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.

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