Tensor and tensor field differences

Question

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?

Answer 1

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.