Confusion about lift in the context of tangent bundle A lift is defined here: https://mathworld.wolfram.com/Lift.html as a tangent vector field $X$ on a manifold, the same way that a section of the tangent bundle gives us $X$ in the context $\dot g = X(g)$. This answer to a different question: https://mathoverflow.net/a/111198/172470 suggests that a path $g$ can be lifted into the tangent bundle space as $\tilde g \in T_x M$, and is a smooth curve there. I am confused about this, because of two reasons:

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*Isn't $T_x M$ the set of vectors tangent to a specific point $x$ (the "tangent space" at $x$)? How can we have a curve there, especially a closed loop, like the cited answer suggests?

*If we can, how is this lift a vector field?

Thanks for any help you can give, including any clarification of definitions I may be wrong about. Pictures are nice too since english isn't my second language, and I've already gone down the wrong road with some ideas in differential geometry because I had formed the wrong picture. Thank you!
 A: A lift of an injective path $\gamma:I\to X$ is as you suggest a map $\widetilde{\gamma}:I\to TX$ so that $\pi:TX\to X$ satisfies $\pi\circ \widetilde{\gamma}=\gamma$. Actually, such a lifting is equivalent to specifying a tangent vector along each point of the curve $\gamma$, so you can think of this as a section of the bundle $TX$ restricted along $\gamma$ (aka $\gamma^*TX$). Anyway, the point is that such a lift is the smooth choice of a tangent vector at each point to the curve, i.e. a vector field along a curve like one sees in multivariable calculus.
Now, if we have the additional structure of an affine connection $\nabla$ on $TX$, then we get the definition of which vector fields are parallel along $\gamma$, meaning they satisfy a differential equation $\nabla_{\gamma'}X=0$, where $\gamma'$ is the trajectory vector field of $\gamma$. If we start with a loop $\gamma:I\to X$, (such that $\gamma(0)=\gamma(1)$ and let's say without other self-intersection) then we might ask when a lifting $\widetilde{\gamma}:S^1\to TX$ is a loop in the tangent bundle. In particular, this occurs exactly when we can define a vector field on the image of $\gamma$ by $X_{\gamma(t)}=\widetilde{\gamma}(t)$.
On the other hand, if we fix a loop $\gamma:I\to X$ and choose a tangent vector $X_{\gamma(0)}\in T_{\gamma(0)}X$, parallel transport allows us to define $\widetilde{\gamma}(t)$ to be the unique parallel lift of $\gamma$ with initial condition $\widetilde{\gamma}(0)=X_{\gamma(0)}$. Very often, $\widetilde{\gamma}(1)\ne \widetilde{\gamma}(0)$ and as a consequence we do not get a well-defined vector field along the image of $\gamma$. We can however define a linear transformation of $T_{\gamma(0)}X$ by sending $X_{\gamma(0)}\mapsto \widetilde{\gamma}(1)$, where $\widetilde{\gamma}$ is the lifting with initial condition $X_{\gamma(0)}=\widetilde{\gamma}(0)$. In particular, a loop at $x$ defines a map $P_\gamma\in \operatorname{GL}(T_xX)$ by the above procedure for $x=\gamma(0)$. This leads to the notion of the holonomy group of $X$ at $x$:
$$\operatorname{Hol}_x=\{P_\gamma:\gamma\:\text{is a loop based at}\:x\}.$$
If $X$ is a Riemannian manifold with metric $g$ and Levi-Civita connection $\nabla$ then the holonomy interacts with the curvature. Intuitively speaking, nonzero holonomy detects curvature. The idea being that curvature causes parallel transport of curves to deviate from the identity on $T_xX$. The following picture from Wikipedia illustrates exactly the idea:

If we start at the north pole, and move due south to the equator in any direction, then due west along the equator and then due north back to the north pole, the nontrivial curvature in the region bounded by our path of travel causes the parallel-translated vector from the north pole to rotate.
As a last comment, there is an interaction with representation theory when the connection $\nabla$ is flat. In this case, the transformation depends only on the homotopy class of the loop $\gamma$ based at $x$. Hence, we get a group homomorphism $\pi_1(X,x)\to \operatorname{GL}(T_xX)$ called the monodromy representation of $\pi_1(X,x)$, and defined by $[\gamma]\mapsto P_{\gamma}$.
