I have recently been introduced to the Tensor bundle and to Riemannian metrics. The tensor bundle is : $$\mathcal{T}^{k,\ell}M=\coprod_{p\in M}(T_pM)^{\otimes k}\otimes(T_p^\ast M)^{\otimes \ell}=\bigcup_{p\in M}\{p\}\times(T_pM)^{\otimes k}\otimes(T_p^\ast M)^{\otimes \ell}.$$
The special case we are interested in is $\mathcal{T}^{0,2}M$, and particularly its sections $g\in\Gamma(\mathcal{T}^{0,2}M)$. Now, to define a Riemannian metric, we need additional hypothesis on $g$. This is due to the following two observations.
One the one hand, from universal property of the tenor product, a bilinear map $B:E\times F\to\mathbb{R}$ corresponds uniquely to a linear map $\tilde{B}:E\otimes F\to\mathbb{R}$ :
That is, $\mathcal{B}(E,F;\mathbb{R})\cong(E\otimes F)^\vee$.
On the other hand, the following linear map (defined on pure tensors and extended by linearity, using this principle twice here) is an isomorphism : $$\begin{matrix}\Phi&:&E^\vee\otimes F^\vee&\to&(E\otimes F)^\vee\\&&\varphi\otimes\psi&\mapsto&[u\otimes v\mapsto\varphi(u)\psi(v)]\end{matrix}$$
Therefore, we have finally, in our case : $$\mathcal{B}(T_pM,T_pM;\mathbb{R})\cong T_p^{0,2}M.$$
Thus, a section $g\in\Gamma(\mathcal{T}^{0,2}M)$ is a collection $g:M\to\mathcal{B}(T_\bullet M,T_\bullet M;\mathbb{R})$ of bilinear forms on the tangent spaces, such that this collections varies regularly in the parameter (since we have constructed a topology on $\mathcal{T}^{0,2}M$). A Riemannian metric is therefore a collection of scalar products that fits the previous requirements. My question is the following :
How to interpret positiveness and definiteness of a bilinear form $B:E\times E\to\mathbb{R}$ in terms of its associated element in $E^\vee\otimes E^\vee$ ? That is, in our case, how to define a Riemannian metric only in terms of the tensor bundle $\mathcal{T}^{0,2}M$ ?
P.S. : I know most of the "tensor bundle" thing is unnecessary to answer the main question, that is about interpreting positiveness and definiteness of a bilinear form in terms of its representation in the tensor product of duals. However, I still wanted to include this discussion as a motivation for my question and as an entry point to some more references I couldn't find by searching.
Best Answer
The construction $g_p:T_pM\times T_pM\to\mathbb R$ is defined as $$g_p(X,Y)={g_p}_{\mu\nu}X^{\mu}Y^{\nu},$$ where $X=X^{\mu}\partial_{\mu}$, $Y=Y^{\nu}\partial_{\nu}$ and $\partial_i$ are the coordinate basis for $T_pM$. The metric tensor $g_p$ has components ${g_p}_{\mu\nu}$ which are functions of the coordinates.
At a given point, ${g_p}_{\mu\nu}X^{\mu}Y^{\nu}$ is a scalar quadratic form for which on checks positive definiteness as usually one does is our linear algebra courses.
I would stress that $g$ is a section of the bundle $T^{(0,2)}M$ which assign to each $p$ in $M$ a tensor $g_p$ an element of $T_pM\otimes T_pM$, which is a bilinear map $T_pM\times T_pM\to\mathbb R$.