Rigidity Conditions for the Boundaries of Submanifolds in a Riemannian Manifold

Abstract

Developing A.D. Aleksandrov's ideas, the first-named author of this article proposed the following approach to study of rigidity problems for the boundary of a $C^0$-submanifold in a smooth Riemannian manifold: Let $Y_1$ be a 2-dimensional compact connected $C^0$-submanifold with nonempty boundary in a 2-dimensional smooth connected Riemannian manifold $(X,g)$ without boundary satisfying the condition $${\rho }_{Y_1}(x,y)=\underset{x^{\prime }\to x,y^{\prime }\to y,x^{\prime },y^{\prime }\in \mathrm{Int}{Y}_1}{\mathrm{liminf}}\{[l({\gamma }_{x^{\prime },y^{\prime },\mathrm{Int}{Y}_1})]\}<\infty ,$$ if $x,y\in Y_1$. Here $\inf [l({\gamma }_{x^{\prime },y^{\prime },\mathrm{Int}{Y}_1})]$ is the infimum of the length of smooth paths joining $x^{\prime }$ and $y^{\prime }$ in the interior $\mathrm{Int}{Y}_1$ of $Y_1$. In the present paper, we first establish that ${\rho }_{Y_1}$ is a metric on $Y_1$. Suppose further that $Y_1$ is strictly convex in the metric ${\rho }_{Y_1}$. Consider another 2-dimensional compact connected $C^0$-submanifold $Y_2$ of $X$ with boundary satisfying the condition ${\rho }_{Y_2}(x,y)<\infty$, $x,y\in Y_2$, and assume that $\partial Y_1$ and $\partial Y_2$ are isometric in the metrics ${\rho }_{Y_j}$, $j=1,2$. There appears the following natural question: Under which additional conditions are the boundaries $\partial Y_1$ and $\partial Y_2$ of $Y_1$ and $Y_2$ isometric in the metric ${\rho }_X$ of the ambient manifold $X$? The paper is devoted to the detailed discussions of this question. In it, we in particular obtain a number of new results concerning the rigidity problems for the boundaries of $C^0$-submanifolds in a Riemannian manifold. The case of $\dim Y_j=\dim X=n$, $n>2$, is also considered.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1305.6169