Differential Geometry

Definition

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Definition

• $M$,
  • manifold
• $N$,
  • manifold
• $T_{p}M$,
  • tangent space at $p$
• $v \in T_{p}M$,
  • tanvent vector at $p$
  • $v:C^{\infty}(M) \rightarrow \mathbb{R}$,
    • linear map
• $T_{p}^{*}M$,
  • cotangent space at $p$
• $\omega \in T_{p}^{*}M$,
  • cotangent vector at $p$
  • $\omega:T_{p}M \rightarrow \mathbb{R}$,
    • linear map
• $TM$,
  • tangent bundle

\[\begin{eqnarray} TM & := & \sqcup_{p \in M}T_{p}M \nonumber \\ & = & \sqcup_{p \in M} \{p\} \times T_{p}M \\ & \cong & \sqcup_{p \in M} \{p\} \times \mathbb{R}^{n} \end{eqnarray} .\]

• $T^{*}M$,
  • cotangent bundle

\[\begin{eqnarray} T^{*}M & := & \sqcup_{p \in M}T_{p}^{*}M \nonumber \\ & = & \sqcup_{p \in M} \{p\} \times T_{p}^{*}M \end{eqnarray} .\]

• $(p, v) \in TM$,
  • $p \in M$, $v \in T_{p}M$,
• $(p, \omega) \in T^{*}M$,
  • $p \in M$, $\omega \in T_{p}^{*}M$,
• $F: M \rightarrow N$,
  • smooth map
• $dF_{p}: T_{p}M \rightarrow T_{F(p)}{N}$
  • differential of $F$ at $p$

\[f \in C^{\infty}(N), \ dF_{p}(v)(f) := v(f \circ F) .\]

• $X: M \rightarrow TM$,
  • vector field
  • $p \in M \mapto X_{p} TM$,

\[\begin{eqnarray} X_{p} & = & X^{i}(p) \left. \frac{\partial}{\partial x^{i}} \right|_{p} \nonumber \\ \pi \circ X & = & \mathrm{Id}_{M} \nonumber . \end{eqnarray}\]

• vector bundle
• coframe
• frame

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Definition

\[\begin{eqnarray} \left. \frac{\partial}{\partial x^{1}} \right|_{\phi(p)}, \ldots, \left. \frac{\partial}{\partial x^{n}} \right|_{\phi(p)}, \end{eqnarray}\]

form a basis for $T_{\phi(p)} \mathbb{R}^{n}$.

Let

\[\begin{eqnarray} \left. \frac{\partial}{\partial x^{i}} \right|_{p} & := & (d \phi_{p})^{-1} \left( \left. \frac{\partial}{\partial x^{n}} \right|_{\phi(p)} \right) \nonumber \\ & = & d(\phi^{-1})_{\phi(p)} \left( \left. \frac{\partial}{\partial x^{n}} \right|_{\phi(p)} \right) \nonumber \end{eqnarray}\]

form a basis for $T_{p}M$.

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Reference