Steps to generalize the concept of tensors from Euclidean to non-Euclidean spaces In the book by Pavel Grinfeld Introduction to Tensor Analysis and the Calculus of Moving Surfaces the "tensor property" is introduced on the grounds of defining the covariant basis $\left\{\vec Z_i\right\}$ as the partial derivatives of the position vector $\vec R$ (invariant) with respect to the different coordinates $Z^i$. The book makes perfect note that this only works in Euclidean spaces, where the concept of $\vec R$ as a straight arrow makes sense.
From that point he moves on to show how these covariant basis vectors transform under a change of coordinates, and how the components do. The Jacobian defines the tensor property.
The question I have, and that I can't find explained in the book, is what are the steps (just the sketch - not a full mathematical exposition, which would be out of the scope of a question) that are needed to correct this "original sin" of constructing the entire edifice of tensors on something (position vector) that is unique to Euclidean spaces is corrected to move towards intrinsic geometry and non-embedded manifolds.
 A: The first step in generalizing vectors and tensors on abstract manifolds is to re-frame what we mean when we talk about tangent vectors.
For concreteness, let $M$ be a smooth manifold of dimension $m$. Abstract manifolds are not subspaces of $\mathbb{R}^n$, so we have no way of taking partial derivatives of the position vector, however, abstract manifolds come with charts.  These are homeomorphisms $\phi: U\to \phi(U)\subset \mathbb{R}^m$ where $U\subset M$ and $\phi(U)\subset \mathbb{R}^m$ are open, with the property that for each pair of charts $(U,\phi)$, $(V,\psi)$, the so-called transition maps $(\psi\circ \phi^{-1}): \phi(U\cap V)\to \psi(U\cap V)$ are smooth.
In extrinsic geometry, we define the tangent vectors at $p$ to be the derivative of smooth curves passing through $p$, i.e. if $\gamma: I\to M\subset \mathbb{R}^n$ is a smooth curve with $\gamma(0)=p$ we define the tangent vector to $\gamma$ at $p$ to be $\frac{d}{dt}\vert_{t=0}\gamma(t)\in T_pM$.
For abstract manifolds, we define tangent vectors in a similar way but using the charts to take derivatives of curves. Let $C_p$ be the the set of smooth curves $\gamma: I\to M$ with $\gamma(0)=p$. We define two curves $\gamma,\rho\in C_p$ to be equivalent if there is a chart $(\psi, U)$ with $p\in U$ such that $$\left.\frac{d}{dt}\right|_{t=0} \psi(\gamma(t))=\left.\frac{d}{dt}\right|_{t=0}\psi(\rho(t))$$
One must verify that this relation gives an equivalence relation on $C_p$. After doing so, we then define $T_pM:=C_p/\sim$.
Of course,  a priori, $T_pM$ does not have a linear structure but one can again use the charts to give it one.  One can show that for each chart $(U,\psi)$ with $p\in U$, the map $d_p\psi: T_pM\to \mathbb{R}^m$ defined by $$[\gamma]\mapsto \left.\frac{d}{dt}\right|_{t=0} \psi(\gamma(t)) $$ is well defined and a bijection. At this point we can give $T_pM$ a vector space structure by using this bijection, but this procedure required a choice of chart. One must then prove that the linear structure does not depend on the choice of chart.
Since we gave $T_pM$  a vector space structure by way of the bijection $d_p\psi$, we automatically know that $\{(d_p\psi)^{-1}(e_i)\}$ forms a basis of $T_pM$, but we would like to relate this basis to the basis induced by a chart $(\phi,V)$. we have $$(d_p\phi)^{-1}(e_i)= ((d_p\psi)^{-1}d_p\psi)(d_p\phi)^{-1}(e_i)=(d_p\psi^{-1})(d_p\psi (d_p\phi)^{-1})(e_i)$$ It turns out that $d_p\psi d_p\phi^{-1}=D(\psi\circ \phi^{-1})(p)$ is the Jacobian matrix of the chart transitions. This is what generalizes the ideas of change of coordinates from extrinsic geometry.
From here you can go about defining tensors and whatnot analogously to the extrinsic case.
