11-1. Covectors

Tangent Covevtors on Manifolds

Definition

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Covector Fields

Definition Cotangent bundle

• $M$,
  • smooth manifold with or without boundary

\[TM := \sqcup_{p \in M} T_{p}^{*}M .\]

is called the cotangent bundle of $M$. $\pi: T^{*}M \rightarrow M$.

\[\pi((p, \omega)) = p .\]

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Definition Covector field

• $M$,
  • smooth manifold with or without boundary
• $\omega: M \rightarrow T^{*}M$,

$\omega$ is said to be covector field if

\[\pi \circ \omega = \mathrm{Id}_{M} .\]

Let $\lambda^{i}: TM \rightarrow \mathbb{R}$ be a basis of $T^{*}M$.

\[\omega(p) = \omega_{i}(p) \left. \lambda^{i} \right|_{p}\]

where $\omega_{i}: M \rightarrow \mathbb{R}$ be a component function.

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Coframes