When does the conormal bundle sequence split? 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.
 A: 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.
A: 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 $\mathcal L$.  
