# Sub-manifolds

Let $P$ and $Q$ be sub-manifolds of $M$. Then $\forall x \in P$ and $\forall y \in Q$ there are charts:

• $\phi : U \longrightarrow \mathbb{R}^m$ for $M$ about $x$ adapted to $P$
• $\psi : V \longrightarrow \mathbb{R}^m$ for $M$ about $y$ adpapted to $Q$

such that

• $\phi (U \cap P) = \phi (U) \cup (\mathbb{R}^p \times \{0\})$
• $\psi (V \cap Q) = \psi (U) \cup (\mathbb{R}^q \times \{0\})$

Now let $j : P \longrightarrow Q$ be the injection map. Consider the commutative diagram below.

Then $\psi \circ j \circ \phi^{-1} |_{\phi (U) \cap \{\mathbb{R}^p \times \{0\})} = \psi \circ \phi^{-1} |_{\phi (U) \cap \{\mathbb{R}^p \times \{0\})}$ which is smooth. Hence $j$ is smooth.

# Integral Curves

Let $\alpha, \beta : I \longrightarrow V$ be integral curves in a manifold $M$ then $A = \{ t : \alpha (t) = \beta (t) \}$ is closed.

Proof:

Let $\{t_n\}$ be have a limit $t$ in $I$. Let $\xi : U \longrightarrow \mathbb{R}^m$ be a chart then $\xi (\alpha (t_n))$ converges to $\xi (\alpha (t))$ by continuity. But $\xi (\beta (t_n))$ also converges to $\xi (\alpha (t))$. Hence $\xi (\alpha (t)) = \xi (\beta (t))$ and so $t \in A$. Thus $A$ is closed.