A tensor field is smooth if it is smooth as function between manifolds. The sets of smooth tensor fields are denoted by \(\Gamma(T^kTM)\), \(\Gamma(T^kT^*M)\) or \(\Gamma(T^{(k,l)}TM)\).

Covariant tensor fields are especially important. They have another notation, i.e. \(\mathcal{T}^k(M)=\Gamma(T^kT^*M)\).

Note, \(\mathfrak{X}(M)=\Gamma(TM)\).

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  • A tensor field is smooth if and only if the components are smooth for every given local chart.
  • Smooth tensor field \(\iff \) \(C^\infty (M)\)-multilinearity (see (0x66d95c16) ).