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• Links:
  • vector bundle
  • de Rham cohomology
  • Riemannian metric
  • Hermitian K theory

tangent bundle

Tangent vectors:

Exterior powers:

• \({\mathbf{T}}_X = \mathop{\mathrm{span}}_k\left\{{{\partial}x_1, \cdots, {\partial}x_n}\right\}\) 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. \(D(xy) = D(x) y + x D(y)\).

Cotangent bundle

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