I am confused about translating the definition of the inertia tensor I know into the language of differential geometry. Part of this confusion arises because in every physics textbook I read, the term "coordinates" is used in different places with different meaning (even within the same book).
Anyway, let me try to illustrate what I am familiar with first, and then try to explain my doubts. For example, the definitions I am familiar with are:
Given a (say real) vector space $V$, an $(r,s)$ tensor over $V$ is a multi-linear map $T: \underbrace{V^* \times \dots \times V^*}_{\text{$r$ times}} \times \underbrace{V \times \dots \times V}_{\text{$s$ times}} \to \Bbb{R}$.
and
Given a smooth manifold $M$, a smooth $(r,s)$ tensor field on $M$ is a smooth section $\xi : M \to T^r_s(M)$ of the $(r,s)$ tensor bundle over $M$. (i.e at each point $p$ of the manifold, we have an $(r,s)$ tensor $\xi(p)$ over the tangent space $T_pM$, such that the association $p \mapsto \xi(p)$ is smooth).
Of course, I know a few examples of tensors and tensor fields from basic linear algebra. For example, if $V$ is a finite-dimensional vector space, we can always equip it with an inner product (a $(0,2)$ tensor over $V$). Also, a typical example of a tensor field is a metric tensor field $g$ on a smooth manifold (a $(0,2)$ tensor field). I'm also "acquainted"(i.e I've seen them but haven't had any practice with them) with other examples of tensor fields from physics, such as the stress-energy tensor field in the context of electromagnetism.
Now, the reason I'm not so confused in these cases is because I know precisely what the spaces $V$ and $M$ are, and I know the exact definition (i.e the rule for the map). When it comes to the inertia tensor however, I'm not so sure.
Now, given a rigid body, here's the definition I know (from Landau and Lifshitz, Volume $1$, $\S 32$).
We take a "moving system of coordinates $x_1, x_2, x_3$, which is supposed to be rigidly fixed in the body, and to participate in its motion", and in these coordinates, we define \begin{align} I_{ij} &= \int_{\text{Body}}(\delta_{ij} \lVert x\rVert^2 - x_i x_j) \cdot \rho \, dV. \end{align} (this discussion occurs over a few pages, so I'm summarizing the essentials).
I understood their motivation for such a definition: namely by defining $I_{ij}$ like this, the rotational kinetic energy can be expressed as $T_{\text{rot}} = \dfrac{1}{2}I_{ij} \Omega^i \Omega^j$ ($\vec{\Omega}$ being the angular velocity of the rigid body). However, here are some things I am unclear about:
$1$. What does it mean in a precise and technical manner to have a system of coordinates attached to a point in the body? I understand the intuitive notion that I'm supposed to think of myself as being anchored to a point in the rigid body and "describe how I see things". But I'm having trouble precisely formulating this "simple" idea as a precise mathematical definition. I'm hoping someone can fill in something along the lines of "A coordinate system (____ on a certain space ____ )is a (_____ type of object ____). And a coordinate system attached to a point is (____ something_____)"
$2$. My next doubt is whether the inertia tensor is actually a tensor over a fixed vector space $V$ (if so which vector space? is it $\Bbb{R}^3$? the tangent space at a point of the rigid body? ) or whether it is actually a tensor field over a certain manifold $M$ (if so which one?). The reason I ask this is because I'm aware that in the physics literature, it's not uncommon to leave out the term "field", because it's usually clear from context... but unfortunately it isn't clear to me $\ddot{\frown}$.
$3.$ Following up with ($2$), what is the type/rank of the inertia tensor(field?) (i.e what are $r$ and $s$)? My guess is that based purely on the way it is written, it is a $(0,2)$ tensor (field?) based on the index structure. But I'm not sure because this is only Volume $1$ of Landau and Lifshitz, and from my understanding, at this point they do not make any distinction between upper vs lower placement of indices. Another reason I ask this is because people usually identify $(2,0), (1,1), (0,2)$ tensor (fields) all together using an inner product/ Riemannian metric tensor field. So, I'm wondering, which "type" is the most natural to begin with.
$4.$ Is it possible to define the inertia tensor in a manner which manifestly makes it clear that it is actually a tensor (field?). For example, if we consider $M = \Bbb{R}^4$ as a smooth manifold, then with the identity chart $(\Bbb{R}^4, \text{id}_{\Bbb{R}^4})$, where we denote its four component functions as $\text{id}_{\Bbb{R}^4}(\cdot) = \left( t(\cdot), x^1(\cdot), x^2(\cdot), x^3(\cdot)\right)$, we can define $g := -dt \otimes dt + \delta_{ij}dx^i \otimes dx^j $. Written in this manner, although we have used the component functions of the identity chart, all the operations used (exterior derivative, tensor product) etc are all clearly chart-independent and purely geometric operations. So, the result is very clearly chart-independent and is indeed easily seen to be a (symmetric) $(0,2)$-tensor field on the manifold $M$. So, my question is if we can describe the inertia tensor (field?) in a similar terminology.
So, really, my issue is one of translating between terminology in the math books I'm familiar with and the physics texts which I also read simultaneously, and one about the geomtric way of defining such objects.