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) $
