Pullback of (covariant) Tensor Fields
The construction of the previous section generalizes immediately to tensor bundles of rank (0,s) for any natural number s: a (0,s) tensor field on a manifold N is a section of the tensor bundle on N whose fiber at y in N is the space of multilinear s-forms
By taking Φ equal to the (pointwise) differential of a smooth map φ from M to N, the pullback of multilinear forms can be combined with the pullback of sections to yield a pullback (0,s) tensor field on M. More precisely if S is a (0,s)-tensor field on N, then the pullback of S by φ is the (0,s)-tensor field φ*S on M defined by
for x in M and Xj in TxM.
Read more about this topic: Pullback (differential Geometry)
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