Why can't Newton's 1st law be expressed as an autoparallel transportation in space? I'm following this series of lectures on differential geometry and general relativity. In the linked lecture (Lecture 9), at around 24:24, professor Frederic Schuller made the conclusion that one can not express Newton's 1st law as an autoparallel transportation in space but can in spacetime, i.e there exists no $\Gamma$ such that the following equation is valid:
$${-g^{\alpha}[x(t)]}~=~{{\Gamma}^{\alpha}_{{\beta}{\gamma}}[x(t)]{\dot{x}}^{\beta}(t){\dot{x}}^{\gamma}(t)}, \qquad \alpha=1,2,3.\tag{1}$$
Could someone explain to me why this is the case? If you could provide an intuitive picture it'll be even better.  
 A: In the question it is mentioned Newton's 1st law, however in lecture 9 the arguing is about Newton's 2nd law. The structure of this latter does not allow for the parallel transport description. Reason being that gravity in general depends on the point, so a parallel transport reading of Newton is valid only in a very limited region of spacetime. In the lecture an example is given of an object falling at the North pole compared to an object falling at the South pole; no way to find a coordinate system to describe both as a parallel transport.
A different case is Newton's 1st law when there is no gravity, i.e. the Newtonian force is zero. In that instance the $\Gamma's$ are zero as well and the parallel transport reduces to the straight line equation in a Euclidean space.
A: If you know GR this question may irks you, since a relativistic point particle in a gravitational field do in fact follow geodesics (which is a special type of autoparallels) in spacetime. 
But the devil is in the details: Prof. Schuller is talking about autoparallels in space, not in spacetime. And he makes the case that the gravitational acceleration $\vec{g}$ (which depends on position, not velocity) cannot be emulated by a term quadratic in velocity, as needed in the autoparallel equation, cf. OP's eq. (1).
Later at 36:06 in the same lecture 9 Prof. Schuller considers the same question in spacetime (as opposed to space), and shows that a point particle in a gravitational field follows an autoparallel in spacetime: In the static gauge $x^0=t$ of spacetime the gravitational acceleration $g^{\alpha}$ can be reproduced the new $\Gamma^{\alpha}_{00}\dot{x}^0 \dot{x}^0$ sector. 
