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Everything generalizes to manifolds with boundary.
A reference for this material is Chapter 1 of John M. Lee. Introduction to smooth
manifolds. Second edition. Grad. Texts in Math., 218. Springer, New York, 2013.
xvi+708pp. ISBN: 978-1-4419-9981-8.
Manifold with boundary An \(n\)-dimensional manifold with boundary is a second-countable
Hausdorff topological space \(M\) such that each point \(p\in M\) admits an open neighborhood \(U \subset M\) and
a homeomorphism \(\varphi : U \to \varphi (U ) \subset \overline { \mathbb {H}^n}\) where \(\varphi (U) \) is an open subset of \(\overline { \mathbb {H}^n}\). Pairs \((U, \varphi )\) as above are called charts.
Let \(A \subset \overline {\mathbb {H}^n}\) be an open set. We say a function \(f : A \to \mathbb {R}^k\) is smooth if there is an open set \(U \subset \mathbb {R}^n\) containing \(A\)
and a smooth function \(\tilde {f} : U \to \mathbb {R}^k\) with \(\tilde {f} \vert _A = f\).
Compatible charts We say two charts \((U, \varphi )\) and \((V, \psi )\) are compatible if the maps
\begin{align*} \varphi \circ \psi ^{-1} &: \psi (U \cap V) \to \varphi (U \cap V) , \\ \psi \circ \varphi ^{-1} &: \varphi (U \cap V ) \to \psi (U \cap V) \end{align*}
are smooth.
Smooth atlases and smooth structures for manifolds with boundary are defined
just like they were defined for manifolds with no boundary. From now on,
we reserve the word “chart” for a chart belonging to the smooth structure.
Boundary Let \(M\) be a smooth \(n\)-dimensional manifold with boundary. The boundary is
defined as the set
where the union is taken over all charts \((U,\varphi )\).
Let \((U,\varphi ) \) be a chart. Then
\[ U \cap \partial M = \varphi ^{-1} ( \partial \mathbb {H}^n ) . \]
Solution:
Exercise. The contention \(\supseteq \) is trivial. For the other one, use the inverse function
theorem. For a manifold with boundary that is not necessarily smooth, the
proposition is still true, but it is more difficult.
Smooth functions \(f: M \to \mathbb {R}\) are defined in the usual way. The same is true for tangent spaces \(T_pM\),
vector fields, and forms \(\Omega ^k (M)\). Bump functions and partitions of unity still work equally
well. The theory of flows is a little different because you can fall off the edge!
Boundary is a manifold Let \(\mathcal {A}\) be a smooth structure on a smooth \(n\)-dimensional
manifold with boundary \(M\). Then the set
\[ \mathcal {A}_{\partial } : = \{ ( U \cap \partial M , \varphi \vert _{U \cap \partial M} ) \, \vert (U ,\varphi ) \in \mathcal {A} \} \]
is a smooth structure on \(\partial M\) (if we delete the
first component of each \(\varphi \vert _{U \cap \partial M }\), since it is identically zero). With this smooth structure, the
inclusion
\[ j _{\partial M} : \partial M \to M \]
is smooth.
Solution:
Exercise.
Orientation Let \(M\) be a smooth \(n\)-dimensional manifold with boundary. An orientation
on \(M\) is an equivalence class of forms \(\eta \in \Omega ^n (M) \) with \(\eta (p) \neq 0\) for all \(p \in M\), where we say that two such forms
are equivalent if one of them can be obtained from the other one by multiplying it by
a positive smooth function.
We say a manifold is oriented if it is equipped with an orientation, and given an
orientation, the forms in this equivalence class are called orientation forms.
Integrals with boundary Let \(M\) be an oriented smooth \(n\)-dimensional manifold with
boundary, \(\eta \in \Omega ^n (M)\) an orientation form form, and \(\omega \in \Omega ^n (M)\) with compact support. Let \(\{ ( U_i , \varphi _i ) \}_{i \in I}\) be a finite set
of charts that cover the support of \(\omega \), and \(\{ \rho _i \} _{i \in I}\) a corresponding partition of unity. If
each \(U_i\) is connected, then the integral of \(\omega \) is defined as
The integral does not depend on the collection of charts nor partition of unity.
Solution:
Same as in the non-boundary case.
Outward pointing vector Let \(M\) be a smooth \(n\)-dimensional manifold with boundary. An
outward-pointing vector is a vector field \(V \in \mathfrak {X}(M)\) such that for any chart \((U , \varphi )\) and any \(p \in U \cap \partial M\), one has
\[ V ( \varphi _1 ) (p ) < 0 , \]
where \(\varphi = (\varphi _1, \ldots , \varphi _n)\).
Let \(M\) be a smooth \(n\)-dimensional manifold with boundary. Then there is at least one
outward pointing vector.
Solution:
Exercise. In \(\overline {\mathbb {H}}^n\), one can take \(V = - \partial _1\). In a manifold, one can construct one using a partion
of unity.
Orientation of boundary Let \(M\) be an oriented smooth \(n\)-dimensional manifold with
boundary, \(\eta \in \Omega ^n (M) \) an orientation form, and \(V \in \mathfrak {X}(M)\) an outward-pointing vector field. The Stokes
orientation of \(\partial M\) is defined to be the one of the form
Smooth manifolds