Let $W=(M \times \mathbb{R}, pr, M)$ be the trivial vector bundle over a compact manifold $M$, and define $$V=T^0_0(M,W) := T^0_0M \otimes W,$$ and $V$ is called "the vector bundle of $W$-valued (0,0)-tensors on $M$".
Apparently we can identify $T^0_0(M,W) = T^0_0(M, \mathbb{R}) = T^0_0(M)$.
My question is, is there a simple way to describe $V$? Is it $V=\mathbb{R}$?
Here, $T^i_k M$ is the $(i,k)$-tensor bundle.
It's not very common to use the language "$(0,0)$-tensor", but by convention a $(0,0)$-tensor on a vector space $V$ is just a scalar, and so a $(0,0)$ tensor field on a manifold $M$ -- that is, an section of $T_0^0(M)$ -- is a smooth function on $M$.
Likewise, it's obvious that the smooth functions on $M$ are exactly the sections of the trivial bundle $W = M \times \mathbb{R}$. Thus $T_0^0(M) \cong W$ as bundles, and both are trivial one-dimensional bundles.
It follows from the definitions that if I take the tensor product of two one-dimensional trivial bundles over $M$, what I get is a one-dimensional trivial bundle. So, $V \cong W \cong T_0^0(M)$ is the trivial rank one bundle over $M$, whose sections are the smooth functions on $M$.