= Solution
The <first fundamental form> of an embedded surface $S\subseteq\mathbb R^3$ is the inner product induced on each tangent plane:
$$
I_p(u,v)=u\cdot v,
\qquad u,v\in T_pS.
$$
For a tangent vector field $V$ along a curve $\alpha$, its <surface covariant derivative> is
$$
\frac{DV}{dt}=\left(\frac{dV}{dt}\right)^\top,
$$
the tangential projection of its ordinary derivative. A curve is a <geodesic> when
$$
\frac{D\dot\alpha}{dt}=0.
$$
A <local isometry> preserves the first fundamental form, so its Christoffel symbols and Levi--Civita covariant derivatives satisfy
$$
\nabla^R_{(\phi\circ\alpha)'}(D\phi V)
=D\phi(\nabla^S_{\alpha'}V).
$$
Taking $V=\alpha'$ shows that if $\alpha$ is geodesic, then $\phi\circ\alpha$ is geodesic.
The converse is false. The dilation
$$
\phi:\mathbb R^2\to\mathbb R^2,
\qquad \phi(x)=2x,
$$
maps every affinely parametrized straight-line geodesic to another such geodesic, but
$$
|D\phi(v)|=2|v|,
$$
so it does not preserve the first fundamental form. This is the <geodesic-preserving homothety that is not a local isometry>.
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