# Tensor bundle interpretation of Riemannian metric

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.

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 $$T_pM\times T_pM\to\mathbb R$$.
• Maybe you mean $T^{(2,0)}M$ instead of $T^{(0,2)}M$? Mar 29, 2022 at 20:43