tangent bundle

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tangent bundle

Tangent vectors:

attachments/Pasted%20image%2020220424180706.png

Exterior powers: attachments/Pasted%20image%2020220424180803.png

  • \({\mathbf{T}}_X = \mathop{\mathrm{span}}_k\left\{{{\partial}x_1, \cdots, {\partial}x_n}\right\}\) and \({\mathbf{T}}_X {}^{ \vee }= \mathop{\mathrm{span}}_k \left\{{dx_1, \cdots, dx_n}\right\}\).
    • This uses the identification \({\mathbf{T}}_{X} \cong \mathop{\mathrm{Der}}_k(C^\infty(X; {\mathbf{R}}), {\mathbf{R}})\) as a space of derivations, i.e. \(D(xy) = D(x) y + x D(y)\).

attachments/Pasted%20image%2020220221004134.png attachments/Pasted%20image%2020220403173357.png

attachments/Pasted%20image%2020220221004200.png attachments/Pasted%20image%2020220221004305.png

A vector field on \(M\) is a global section of \({\mathbf{T}}M\).

Cotangent bundle

attachments/Pasted%20image%2020220424180722.png

There is a pointwise pairing \({{\Gamma}\qty{{\mathbf{T}}M} } \times {{\Gamma}\qty{{\mathbf{T}} {}^{ \vee }M} } \to C^\infty(M; {\mathbf{R}})\) where \(v,\alpha \mapsto \alpha(v)\).

Orientability

attachments/Pasted%20image%2020220424181000.png

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