general relativity - 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.