Glued spaces and lower Ricci curvature bounds

We consider Riemannian manifolds $M_i$, $i=0,1$, with boundary and ${\mathrm{\Phi }}_i\in C^{\mathrm{\infty }}(M_i)$ non-negative such that the pair $(M_i,{\mathrm{\Phi }}_i)$ admits Bakry-Emery $N$-Ricci curvature bounded from below by $K$. Let $Y_0$ and $Y_1$ be isometric, compact components of the boundary of $M_0$ and $M_1$ respectively and assume ${\mathrm{\Phi }}_0={\mathrm{\Phi }}_1$ on $Y_0\simeq Y_1$. We assume that ${\mathrm{\Pi }}_0+{\mathrm{\Pi }}_1=\mathrm{\Pi }\ge 0$ (), and $d{\mathrm{\Phi }}_0({\nu }_0)+d{\mathrm{\Phi }}_1({\nu }_1)\le tr\mathrm{\Pi }$ on $Y_0\simeq Y_1$ (**) where ${\mathrm{\Pi }}_i$ is the second fundamental form and ${\nu }_i$ is inner unit normal field along $\mathrm{\partial }{M}_i$. We show that the metric glued space $M=M_0{\cup }_{\mathcal{I}}{M}_1$ together with the measure $\mathrm{\Phi }d{\mathcal{H}}^n$ satisfies the curvature-dimension condition $CD(K,\lceil N\rceil )$ where $\mathrm{\Phi }:M\to [0,\mathrm{\infty })$ arises tautologically from ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$. Moreover, $(M,\mathrm{\Phi }d{\mathcal{H}}^n)$ is the collapsed Gromov-Hausdorff limit of smooth, $\lceil N\rceil$-dimensional Riemannian manifolds with Ricci curvature bounded from below by $K-\epsilon$ and is also the measured Gromov-Hausdorff limit of smooth, weighted Riemannian manifolds such that the Bakry-Emery $\lceil N\rceil$-Ricci curvature is bounded from below by $K-\epsilon$. On the other hand we show that given a glued manifold as described it satisfies the curvature-dimension condition $CD(K,N)$ only if the condition () and (**) hold. The latter statement generalizes a theorem of Kosovskiĭ for sectional lower curvature bounds and especially applies for the unweighted case where a lower Ricci curvature bound and ${\dim }_{M_i}\le N$ replaces a lower Bakry-Emery $N$-Ricci curvature bound.