I would like to understand whether the following multidimensional (partial) generalization of the A.D. Alexandrov gluing theorem is true and, if yes, whether there is a reference.
(The original Alexandrov's gluing theorem was proven in dimension 2 only but under much weaker assumptions on the regularity of the metrics. It is actually much deeper than the situation I consider right now.) 


Let $M\_1, M\_2$ be two smooth Riemannian manifolds with boundary with sectional curvatures at least $\kappa$. 
Let $f\colon \partial M\_1\tilde\to \partial M\_2$ be a Riemannian isometry between their boundaries. Define a new smooth manifold without boundary $M=M\_1\cup\_f M\_2$ by identifying the boundaries of $M\_1$ and $M\_2$ via $f$. 


One defines on $M$ an intrinsic metric: for two points $x,y\in M$ the distance between them is equal to the infimum of the lengths of all paths connecting them when the length of the parts of the path contained in $M\_1$ (resp. $M\_2$) is computed with respect to the Riemannian metric of $M\_1$ (resp. $M\_2$).


Remark. It is not hard to see that $M$ has a structure of $C^\infty$-smooth manifold, and the above metric is induced by a $C^0$-regular Riemannian metric. 


**Question.** Whether the following is known to be true:
 $M$ has a bounded below curvature in the sense of Alexandrov if and only if the sum of the second fundamental forms of the boundaries of $M\_1$ and $M\_2$ (after the identification via $f$) is non-negative. In that case the curvature of $M$ is at least the above $\kappa$.


A reference would be helpful.



Multidimensional gluing theorem for Riemannian manifolds
Just learned that the answer is positive, at least its main part saying that if the sum of second fundamental forms is non-negative then the curvature of $M$ is at least $\kappa$. The answer is published here:
N. N. Kosovski˘ı, “Gluing of Riemannian manifolds of curvature ≥ κ”, Algebra i
Analiz 14:3 (2002), 140–157.


