= Tensoriality
{title2=$C^\infty(M)\text{-linearity}$}
= Tensorial
{synonym}
A <multilinear> operation on <vector fields> is <tensorial> when it is <linear> over <smooth functions> in every argument. Its value at a point then depends only on the argument <vectors> at that point. To see this, use a <smooth cutoff function> to reduce to local fields, expand them in a <local frame>, and apply <linearity> over <smooth functions> to their coefficients. A smoothly valued <tensorial> operation therefore defines a <tensor field>. The <curvature> of an <affine connection> is <tensorial>, whereas a <covariant derivative> differentiates a <scalar> coefficient in its second argument and is not <tensorial> there.
Back to article page
