A rigidity condition for the boundary of a submanifold in a Riemannian manifold (Q954186)

If \((X,g)\) is an \(n\)-dimensional smooth connected Riemannian manifold without boundary and \(Y_{1},Y_{2}\) are its \(n\)-dimensional compact connected \(C^{0}-\)submanifolds with non empty boundaries \(\partial Y_{1},\partial Y_{2},\) the author considers the case when on each \(Y_{i},\, i\in\{1,2\}\) it exists a metric \(g_{Y_{i}}\) and any two points of \(Y_{i}\) can be joined in this submanifold by a shortest path with respect to the metric \(g_{Y_{i}}.\) In these hypothesis, the author proves that if \(Y_{1}\) is strictly convex with respect to \(g_{Y_{1}}\) and \((\partial Y_{1},g_{Y_{1}}),\,(\partial Y_{2},g_{Y_{2}})\) are isometric, then \(Y_{2}\) is also strictly convex with respect to \(g_{Y_{2}}\). The case \(n=2\) is treated separately.