A $n-1$ dimensional submanifold $N\subset \mathbb{R}^n$ is called a convex submanifold if for every $x\in N$ ,ther is a neighborhood $W$ of $x$ in $N$ such that $W$ entirly lies at one side of $T\_x N$. A (local) diffeomorphism $\phi$ on $\mathbb{R}^n$ is called a convex diffeomorphism if $\phi$ and its inverse preserves the convexity of all codimension $1$ submanifolds. For example every affine linear isomorphism is a convex diffeomorphism but the diffeomorphism $\phi(x,y)=(x,y-x^2+x^3)$ is not a convex diffeomorphism because it maps the convex curve $y=x^2$ to non convex curve $y=x^3$.



> 
> 1.In the above definition, is the word "its inverse", redundant?
> 
> 
> 2. Is there a well known description of the group of all smooth convex diffeomorphisms of $\mathbb{R}^n$?
> 
> 
> 3.Does every manifold admit an atlas whose all transition maps are convex diffeomorphism?
> 
> 
> 


If the answer to the later question is affirmative, then we can define the concept of convexity for any codimension $1$ submanifold of an arbitrary manifold $M$. In particular we can speak of "convex foliation" of a manifold $M$, a codimension $1$ foliation of $M$ whose all leaves are convex submanifold.
In this case the next question would be the following:



> 
> 4.Is there a manifold which admit a codimension $1$ foliation but does not admit any convex foliation?In particular does $S^3$ admit a convex foliation?
> 
> 
> 



The concept of convex foliation
If $n>1$, and a smooth diffeomorphism $f:U\to V$ (where $U$ and $V$ are, say, convex, open subsets of $\mathbb{R}^n$) carries convex sets to convex sets, then it is easy to show that it must carry each intersection $U\cap H$, where $H\subset\mathbb{R}^n$ is a hyperplane, to an intersection $V\cap H'$, where $H'\subset\mathbb{R}^n$ is another hyperplane. Hence $f$ must carry intersections of hyperplanes to intersections of hyperplanes, and hence line segments to line segements. In particular $f$ must be the restriction of a projective transformation of $\mathbb{RP}^n$ to $U\subset\mathbb{R^n}\subset\mathbb{RP}^n$. Conversely any such restriction carries convex sets to convex sets, as does its inverse. This answers Questions 1 and 2.


For Question 3, you are asking whether every smooth $n$-manifold $M$ admits a flat projective structure. The answer is no, for then the associated developing map on the simply connected cover would have a smooth immersion into $S^n$. However, already this is impossible for $n=4$ if one starts with a simply-connected compact $4$-manifold other than the $4$-sphere.


However, note that $n=2$ does work, since, by uniformization, every two-dimensional surface carries a metric of constant Gauss curvature, and the underlying projective structure associated to such a metric is projectively flat.


Finally, $S^3$, which does admit a codimension $1$ foliation and does admit a flat projective structure (as the double cover of $\mathbb{RP}^3$), does not admit a convex foliation. The reason is that the flat projective structure is unique (since $S^3$ is simply-connected) and at least one of the leaves of the foliation would have to be a compact torus because every codimension $1$ foliation of the $3$-sphere must have a Reeb component. However, it is not possible to immerse the standard torus in the standard $S^3$ smoothly as a convex surface (with respect to the projective structure): Endow $S^3$ with its standard 'round' metric of constant sectional curvature $+1$; its underlying projective structure is the flat one. For any torus $T^2\subset S^3$, the Gauss curvature of the induced metric must vanish somewhere (since the integral is zero), and, at such a point, the product of the principal curvatures must be $-1$, by the Gauss equation. In particular, the two principal curvatures are nonzero, with opposite signs, and the torus is not convex on a neighborhood of such a point. Thus, this gives an example of the kind asked for in Question 4.


