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The “with boundary” does get a bit unwieldy when you have to write it more than a couple of times.

I can't seem to find any alternative term on Wikipedia or elsewhere, but surely someone has come up with a more concise one, in a paper that dealt a lot with these?

Else I think I'll have to invent one...

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    $\begingroup$ Start calling them manwibos. $\endgroup$ – Umberto P. Sep 10 '15 at 12:45
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    $\begingroup$ Please don't do what @UmbertoP. said. $\endgroup$ – Alec Teal Sep 10 '15 at 12:47
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A few options:

  • Write somewhere near the beginning "all manifolds are assumed to have a boundary unless indicated otherwise", but then you run into issues when you want to talk about manifolds without boundary. When these are compact you have the shortcut "closed manifold" though, so it's not exactly 50-50.
  • If your category of manifolds is called something like $\mathsf{Man}$ or $\mathsf{Mfd}$, then the category of manifolds with boundary can be called $\mathsf{Man}^\partial$ or $\mathsf{Mfd}^\partial$, and then you can write "Let $M \in \mathsf{Mfd}^\partial$..."
  • The notation "Let $M^3$ be a manifold" is often used to mean that $M$ actually has dimension three, so maybe you can write "Let $M^\partial$ be a manifold" to indicate that $M$ has a boundary? I have never seen this anywhere though, so be sure to explain beforehand.
  • Suck it up and write "with boundary" every time, maybe "w/ boundary" if you want to save two characters. I don't really think a shorthand is necessary, because then you would want a shorthand for "oriented", "framed", etc, and you end up with a monstrosity like $M^{\partial,or}_{fr}$...
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  • $\begingroup$ Maybe one could write "let $(M, \partial M)$ be a manifold"? That'd be equally tedious though. $\endgroup$ – Balarka Sen Sep 10 '15 at 12:54
  • $\begingroup$ Well, a manifold with boundary is not a manifold. (Or is it? I think boundaries are usually precluded when only talking about manifolds.) That's also something I dislike about the term. This is not a problem with oriented manifolds etc.. $\endgroup$ – leftaroundabout Sep 10 '15 at 13:13
  • $\begingroup$ It all depends on what you call a manifold @leftaroundabout. Usually the word "manifold" on its own assumes there's no boundary, but so long as you're clear about it, it's perfectly fine to assume that the manifolds you consider are more general. $\endgroup$ – Najib Idrissi Sep 10 '15 at 14:15
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w/ boundary is as good as it's going to get.

Then:

let $M$ be a manifold w/ boundary....

Short of inventing your own I cannot see one. However it isn't wrong to invent your own as long as:

  1. It makes sense (it isn't the product of an insane and troubled mind)
  2. YOU STATE IT AT THE FIRST OPPORTUNITY

For example:

Here $\mathcal{B}$ shall mean a manifold with boundary, for example:

  • Let $\mathcal{B}M$ be a manifold

Tells us this is a manifold with boundary

But I'll be honest, that doesn't seem great. It's only worth it if you write this A LOT on the same thing IMO.


Initial answer

At first I said to use this:

Let $\partial M$ be a manifold

While this implies that $M$ has a boundary, it isn't the most obvious. Make it clear at the start you use $\partial M$ to denote a manifold with boundary if you use this.

I don't like it because the boundary of this manifold is then $\partial\partial M$ unless you assume $M$ denotes the not-boundary part and $\partial M$ the boundary.

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  • $\begingroup$ As a fun fact I'm currently deciding whether or not to use "manifold with a countable basis" where I used to say "manifold" .... that's a long one! $\endgroup$ – Alec Teal Sep 10 '15 at 12:45
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    $\begingroup$ This is a really weird notation considering $\partial M$ is usually the boundary of $M$ itself... It looks like you define something whose boundary is a manifold, but the thing itself isn't! $\endgroup$ – Najib Idrissi Sep 10 '15 at 12:49
  • $\begingroup$ @NajibIdrissi that's actually a really good point.... what was I thinking? Should I delete this? $\endgroup$ – Alec Teal Sep 10 '15 at 12:49
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    $\begingroup$ @NajibIdrissi: it does make some sense, because if $\partial M$ exists them $M$ has to be a manifold with boundary... but I think this way of writing is confusing. (Mind, C programmers might disagree... by writing int *p, they declare that p is a pointer because what it points to is an integer.) $\endgroup$ – leftaroundabout Sep 10 '15 at 12:50
  • $\begingroup$ @NajibIdrissi see edits $\endgroup$ – Alec Teal Sep 10 '15 at 12:56
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Some ideas.

  • Turn the “with boundary” suffix into an adjective prefix, e.g. bounded manifold (which might be ambiguous), boundered manifold (even though “boundered” is not a word), or $B$-manifold or $b$-manifold.
  • Declare that all manifolds in a particular section (or even in the whole text) may have boundary, and when you need a boundaryless manifold in a particular situation, simply add $∂M = ∅$, e.g. “Let $M$ be a manifold with $∂M = ∅$”.
  • Just don't make it shorter. I think that writing “Let $M$ a manifold with boundary” is ok even when written multiple times.

I also think that it is not a good idea to make changes in the name of a defined object in order to declare its type (e.g. proposed notation $∂M$, $\mathcal{B}M$, $M^∂$). It is ok when a name of an object indicates its type – e.g. to use name $\vec{x}$ for a vector, $\bar{x}$ for a tuple, small letters for points while using capital letters for sets. But I don't think it is ok when such indications replaces the actual definition. An object is defined and then assigned to a name, but the definition shouldn't be dependent on the name.

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