- #1
cianfa72
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- TL;DR
- About the equivalence of two different definitions of tensor field
As far as I know, there are two definitions of tensor field on a differential manifold ##M##. Just to fix ideas consider the two-times covariant tensor field ##T_{ab}## on it.
First definition: take the tensor product ##T_p^{*}M \otimes T_p^{*}M## as the fiber over the point ##p \in M## and define/build the fiber bundle over ##M##. Then ##T_{ab}## is a smooth section of such tensor product bundle on ##M##.
Second definition: given the tangent and cotangent bundles over ##M##, then ##T_{ab}## is a ##C^{\infty}##-multilinear map $$T_{ab} : \Gamma(T^{*}M) \times \Gamma(T^{*}M) \to C^{\infty}(M)$$ where ##\Gamma(T^{*}M)## is the ##C^{\infty}(M)##-module of the set of smooth sections of ##T^{*}M##.
Are the two definitions actually equivalent ?
First definition: take the tensor product ##T_p^{*}M \otimes T_p^{*}M## as the fiber over the point ##p \in M## and define/build the fiber bundle over ##M##. Then ##T_{ab}## is a smooth section of such tensor product bundle on ##M##.
Second definition: given the tangent and cotangent bundles over ##M##, then ##T_{ab}## is a ##C^{\infty}##-multilinear map $$T_{ab} : \Gamma(T^{*}M) \times \Gamma(T^{*}M) \to C^{\infty}(M)$$ where ##\Gamma(T^{*}M)## is the ##C^{\infty}(M)##-module of the set of smooth sections of ##T^{*}M##.
Are the two definitions actually equivalent ?
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