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This is a cross-post of this MSE post that users commented that it is appropriate for MO.

I want to know

Question: Is the 2-dimensional Gauss-Bonnet theorem applicable (any topological or geometrical obstruction) in higher dimensions?

My idea is that one can consider 2-dimensional embedded submanifolds of $(M^n,g)$ and then applying Gauss-Bonnet theorem to all of such submanifolds then collecting these information together somehow and finding a topological or geometrical property (like fundamental group, Homology groups, etc.). Is that possible at all?

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    $\begingroup$ Isn't the Chern-Gauss-Bonnet Theorem a higher dimensional result??? $\endgroup$
    – abx
    Commented May 26, 2020 at 12:06
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    $\begingroup$ I am not aware of any proof of Chern--Gauss--Bonnet, or any other theorem, that arises by simultaneous gluing of all surfaces inside the manifold. I think that nothing is known about that question. $\endgroup$
    – Ben McKay
    Commented May 26, 2020 at 12:29
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    $\begingroup$ I don't know why this question has been down-voted? What kind of details needed to clarify the problem and prevent from closing? Is this a vague question? $\endgroup$
    – C.F.G
    Commented May 26, 2020 at 15:13
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    $\begingroup$ I think it is down-voted for being too vague. In the present form, it is not answerable. As for how many details you need to add, I do not know, since I am not sure what the question really is. $\endgroup$
    – Moishe Kohan
    Commented May 26, 2020 at 18:33
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    $\begingroup$ I think the wording of the question is unclear. You're asking whether the 2-dimensional Gauss-Bonnet theorem is ever used to prove a theorem about higher dimensional manifolds. $\endgroup$
    – Deane Yang
    Commented May 26, 2020 at 21:38

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There are some applications of Gauß-Bonnet in foliations and laminations of 3-manifolds.

The first result that comes to mind is Candel’s Uniformization Theorem, which gives necessary and sufficient conditions for a lamination to admit a leafwise hyperbolic metric. The proof uses Gauß-Bonnet in a nontrivial way. (A shorter exposition is in Chapter 7 of Calegari: “Foliatuons and the Geometry of 3-manifolds”.)

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