, so I hope I understood this right):

Theorem: An immersed submanifold is initial if and only if it has the universal property.

I don't know whether initial submanifolds are a 'thing'; however, they are also known as weakly embedded submanifolds and as such have been asked about on Math.SE here.

They're important as:

1. The inverse image of an intial submanifold by a transversal map is again initial and with the same codimension as the original submanifold has.

2. Integral submanifolds of an distribution are always initial submanifolds

3. The leaves of a singular foliation are also initial submanifolds.

4. Lie subgroups are always initial submanifolds

These examples are all found in the book by Michor, Kolar & Slovak, Natural Operations in Differential Geometry. In fact, they give a short proof that a Lie subalgebra integrates to a Lie subgroup by using the initial properties of a distribution defined by the subalgebra.

No, it's not a thing. The book by Kolář, Michor and Slovák is an attempt to present differential geometry in a language with a category theoretic flavour. They (and their PhD students, if any) are probably the only ones preoccupied by this problem. Outside of their niche, nobody cares. In science, an answer is good if it answers an important question. These authors have an answer, but it doesn't answer any question... Please note that they don't introduce any conceptually new insight in their work, but only a somewhat new language. 20 years later we can safely say that their effort has been graciously ignored by the differential geometry community.

What KMS are doing is noticing (2.10. page 12) that "initial morphisms" are too difficult to describe in their setting, therefore they come up with a milder alternative that they call "initial submanifolds".

In Choquet-Bruhat's article, the word "initial" is used with an entirely different meaning: she works with a space-time $$V$$ in which she considers a space-like submanifold $$S$$ of codimension $$1$$. The pseudo-Riemannian tensor of $$V$$ is the solution of Einstein's equations, while $$S$$ plays the role of an initial condition, hence the word "initial".