I have a lot of confusion understanding manifolds, charts, tangent and cotangent vector spaces…
Please let me know where I am wrong.
Let’s take a flat minkosky space in 4 dimensions and call it $M$, and take a standard orthonormal basis $\{e_\mu\}$.
I can think of this space as a flat manifold, so I can build tangent and cotangent spaces for every point $p$ on top of it. Because of it being flat I can identify the tangent space (which would have as standard basis $\partial_\mu$) as the space M itself (so $\partial_\mu$ and $e_\mu$ coincide). (How can I prove this?)
So now I have the components of a point on M as coordinates $x=x^\mu e_\mu$ and components of a vector on $T_pM$ as $V=V^\mu e_\mu$ with the same basis $e^\mu$. Because I can identify $T_pM$ with $M$ for every $p$ therefore under a basis transformation: $$x^i=\lambda^i_j x^j$$ also the components of the tangent spaces will change in the same way: $$V^i=\lambda^i_j V^j$$ (Couldn’t I get the same result by stating that every vector in the tangent space transforms with the Jacobian of the transformation?)
In the end I have two different objects (vector on $M$ and vector in $T_pM$) that behave in the same way under a transformation. Is this correct?
Thank you for your help and sorry for my confusion.