Nsryjdtyk

The following theorem is commonly attributed to Hadamard.

Assume $Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $Sigma$ is embedded and bounds a convex set.

Many authors refer to Hadamard's Sur certaines propriétés des trajectoires en Dynamique (1897)
(for example, J.J.Stoker in his Über die Gestalt der positiv... (1936)).

Likely the statement is there, but the paper is long, it is in French and often the statements are not clearly marked; I was searching for it for several days. I asked a friend and she said that it was there 20 years ago, but she could not find it; she also said that it was not easy to extract it from what is written ( = one has to think). [For sure the word immersion is not there.]

I hope someone here knows this paper and can help me.

I think the relevant location is item 23, page 352, but what Hadamard aims to is stated as follows:

A smooth, co-orientable surface of $mathbb{R}^3$ with Gauss curvature bounded below by some $kappa >0$ is simply connected. (implicitly, the surface is compact without boundary)

("Or une surface à deux côtés et sans points singuliers, à courbure partout positive (la valeur zéro et les valeurs infiniment petites étant exclues) est toujours simplement connexe.")

The goal is to use the Gauss-Bonnet Formula to deduce that when curvature is positive, any two closed geodesics must meet (otherwise they would together bound a total curvature 0 region of the surface).

What is not clear from the text of item 23 is whether the surface assumed to be immersed or embedded. He basically says that the normal map is a global diffeomorphism, because positive curvature makes it a covering of the sphere. It seems the argument does provide the statement attributed to this paper, although it seems not explicitly stated.