Let $X\subset \mathbb{P}^n$ be a smooth projective variety with ideal sheaf $I_X$. The conormal sequence is given by
$$ 0\to I_X/I_X^2\to \Omega_{\mathbb{P}^n}|_X\to \Omega_{X}\to 0. $$ For which varieties $X$ is the sequence above split?
If I'm not mistaken, if $X$ a hypersurface, the sequence is split if and only if $X$ has degree 1.
You are right. This is a result due to Van de Ven.
[A. Van de Ven, A property of algebraic varieties in complex projective spaces. In: Colloque Géom. Diff. Globale (Bruxelles, 1958), 151–152, Centre Belge Rech. Math., Louvain 1959. MR0116361 (22 #7149) Zbl 0092.14004]
Even more is true. Recently Ionescu and Repetto proved the following generalization of Van de Ven's Theorem.
Let $X \subset \mathbb P^n$ be a smooth subvariety. If there exists a curve $C \subset X$ such that the restriction to $C$ of the conormal sequence of $X$ splits then $X$ is linear.
Let me sketch a short elementary proof (of Van de Ven's result not its generalization) in the case of hypersurfaces. I will phrase it in the analytic category but once it is translated to the algebraic category, working with infinitesimal neighborhoods, I believe that what will emerge is one of the proofs in the literature.
If the normal sequence splits then we can define a foliation $\mathcal L$ by (germs of) lines everywhere transverse to $X$ at a neighborhood $U$ of $X$. Since the complement of $X$ is Stein we can extend $\mathcal L$ to the whole $\mathbb P^n$. Therefore $\mathcal L$ is defined by a global section of $T \mathbb P^n(d-1)$ for some $d \ge 0$. With the help of Euler's sequence, this section can be presented as a homogeneous vector field $v$ on $\mathbb C^{n+1}$ with coefficients of degree $d$. To compute the tangencies between $\mathcal L$ and $X$ we have just to contract the differential $dF$ of a defining equation $F$ of $X$ with $v$. If $F$ is not linear then the divisor on $X$ defined by the tangencies between $\mathcal L$ and $X$ (defined by $F=dF(v)=0$) will be non-empty contradicting the transversality between $X$ and $\mathcal L$.
Here's a partial answer, which confirms your last sentence at least.
Lemma: If $X\subset \mathbb{P}^n$ is smooth and the conormal sequence splits, then $X$ is a projective space.
Proof: This implies that the tangent bundle $T_{\mathbb P^n}|_X$ surjects onto $T_X$. Therefore $T_X$ is ample. Now apply Mori's solution to the Hartshorne conjecture.
