How does the cotangent-space definition change for a general manifold M over a field F?

Conditions

• $M$ is a smooth manifold.
• $F$ is the field over which the manifold is considered.

Reasoning, step by step

1. Replace the ambient space $\mathbb{R}^n$ with a general smooth manifold $M$.
2. Replace the field $\mathbb{R}$ with a general field $F$.
3. Define the cotangent space $T_p^*M$ as $\operatorname{Hom}(T_pM, F)$.

Example

The video displays the generalization $T_p^*M = \operatorname{Hom}(T_pM, F)$ and the narrator explains that $\mathbb{R}^n$ is the manifold $M$ and $\mathbb{R}$ is the field $F$ in the earlier example.

Common misconceptions

• Thinking the generalization only applies to $\mathbb{R}^n$.
• Confusing the manifold $M$ with the field $F$.

Watch the explanation

Connected concepts

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