Tensor Field - Notation

Notation

The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent bundle TM = T(M) might sometimes be written as

to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold M. Do not confuse this with the very similar looking notation

;

in the latter case, we just have one tensor space, whereas in the former, we have a tensor space defined for each point in the manifold M.

Curly (script) letters are sometimes used to denote the set of infinitely-differentiable tensor fields on M. Thus,

are the sections of the (m,n) tensor bundle on M which are infinitely-differentiable. A tensor field is an element of this set.

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