We consider Riemannian manifolds \(M_i\) , \({i=0,1}\) , with boundary and \(\Phi _i\in C^{\infty }(M_i)\) non-negative such that \((M_i, \Phi _i)\) has 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 \(\Phi _0=\Phi _1\) on \(Y_0\simeq Y_1\) . We assume that \(\Pi _0+\Pi _1=:\Pi \ge 0\) (*), and \(d\Phi _0(\nu _0)+ d\Phi _1(\nu _1)\le {{\,\textrm{tr}\,}}\Pi \) on \(Y_0\simeq Y_1\) (**) where \(\Pi _i\) is the second fundamental form and \(\nu _i\) is inner unit normal field along \(\partial M_i\) . We show that the metric glued space \(M=M_0\cup _{\mathcal {I}}M_1\) together with the measure \(\Phi d\mathcal {H}^n\) satisfies the curvature-dimension condition CD(K, N) where \(\Phi : M\rightarrow [0,\infty )\) arises tautologically from \(\Phi _1\) and \(\Phi _2\) . Moreover, \((M, \Phi d\mathcal {H}^n)\) 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 case \(\Phi \equiv 1\) where a lower Ricci curvature bound and \(\dim _{M_i}\le N\) replaces a lower Bakry-Emery N-Ricci curvature bound.
