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<br>The tensor bundle is just not a differentiable manifold in the normal sense, since it's infinite dimensional. It is however an algebra over the ring of scalar functions. Each tensor is characterized by its ranks, which indicate how many tangent and cotangent components it has. Sometimes these ranks are referred to as covariant and contravariant ranks, signifying tangent and cotangent ranks, respectively. A frame (or, in additional precise terms, a tangent frame), is an ordered basis of specific tangent area. Likewise, a tangent body is a linear isomorphism of Rn to this tangent area. A shifting tangent body is an ordered checklist of vector fields that give a basis at every point of their area. One may regard a transferring body as a piece of the body bundle F(M), a GL(n, R) principal bundle made up of the set of all frames over M. 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