Allendoerfer and Weil's generalization of Gauss-Bonnet Theorem

Question

In Peter Petersen's Riemannian Geometry (reference, p. 98), he says that

The theorem is now called the Chern-Gauss-Bonnet Theorem despite the fact that Allendoerfer and Weil were the first to prove it in complete generality in higher dimensions.

But on Wikepedia (reference), it is said that

In 1943, Carl B. Allendoerfer and André Weil proved a special case for extrinsic manifolds. In a classic paper published in 1944, Shiing-Shen Chern proved the theorem in full generality connecting global topology with local geometry.

Who was really the first to prove the "full generality"? I don't have enough knowledge of differential geometry to judge it myself, but I want to know the history.

Answer 1

Every Riemannian manifold $(X,g)$ can be isometrically embedded into $\mathbb{R}^N$ for $N$ sufficiently large. So in principle, when one wants to prove something about Riemannian manifolds, it suffices to consider isometrically embedded submanifolds $X\subset\mathbb{R}^N$. However, this is not the most natural way to study the geometric object $(X,g)$, because it requires a choice of embedding into $\mathbb{R}^N$. Hence, this is called "extrinsic". The text states that Carl B. Allendoerfer and André Weil proved the theorem in full generality for embedded submanifolds $X\subset\mathbb{R}^N$, and thus for every Riemannian manifold - but not in the most natural way, because it requires a choice of embedding.

There is also a way to prove the same theorem without making reference to any embedding into Euclidean space. This is called "intrinsic", and the text states the Chern was the first to prove the theorem in full generality without making reference to such an embedding. Doing this does not prove anything new, because every Riemannian manifold can already be viewed as a submanifold of some $\mathbb{R}^N$. This is why the text says "despite the fact that Allendoerfer and Weil were the first to prove it in complete generality in higher dimensions." - Chern wasn't the first to prove the theorem, but he gave a more natural proof, which is why the result now bears his name.

Answer 2

1. As Arctic Char on strike stated, Allendoerfer and Weil proved in 1943 the theorem in full generality, in the contrary to what is written in Wikipedia. They were the first to do that, to the best of my knowledge.

2. Their proof used a result on existence of isometric imbeddings of Riemannian manifolds into Euclidean space available at the time (which is weaker than the Nash theorem).

3. In 1944 Chern gave another proof which did not use any imbeddings into Euclidean space. However Chern's proof worked only for closed manifolds, while Allendoerfer and Weil proof worked in greater generality: for manifolds with boundary and corners.