The Gauss–Bonnet theorem states for any compact surface (S,g) that the quantity Q^{S}_{GB}(S)=\int_{S} \operatorname{Sc}(S,s)\,\mathrm{d}s+\int_{\partial S}\mathrm{mean.curv.}(\partial S,b)\,\mathrm{d}b-4\pi\chi(S) vanishes identically. Let (X,g) be a compact Riemannian manifold of dimension n\geq 3 with smooth boundary, associated with a continuous map {f=(f_1,\ldots,f_{n-2}) \colon X\to [0,1]^{n-2}} , where \operatorname{Lip}f_{i}\leq d_{i}^{-1} for positive constants d_{i} . For a universal constant C_{n}(d_{i}) depending only on d_{i} and n , we show that there is a compact surface \Sigma homologous to the f -pullback of a generic point such that each component S of \Sigma satisfies Q_{GB}^{X}(S)\leq C_{n}(d_{i})\cdot\operatorname{area}(S) , where Q^{X}_{GB}(S)=\int_{S} \operatorname{Sc}(X,s)\,\mathrm{d}s+\int_{\partial S}\mathrm{mean.curv.}(\partial X,b)\,\mathrm{d}b-4\pi\chi(S). As corollaries, if X has “large positive” scalar curvature, we prove in a variety of cases that if X “spreads” in (n-2) directions “ distance-wise ”, then it cannot much “spread” in the remaining 2-directions “ area-wise ”.