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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).

I refer to this page.

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

A. I consider manifolds as manifolds with boundary with empty boundary. Thus, with $M$ a "manifold with boundary", $\partial M$ could be empty.

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

(Here are Questions 1 and 2)

Question 3. Should I interpret the following sentence

"This example shows that the image of a smooth nonregular curve need not be smooth" $\tag{A}$

as

"...need not be a smooth immersed submanifold with boundary" $\tag{B}$

rather than either of the following?

  • "...need not be a smooth regular/embedded submanifold with boundary" $\tag{C}$

  • or "...need not be a smooth manifold with boundary" $\tag{D}$

Explanation of the question:

  • 3.1. $(A)$ suggests that

    • "The image of a smooth regular curve is smooth." $\tag{E}$

  • 3.2. Since regular parametrized curves are equivalent to immersed parametrized curves, images of smooth regular curves are

    • 3.2.1. necessarily immersed submanifolds with boundary

    • 3.2.2. not necessarily regular/embedded submanifolds with boundary

    • 3.2.3. not necessarily manifolds with boundary.

  • 3.3. A regular curve $c$ is an embedding if the image is a 1-submanifold with boundary (or 1-manifold with boundary. I'm not really sure) with non-empty boundary or equivalently if $c$ is injective. (See here)

  • 3.4. By (3.2) and (3.3), I think either that

    • 3.4.1. $(E)$ is what is meant and that "is smooth" should be interpreted as "is a smooth immersed submanifold with boundary"

    • 3.4.2. $(E)$ is what is meant and that "is smooth" is interpreted as "is a smooth regular/embedded submanifold with boundary". If this is the case, then I not only find it weird that we don't have an immersed submanifold with boundary for an example but also think this is wrong unless injective is somehow included in the definition of regular.

    • 3.4.3. $(E)$ is not what is meant. In this case, I find "This example shows that the image of a smooth nonregular curve need not be smooth" pointless. I think $(E)$ is what is meant.

Related, I think: Are geometric curves immersed submanifolds with boundary, manifolds with boundary or neither?

  • @reuns What do you mean? I'm asking what is meant by "smooth" in $(A)$ –  Aug 02 '19 at 06:54
  • I mean what I wrote. If $f$ is a smooth function from $(-1,1)$ to a real manifold what is its local behavior and if it is good what is its global behavior – reuns Aug 02 '19 at 06:57
  • @reuns I think your comments have too many grammatical errors. Please clarify. I think your first comment is meant to be "Let $M$ be a manifold (with dimension). Let $f:(−1,1) \to M$ be a smooth map. Let $a \in (-1,1)$ and $\varepsilon > 0$ such that $[a−\varepsilon,a+\varepsilon] \subseteq (-1,1)$. Consider the following questions 1. Is $f([a−\varepsilon,a+\varepsilon])$ a one-dimensional submanifold with boundary, of $M$? 2. Can the image $f(−1,1)$ be a triangle? 3. If some derivative of $f$ at a, $f^{(k)}(a)$, is non-zero, then do those submanifolds glue together ?" –  Aug 05 '19 at 04:18
  • @reuns I have questions about your questions, but first please clarify that I understand correctly so far. –  Aug 05 '19 at 04:21
  • @reuns By "what do you mean?", I mean "what are these questions leading to?" I mean: are you trying to show that the image of a regular curve need not a 1-submanifold with boundary? Are you trying to show the image of a regular curve need not be an immersed submanifold with boundary? –  Aug 05 '19 at 04:21
  • Smoothness is a local property, so it is natural to ask for the local behavior. If some derivative at $a$ is non-zero then $f(a+t) = f(a)+t^n u+O(t^{n+1})$ where $u \ne 0$ (the addition makes sense in a local chart). Thus $f([a-\epsilon,a+\epsilon])$ is a smooth submanifold and the problem reduces to that of the self-intersections. If all the derivatives vanish at $a$ then $f([a-\epsilon,a+\epsilon])$ doesn't have to be locally smooth, for example it can be $V$ shaped – reuns Aug 05 '19 at 04:25
  • @reuns "doesn't have to be locally smooth" What exactly does it mean for a set (that is a subset of a manifold) to be "locally smooth" or even "smooth"? This is precisely what I'm asking, I believe. I understand that the image of regular curves are smooth in the sense of being (smooth) immersed submanifolds with boundary, but I'm clarifying as to whether or not this is what is meant. I mean, an injective regular curve is an embedding (3.3), so its image is a 1-submanifold with boundary. However, its image is only an immersed submanifold with boundary without "injective". (continued) –  Aug 05 '19 at 05:03
  • @reuns (continued) In either case, the image is either a "smooth" 1-submanifold with boundary or a "smooth" immersed submanifold with boundary, but in the latter case the image is not a "smooth" manifold with boundary, so I wouldn't think the image of regular curves are "smooth" manifolds with boundary unless injective is actually somewhere in the definition of regular that I missed or regular somehow implies injective. –  Aug 05 '19 at 05:06
  • @reuns Therefore, I think images of regular curves are "smooth" in the sense of being "smooth" immersed submanifolds with boundary where the "smoothness" is that we say $A$ is a "smooth" immersed submanifold with boundary of the smooth manifold with boundary $M$ if there exists a smooth manifold with boundary $N$ and a smooth map $F: N \to M$ such that $F$ is an immersion with image $F(N)=A$. –  Aug 05 '19 at 05:08

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