Talk:Tensor bundle

Is this correct?

Upon searching google books, I found various definitions. Many of them defined tensor bundle of type bla, where bla would be something like (r,s). Other definitions were much more general than the one given in this article, like in the book "Heat kernels and Dirac operators" where a tensor bundle is defined to be any vector bundle arising from a representation of GL(n). RobHar (talk) 16:05, 2 August 2008 (UTC)

The article seems to be referring to "the tensor bundle", which is common parlance for the bundle whose fibre is the full tensor algebra. However, tensors arising as associated bundles by the pushout of a GL(n) representation are a more suitable topic for an encyclopedia article (tensors of type ρ). Unfortunately, there seems to be little agreement as to where one draws the line. For instance, must all tensor bundles be associated bundles of the tangent bundle? Or is one allowing a more general principal bundle besides the frame bundle? The former requirement is clearly the case in most uses of the term "tensor" in physics and analysis, where the tensor intertwines the action of diffeomorphisms with a linear transformation of the fibre. To make matters worse, determinantal representations of GL(n) are often ruled out (and called "density bundles" rather than "tensor bundles", and tensor products thereof are termed "tensor density bundles" or similar). siℓℓy rabbit (talk) 17:29, 2 August 2008 (UTC)

I see. I don't see any reason to not mention this panoply of definitions, but one thing to keep in mind is the fact that various articles link to this page, and they don't all refer to the current definition in here, e.g. Tensor field wants tensor bundle to refer to a vector bundle of the form T(M)^⊗r ⊗ T*(M)^⊗s, Algebra bundle wants the tensor bundle to be the tensor algebra of any vector bundle, and Differentiable manifold likes the definition as it is. —Preceding unsigned comment added by RobHar (talk • contribs) 17:39, 2 August 2008 (UTC)

Calculus?

While it is certainly true that doing calculus for arbitrary tensor fields requires a connection, the exterior derivative of an antisymmetric differentiable tensor field is well defined even without one. Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:09, 20 March 2014 (UTC)

Why Direct Sum?

Shouldn't the tensor bundle be defined as the tensor products of the tangent bundle and the cotangent bundles instead of the direct sum of those? Pratyush Sarkar (talk) 03:22, 24 June 2014 (UTC)