8 Boundary

Definition 8.1 (Boundary of half space) Recall the half space is defined as \[ \mathbb H^n = \{ (x^1, \dots, x^n) : x^n \ge 0 \} \] Its boundary is defined as the set \[ \mathbb \partial H^n = \{ (x^1, \dots, x^n) : x^n = 0 \} \] Its interior is the remainder of the set \[ \mathbb (H^n)^{\circ} = \{ (x^1, \dots, x^n) : x^n > 0 \} \]

Theorem 8.1 Let \(U, V\) be open sets of \(\mathbb H^n\). If \(f: U \to V\) is a smooth function in \(\mathbb H^n\), then \(f\) maps interior point of \(\mathbb H^n\) to interior point of \(\mathbb H^n\), and boundary points to boundary points.

::: {.definition name=“Boundary of manifolds”}

If \(M\) is a manifold of dimension \(n\), its boundary \(\partial M\) is defined as the set of all points that is mapped to \(\partial \mathbb H^n\) by some local chart of \(\mathbb M\).

If \(\{(U, \phi) \}\) is an atlas of \(M\), then \(\{ (U \cap \partial M, \phi |_{U \cap \partial M}) \}\) can serve as an atlas of \(\partial M\), where \[ \phi|_{U \cap \partial M}: (U \cap \partial M) \to \partial \mathbb H^n = \mathbb R^{n - 1} \] Hence \(\partial M\) has dimension \(n - 1\).

The above atlas maps neighborhoods of \(\partial M\) to \(\mathbb R^{n-1}\) not \(\mathbb H^{n-1}\), violating our definition. This is fixed by a diffeomorphism between \(\mathbb R^{n-1}\) and any open ball \(B\) of \(\mathbb R^{n-1}\) that resides in \(\mathbb H^{n-1}\).

8.1 Interior Multiplication

Definition 8.2 (Interior multiplication) If \(\beta\) is a \(k\)-covector on a vector space \(V\), the interior multiplication, or contraction of \(\beta\) with a vector \(v \in V\) is the \((k - 1)\)-covector \[ \iota_v \beta (v_2, \dots, v_k) = \beta(v, v_2, \dots, v_k) \]

Define \(\iota_v(\beta) = 0\) for any \(0\)-covector \(\beta\).

Proposition 8.1 For covectors \(\alpha^1, \dots, \alpha^k\) on a vector space \(V\), and \(v \in V\) \[ \iota_v(\alpha^1 \wedge \dots \wedge \alpha^k) = \sum^k_{i=1} (-1)^{i-1} \alpha^i(v) \alpha^1 \wedge \dots \wedge \hat {\alpha^i} \wedge \dots \wedge \alpha^k \]

8.2 Boundary Orientation

Proposition 8.2 The inclusion map \(i: \partial M \to M\) is smooth, its differential is an injection from tangent spaces of \(\partial M\) to those of \(M\). However, such injection is not a surjection, and in particular \(\partial/\partial x_n\) is not in the range for any local chart \((x^1, \dots, x^n)\) of \(M\).

At a point \(p \in \partial M\), identify the tangent space \(T_p(\partial M)\) as a subspace of \(T_pM\) by the above proposition.

Definition 8.3 (Outward pointing vector) A vector \(X\) of \(T_pM \setminus T_p \partial M\) at \(p \in \partial M\) is said to be inward pointing if there exists curve \(c: (-1, 1) \to M\) such that

1. \(c(0, 1) \subset M^\circ\)
2. \(c(0) = p\)
3. \(c'(0) = X\)

The vector \(X\) is outward pointing if \(-X\) is inward pointing.

Definition 8.4 (Smooth outward pointing vector field) A vector field along \(\partial M\) assigns each \(p \in \partial M\) a vector \(X_p \in T_pM\). Such vector field is said to be smooth at \(p\) if written in local coordinates of \(M\), \[ X = \sum a_i \frac{\partial}{\partial x^i} \] \(a_i\) is smooth at \(p\) in \(\partial M\), i.e., \(a_i \in C^\infty_p(\partial M)\).

Theorem 8.2 There exists a smooth outward pointing vector field along \(\partial M\) for any smooth manifold \(M\).

Theorem 8.3 If \(\omega\) is an orientation form of \(M\), and \(X\) is a smooth outward pointing vector field along \(\partial M\), then \(\iota_X\omega\) is an orientation form of \(\partial M\).