This is a cross-post of [this](https://math.stackexchange.com/q/3691891/272127) MSE post that users commented that it is appropriate for MO.



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> > **Question:** Is the 2-dimensional Gauss-Bonnet theorem applicable (any topological or geometrical obstruction) in higher dimensions?
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> 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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Is the 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions?
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](http://www.numdam.org/article/ASENS_1993_4_26_4_489_0.pdf), 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”.)


