Tensor and tensor field differences

I have recently tried to understand the differences between tensors and tensor fields. Am I correct in the statement that a tensor is defined as a linear multilinear map on a set of vector spaces and/or dual vector spaces to a field $$\mathbb{R}$$ or $$\mathbb{C}$$ (as is written in wikipedia). On the other hand a tensor field is defined as a linear multilinear map on a set of tangent vector spaces and/or dual tangent vector spaces to a field $$\mathbb{R}$$ or $$\mathbb{C}$$ (as is written in wikipedia).

If this is correct does that mean that a tensor gives as an output a global property of the vector space we put in and sharing all symmetries of the vector space? And on the other hand the tensor field tells us a local property (such as curvature at point x) of the manifold?

• It's the same as the difference between a vector and a vector field. Sep 23, 2021 at 12:36

A $$(p,q)$$ tensor field $$T$$ on a smooth manifold $$M$$ is defined as a multilinear map: $$T : \underbrace{\Omega(M)\times\dots\times\Omega(M)}_{p\ times} \times\underbrace{\mathfrak{X}(M)\times\dots\times\mathfrak{X}(M)}_{q\ times} \to C^{\infty}(M)$$ where $$\Omega(M)$$ is the set of all covector fields on $$M$$ and $$\mathfrak{X}(M)$$ is the set of all vector fields on $$M$$.

Unpacking the above, $$T$$ sends $$p$$ covector fields $$H_1,\ldots,H_p$$ and $$q$$ vector fields $$X_1,\ldots,X_q$$ to some $$C^{\infty}$$ function $$f$$ on $$M$$. i.e., $$T(H_1,\ldots,H_p,X_1,\ldots,X_q)$$ is a $$C^{\infty}$$ function on $$M$$. Notationally, its effect for a point $$x\in M$$ is $$T_x(H_1(x),\ldots,H_p(x),X_1(x),\ldots,X_q(x))=f(x)$$ where $$H_i(x)$$ and $$X_i(x)$$ are respectively covectors and vectors in cotangent and tangent spaces at $$x$$. $$T_x$$ is defined locally at $$x$$ as: $$T_x:\underbrace{T^*_xM\times\dots\times T^*_xM}_{p\ times} \times\underbrace{T_xM\times\dots\times T_xM}_{q\ times} \to F$$

Thus you can see how the tensor field $$T$$ selects for each point $$x$$ a locally defined tensor acting on cotangent and tangent spaces at $$x$$.

To address what you wrote in the question: "a tensor is defined as a linear multilinear map on a set of vector spaces and/or dual vector spaces to a field..." - you hopefully see why this isn't accurate. Similarly, "tensor field is defined as a linear multilinear map on a set of tangent vector spaces and/or dual tangent vector spaces to a field" isn't accurate either.

• Ok Thanks. So physically one might say a tensor field at point $x$ is a locally defined tensor, which in your notation is $T_x$ acting on (co-)tangent vector spaces of the point $x$ with respect to the manifold. Sep 23, 2021 at 11:31
• @krabbypatty: That's correct. However, personally I wouldn't call it a "physical" description. More of an intuitive mathematical one. Sep 23, 2021 at 11:37
• Your definition of tensor field is missing a specification of what kind of map $T$ is. Your description of $\Omega(M)$ as a set, suggests we are in the category of sets, which clearly is not enough. Sep 23, 2021 at 15:36
• @mmeent: Thanks for pointing it out! By "kind of map", I'm taking it to mean that I've failed to mention that $T$ is multilinear (maybe you meant something else by that?). As for $\Omega(M)$, I remember first learning it as a "collection of all covector fields on a smooth manifold". If I've missed a subtlety in description, could you let me know and I'll update my answer asap Sep 23, 2021 at 15:48
• Is multilinear enough here? Doesn't it need to be a morphism of $C^\infty(M)$-modules? Sep 23, 2021 at 15:57