While reviewing the proof of Gauss-Bonnet in John Lee's book, I noticed the following paragraph:

" ...In a certain sense, this might be considered a very satisfactory generalization of Guass-Bonnet. The only problem with this result is that the relationship between the Pfaffian and sectional curvature is obscure in higher dimensions, so no one seems to have any idea how to interpret the theorem geometrically! For example, it is not even known whether the assumption that $M$ has strictly positive sectional curvatures implies that $\chi(M)>0$.... (page 170) "

May I ask if this "interpretation problem" has been resolved? I felt much the same way when I read Milnor's proof of Chern-Gauss-Bonnet using chern classes, and Chern's statement in his own book using Lipschitz-Killing curvature. Neither has a geometric meaning that is "self-transparent" to me. When I had a class in the index theorem, our proof basically showed Gauss-Bonnet is a special case of Atiyah-Singer using an appropriate Dirac operator. And I do not recall that it involved much geometry, but instead a lot of algebraic manipulations.

Chern suggested the following way to look at it in his book: Consider the exterior $2n$-form $$ \Omega=(-1)^{n}\frac{1}{2^{2n}\pi^{n}n!}\delta^{i_1 \cdots i_{2n}}_{1\cdots 2n}\Omega_{i_1i_2}\cdots \Omega_{i_{2n-1}i_{2n}}, \Omega=K d\sigma $$ Then the "key" to prove Chern-Gauss-Bonnet is to represent $\Omega$ on the sphere bundle of $M$ so that one has $\Omega=d\prod$, where $\prod$ is a $2n-1$-form. However, I still do not know how this shed any light on the picturesque side of the equation so that I can visualize it. So I decided to ask. I suppose that this might be one of those topics well known to experts but not written down in introductory level textbooks.

Reference:

John M. Lee: Riemannian Manifolds, page 170

Chern: Lectures on Differential Geometry, page 171

Milnor&Stasheff: Characteristic Classes, appendix A?

For a definition of Pfaffian, see here from wikipedia.

While reviewing the proof of Gauss-Bonnet in John Lee's book, I noticed the following paragraph:

" ...In a certain sense, this might be considered a very satisfactory generalization of Gauss-Bonnet. The only problem with this result is that the relationship between the Pfaffian and sectional curvature is obscure in higher dimensions, so no one seems to have any idea how to interpret the theorem geometrically! For example, it is not even known whether the assumption that $M$ has strictly positive sectional curvatures implies that $\chi(M)>0$.... (page 170) "

May I ask if this "interpretation problem" has been resolved? I felt much the same way when I read Milnor's proof of Chern-Gauss-Bonnet using Chern classes, and Chern's statement in his own book using Lipschitz-Killing curvature. Neither has a geometric meaning that is "self-transparent" to me. When I had a class in the index theorem, our proof basically showed Gauss-Bonnet is a special case of Atiyah-Singer using an appropriate Dirac operator. And I do not recall that it involved much geometry, but instead a lot of algebraic manipulations.

Chern suggested the following way to look at it in his book: Consider the exterior $2n$-form $$ \Omega=(-1)^{n}\frac{1}{2^{2n}\pi^{n}n!}\delta^{i_1 \cdots i_{2n}}_{1\cdots 2n}\Omega_{i_1i_2}\cdots \Omega_{i_{2n-1}i_{2n}}, \Omega=K d\sigma $$ Then the "key" to prove Chern-Gauss-Bonnet is to represent $\Omega$ on the sphere bundle of $M$ so that one has $\Omega=d\prod$, where $\prod$ is a $2n-1$-form. However, I still do not know how this shed any light on the picturesque side of the equation so that I can visualize it. So I decided to ask. I suppose that this might be one of those topics well known to experts but not written down in introductory level textbooks.

Reference:

John M. Lee: Riemannian Manifolds, page 170

Chern: Lectures on Differential Geometry, page 171

Milnor & Stasheff: Characteristic Classes, appendix A?

For a definition of Pfaffian, see here from wikipedia.