# How are regular curves (parametrized curves that are immersions) related to regular points of smooth maps?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An Introduction to Manifolds by Loring W. Tu (Volume 1).

Convention for this question (See this post which forms the basis for these conventions):

1. I consider manifolds as manifolds with boundary, namely manifolds are equivalent to manifolds with boundary with empty boundary. Thus, when I say that $$M$$ is a "manifold with boundary", we may or may not have that $$\partial M = \emptyset$$.

2. "Manifold with boundary" means "Manifold with boundary with dimension". Objects like $$[0,1) \cup \{2\}$$ are manifolds with boundary with no (uniform) dimension.

Question: How are regular curves (from Volume 3) related to regular points (from Volume 1)?

Here is what I thought of so far:

1. A smooth map where each point in its domain is regular is equivalent to a submersion.

2. "Regular curve" and "regular point" are related to each other by that parametrized curves are regular if and only if they are immersions and by (1).

3. If the range $$M$$ (I really mean $$M$$ and not the image $$c[a,b]$$) of a parametrized curve $$c:[a,b] \to M$$ has dimension 1, then $$c$$ is regular/an immersion if and only if and only if $$c$$ is a submersion if and only if $$c$$ is a local diffeomorphism, by this. If $$c$$ is regular/an immersion with $$\dim M =1$$, then $$c[a,b]$$ is a submanifold with boundary, with $$1 = \dim M = \dim c[a,b]$$.

4. By (3), "regular curve" and "regular point" are also related to each other by that if $$\dim M = 1$$, then we have both that if $$c$$ is regular/an immersion, then each point in the domain of $$c$$, which is $$[a,b]$$, is a regular point and that conversely if $$c$$ is a submersion, then $$c$$ is regular/an immersion.

5. If the image $$c[a,b]$$ is a 1-submanifold with boundary and $$c$$ is regular/an immersion, then $$\tilde c: [a,b] \to c[a,b]$$ is also regular/an immersion. Then $$\tilde c$$ is a submersion and local diffeomorphism by this. Thus, $$c$$ is a local diffeomorphism onto image.

6. If the image $$c[a,b]$$ is a 1-submanifold with boundary with non-empty boundary and if $$c$$ is regular/an immersion, then both $$\tilde c$$ and $$c$$ are injective. Then $$\tilde c$$ is a diffeomorphism and $$c$$ is an embedding.

That's about all I can think of, assuming they are correct.