Results that hold on smooth manifolds, but not on smooth manifolds with boundary?

Question

The only one I can think of is that the product of two smooth manifolds is a smooth manifold. This isn’t the case for smooth manifolds with boundary. Are there other results like this?

Answer 1

I'll share some examples that come to mind.

Rank theorem, case of immersions

Suppose $M$ and $N$ are manifolds (without boundary) and $F : M \to N$ is a smooth immersion. For each $p \in M$, there exist a smooth chart $(U, \phi)$ for $M$ centred at $p$ and a smooth chart $(V, \psi)$ for $N$ centred at $F(p)$ such that $F(U) \subset V$, in which F has a coordinate representation of the form $$ \psi \circ F \circ \phi^{-1} (x^1, \dots, x^m) = (x^1, \dots, x^m , 0, \dots , 0).$$

Now let's see what can happen if $N$ has a boundary.

Take $M = \{ t \in \mathbb R \}$ and $N = \{ (u_1, u_2) \in \mathbb R^n : u_2 \geq 0 \} $. Let $F: M \to N$ be the map $F : t \mapsto (u_1, u_2) = (t, t^2)$. This is a smooth immersion. However, the conclusion of the rank theorem (above) does not hold for the point $t = 0$. (It fails to hold even if we allow the local representation of $F$ in the conclusion of the rank theorem to shuffle the $0$'s around, and/or replace the $0$'s with constants.)

Rank theorem, case of submersions

Suppose $M$ and $N$ are manifolds (without boundary) and $F : M \to N$ is a smooth submersion. For each $p \in M$, there exist a smooth chart $(U, \phi)$ for $M$ centred at $p$ and a smooth chart $(V, \psi)$ for $N$ centred at $F(p)$ such that $F(U) \subset V$, in which F has a coordinate representation of the form $$ \psi \circ F \circ \phi^{-1} (x^1, \dots, x^m, x^{m+1}, \dots x^n) = (x^1, \dots, x^m).$$

Let's see what can go wrong if $M$ is a manifold with boundary.

Take $M = \{ (s_2, s_2) \in \mathbb R : s_2 \geq 0 \}$ and $N = \{ u \in \mathbb R \} $. Let $F: M \to N$ be the map $F : (s_1, s_2) \mapsto u = s_2 - s_1^2$. This is a smooth submersion. But the conclusion of the rank theorem does not hold for the point $(s_1, s_2) = (0, 0)$. (It fails to hold even if we allow the local representation of $F$ in the conclusion of the rank theorem to permute some coordinates).

Global flows

Suppose $V$ is a smooth vector field on a compact manifold $M$ (without boundary). Then every maximal integral curve of $V$ is defined for $t \in \mathbb R$.

Let's see what can happen if $M$ has a boundary. Suppose $M = \{ x \in \mathbb R : x \in [-1, 1] \} $ and consider the vector field $V = \frac \partial {\partial x}$. This has a integral curve $\gamma(t) = t$, but this integral curve is only defined for $t \in [-1, 1]$. It's not defined for $t \in \mathbb R$.

Positively oriented atlases

Let $M$ be a manifold (without boundary) that is orientable. Then there exists an atlas for $M$ consisting entirely of positively oriented charts.

Let's see what can happen if $M$ has a boundary. Suppose $M = \{ u \in \mathbb R : u \in [-1, 1] \} $, and pick the orientation such that $(\frac \partial {\partial u})$ is a positively oriented at each point. The boundary chart $\phi : (-1, 1] \subset M$ defined by $\phi : u \mapsto x^1 = (1 - u)$ is negatively oriented. We can't "flip" this chart to make it positively oriented, because of our conventions that the final local coordinate for a boundary chart must be positive.

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Edit: It might be helpful if I also list some important theorems that do hold for manifolds with boundary in the same way that they hold for manifolds without boundary.

• A manifold (with or without boundary) is locally path connected. (Hence its connected components are the same as its path components, and each connected component is an open subset.)
• Given an open cover for a manifold (with or without boundary), there exists a smooth partition of unity subordinate to this open cover. (This allows us to do many useful things. For example, this enables us to construct smooth extensions of functions that are only defined on a closed subset. This also enables to us to come up with a reasonable definition of the integral of differential forms on the manifold. All these constructions work fine for manifolds with boundary just as they do for manifolds without boundary.)