There is a known formula for Euler characteristic in terms of Ricci scalar: \begin{equation} \chi(M)=\frac{1}{4\pi} \int_M \sqrt{g} \,R\,d^2x\,. \end{equation} I am sure that this formula holds for two dimensional manifolds, but what about higher dimensional ones? What integral can define Euler characteristic in higher dimensions?
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You want the Chern-Gauss-Bonnet theorem. See also Chern-Weil theory.
Qiaochu Yuan
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Links may be useful but explain something as a hint to OP. Can you please add some more details? – Bhaskara-III Nov 05 '15 at 20:03
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@Bhaskara: the details are all in the links. On a closed oriented Riemannian manifold of dimension $2n$ there is a particular $2n$-form you can write down, which is a fairly explicit function of the curvature $2$-form, whose integral gives the Euler characteristic. There's even an example given when $n = 2$. – Qiaochu Yuan Nov 05 '15 at 20:07