I am trying to re-learn differential geometry and I ran into a conceptual wall regarding Christoffel symbols. I think I understand their meaning in the context of a change of coordinates, but I fail to see their meaning wihtin the context of diffrentiable manifolds.
As far as I understand, if we have a change of coordinates, the variation of the basis vectors $v_k$ is itself a vector of the same space whose components are the Christoffel symbols when that vector is expressed in that same basis ( the $v_k$'s).
On the other side, if I have a differentiable manifold, I think I understood that the Christoffel symbols can be seen (in a similar fashion as before) as the components of the variation (i.e. of the derivative) of the basis vectors of the tangent space, expressed in those same basis vectors.
But how are we sure that the derivative of a tangent vector lives/stays on the tangent space? If it does not (and I think in general it does not) how can it be expressed as a combination of the basis vectors of the tangent plane?
To clarify my question: the Christoffel symbols are the component of what vector, in the context of a differentiable manifold?