multilineal lección six


tensor fields on a surface and its covariant derivatives

the collection T\Sigma of all the tangent spaces T_p\Sigma is called the tangent bundle of the surface, i.e.

T\Sigma=\bigsqcup_pT_p\Sigma

A vector field in a surface is a mapping X:\Sigma\to T\Sigma with the condition p\mapsto X\in T_p\Sigma and since \partial_1,\partial_2 span T_p\Sigma then

X=X^s\partial_s

This construction determines a contravariant tensor field of rank one, which is taken as the base to ask how other tensor fields -of any rank and any variance- vary…

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Filed under differential geometry, fiber bundle, math, multilinear algebra

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