> [!definition] > > Let $X$ be an $E$-[[Manifold|manifold]] of class $C^p$ and $Y \subset X$. $Y$ is a **submanifold** if: > 1. There exists [[Split Subspace|split subspaces]] $E_1$ and $E_2$. > 2. For each $y \in Y$, there exists a [[Atlas|chart]] $(U, \psi)$ at $y$, such that $\psi: U \to U_1 \times U_2$, where $U_1$ is [[Open Set|open]] in $E_1$, and $U_2$ is open in $E_2$. > 3. There exists $a_2 \in U_2$ such that $\psi(Y \cap U) = U_1 \times a_2$ looks like a [[Level Set|level set]]. > > where charts satisfying the above two conditions are known as **slice charts**. > > Then $Y$ is [[Locally Closed|locally closed]] in $X$, and the map $\psi$ induces a bijection > $ > \psi_1: Y \cap V \to V_1 \quad y \mapsto \psi(y)_1 > $ > The collection of all pairs $(Y \cap U, \psi_1)$ obtained above forms an [[Atlas|atlas]] of class $C^p$ for $Y$, making $Y$ a $E_1$-manifold of class $C^p$. > > *Proof*. Let $y \in Y$ and $(U, \psi)$ be a chart at $y$. Since $\psi(Y \cap U) = U_1 \times a_2$ is locally closed, for any $\hat p \in U_1 \times a_2$, there exists an open set $V \subset E$ such that $U_1 \times a_2$ is closed in the relative topology. Therefore $Y \cap \psi^{-1}(V)$ is closed in $\psi^{-1}(V)$. > > Let $(U, \psi)$ and $(V, \varphi)$ be two charts with $U \cap V$ intersecting. Let $(Y \cap U, \psi_1)$ and $(Y \cap V, \varphi_1)$ be the induced charts, then there exists open sets $U_2$ and $V_2$ such that $U_1 \times U_2 \subset \widehat U$ and $V_1 \times V_2 \subset \widehat V$. > > The translation map of the original charts > $ > \varphi \circ \psi^{-1}: U_1 \times U_2 \to V_1 \times V_2 > $ > is a [$C^p$-morphism](Space%20of%20Continuously%20Differentiable%20Functions) with respect to the product open sets. We can express the translation map $\varphi \circ \psi_1^{-1}$ as > $ > U_1 \to U_1 \times U_2 \to V_1 \times V_2 \to V_1 > $ > a composition of the translation map with inclusion before and projection after, making it a $C^p$-isomorphism as well. > [!theorem] > > Let $X$ be an $E$-manifold of class $C^p$ and $Y \subset X$ be a submanifold. Then the inclusion map $\iota: Y \to X$ is a $C^p$ [[Manifold Embedding|embedding]], and $Y$ has the [[Relative Topology|induced topology]] from $X$. > > *Proof*. Let $y \in Y$, $(U, \psi)$ be a chart at $y$ that induces a chart $(U^Y, \psi^Y)$ on the submanifold. If $E$ splits as $E_1 \oplus E_2$ with $\psi(Y \cap U) \subset E_1 \times a_2$, and $\iota_1: E_1 \to E$ with $x \mapsto x + a_2$, then the interpretation of $\iota$ with respect to these maps is > $ > \iota_{U^Y, U}: E_1 \to E \quad \iota_{U^Y, U} = \psi \circ \iota\circ \paren{\psi^{Y}}^{-1} = \text{Id} \times a_2 > $ > a $C^p$-morphism, with $D(\iota_{U^Y, U})(y) = \text{Id} \times 0$ being injective, and its image splitting $E$ as $E_1 \oplus E_2$. Since the coordinate representation of the inverse restricted to $Y$ is just the identity map, $\tau^{-1}$ is continuous, and $\tau$ is an [[Embedding|embedding]]. > [!definition] > > Let $X$ be an $E$-manifold and $Y \subset X$ be an $F$-submanifold. Let $\catc$ be a subcategory of Banach spaces containing $E$ and $F$, and $\lambda: \catc \to \mathfrak{D}$ be a [[Differentiable Functor|differentiable functor]]. If $\omega \in \mathbf{S}(X, \lambda(TX))$ is a [[Sections of Vector Bundles|section of]] the image of the tangent bundle, then > $ > \begin{CD} > Y @>{\iota}>> X \\ > @V{\omega^*}VV @VV{\omega}V \\ > \lambda(TY) @<{\lambda(d\iota)}<< \lambda(TX) > \end{CD} > $ > the **induced section** $\omega^* \in \mathcal{S}(X, \lambda(TX))$ is the [[Pullback of Sections|pullback]] of $\omega$ by the inclusion map. > > If the context is clear, the induced section will simply be denoted as $\omega$ as well.