Varieties with the property that the cotangent bundle restricted to a complete nonsingular curve is free Let $X$ be a $d$-dimensional smooth projective connected variety with cotangent sheaf $\Omega^1_X$ over $\mathbb C$. Suppose that for any nonsingular complete curve $C$ and non-constant morphism $\gamma:C\to X$, we have that $$\gamma^\ast \Omega^1_X = \mathcal O_C^{\oplus d}.$$ 
What can we say about $X$?
I know that if $X$ is an abelian variety, it has the above property.
Are there any other varieties which satisfy the above property? For instance, what if the determinant of $\Omega^1_X$ is trivial, i.e., $\omega_X = \mathcal O_X$. Do we have the above property? 
What if $X$ is a K3 surface? (This is a special case of the above question.)
 A: This is only a partial answer, but it is not true in general even for $K3$ surfaces.  Let $S$ be a smooth algebraic $K3$, containing a smooth rational curve $C \stackrel{\gamma}{\hookrightarrow} S$.  Then by adjunction, the normal bundle is $\mathcal{N}_{C|S} \cong \mathcal{O}_C(-2)$.  There is a short exact sequence
$$
0 \to \mathcal{N}_{C|S}^{\check{}} \to \gamma^*\Omega^1_S \to \Omega^1_C \to 0 ~,
$$
which becomes
$$
0 \to \mathcal{O}_C(2) \to \gamma^*\Omega^1_S \to \mathcal{O}_C(-2) \to 0 ~.
$$
Since $\mathcal{O}_C(2)$ doesn't inject into $\mathcal{O}_C^{\oplus 2}$, $\gamma^*\Omega^1_S$ must be non-trivial.

As suggested by Tom, the above can be generalised (and it can probably be generalised further by somebody who knows their stuff):
If $X$ is a smooth variety containing a smooth rational curve $\gamma : C \hookrightarrow X$, then $X$ does not have the property given in the OP.  To see this, start with the following exact sequence:
$$
 0 \to {N}_{C|X}^{\check{}} \to \gamma^*\Omega^1_X \to \Omega^1_C \to 0
$$
where $N_{C|X}^{\check{}}$ is the conormal bundle. Taking determinants, we find
$$
\gamma^* \omega_X \cong \omega_C\otimes\det (N_{C|X}^{\check{}}) \cong \mathcal{O}_C(-2)\otimes\det (N_{C|X}^{\check{}})
$$
So if $\gamma^*\Omega^1_X$ is trivial, we must have $\det (N_{C|X}^{\check{}}) \cong \mathcal{O}_C(2)$.  As $N_{C|X}^{\check{}}$ is isomorphic to a sum of line bundles, at least one of them must have positive degree.  But then it cannot inject into a trivial bundle, so we have a contradiction.
