How to get from a tangent space to the underlying manifold? In a Lie group like $SO(3)$, it is possible to create a tangent space $\mathcal{T}_pSO(3)$ at a point $p \in SO(3)$. The tangent space has its basis vectors, which span a local linear coordinate system.
Using the exponential map $\text{exp}(\cdot)\colon \mathcal{T}_pSO(3) \to SO(3)$, I can map an arbitrary point $q \in \mathcal{T}_pSO(3)$ from the linear tangent space onto the (nonlinear) surface of the Lie group's manifold $SO(3)$. (I imagine this operation as "wrapping" the tangent hyperplane onto the surface of the $SO(3)$ manifold.)
The question:
Suppose there is a surface given by the constraint $\mathcal{S}\colon x^2 + 3y^2 - z = 0$. Moreover, let there be a point $r \in \mathcal{S}$ on that surface and let us construct a tangent plane $\mathcal{T_rS}$ at the point $r$.
Is it possible to construct a similar $\text{exp}(\cdot)$ operation for the points on the tangent plane $\mathcal{T_rS}$? In other words: is it possible to come up with a similar function $\text{exp}(\cdot)\colon \mathcal{T_rS} \to S$? How would I proceed with deriving such operation?

Yes, you can always construct an exponential map. Given a complete Riemannian manifold $M$, a point $r$ on the manifold and a tangent vector $\xi \in T_r M$, there is a unique geodesic $\gamma_\xi$ in $M$, starting at $r$, with initial derivative $\xi$. The exponential map $\mathrm{exp}:T_r M \to M$ can be defined as $\mathrm{exp}(\xi):=\gamma_\xi(1)$ (i.e. travel along this geodesic for time 1 at speed $|\xi|$, or travel along a unit-speed geodesic in the same direction for time $|\xi|$). This doesn't really have much to do with the exponential map, except in the special case of Lie groups (e.g. the group of invertible real $n\times n$ matrices).
