# Integral scalar curvature of the submanifold

Let $$(X,g)$$ be a closed Riemanian manifold, and $$Y\to X$$ be an embeded submanifold. We denote by $$N(Y)$$ the tubular neighborhood of $$Y$$ and $$Z=\partial N(Y)$$.
On $$TY$$ we can calculate the scalar curvature $$Scal^0$$ of $$X$$ by the induced metric $$g_0=g\big|_{Y}$$ of $$TX\big|_Y$$, we equip $$TX\big|_{N(Y)}$$ with this induced metric, i.e. $$g_{N(Y)}=\pi ^*g_0$$, where $$\pi :N(Y)\to Y.$$ We know that $$Z$$ has a collar neighborhood $$(-\epsilon,\epsilon)\times Z$$. We extend this collar by $$(-T,T)\times Z$$ equipped with the product metrict. Then for any $$T>0$$, we can find a metric $$g'$$ on $$X$$, such that over $$(-T+1,T-1)\times Z$$ the metric is the product metric, i.e. $$g'\big|_{(-T+1,T-1)\times Z}=dt^2+\pi^*g^0,$$ and $$g'\big|_{X\setminus(-T,T)\times Z}=g\big|_{X\setminus(-T,T)\times Z}.$$

Q Suppose $$\int_YScal^0dvol(g^0)=A$$, for any $$T>0$$, we have a manifold $$X(T)$$ with the metric $$g'$$. Can we have $$\int_X(T)Scal^{g'}dvol(g')= T\cdot A+C,$$ where $$C=\int_{X\setminus N(Y)}Scal^gdvol(g)$$