Shortest paths on convex hypersurfaces of Riemannian spaces
Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part II, Tome 66 (1976), pp. 114-132
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A convex hypersurface $\mathscr F$ in a Riemannian space $M^m$ is part of the boundary of an $m$-dimensional locally convex set. It is established that there exists an intrinsic metric of such a hypersurface $\mathscr F$ and it has curvature which is bounded below in the sense of A. D. Aleksandrov; curves with bounded variation of rotation in $\mathscr F$ are shortest paths in $M^m$. For surfaces in $R^m$ these facts are well known; however, the constructions leading to them are in large part inapplicable to spaces $M^m$. Hence approximations to $\mathscr F$ by smooth equidistant (not necessarily convex) ones and normal polygonal paths, introduced (in the case of $R^3$) by Yu. F. Borisov are used.
@article{ZNSL_1976_66_a2,
author = {S. V. Buyalo},
title = {Shortest paths on convex hypersurfaces of {Riemannian} spaces},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {114--132},
publisher = {mathdoc},
volume = {66},
year = {1976},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1976_66_a2/}
}
S. V. Buyalo. Shortest paths on convex hypersurfaces of Riemannian spaces. Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part II, Tome 66 (1976), pp. 114-132. http://geodesic.mathdoc.fr/item/ZNSL_1976_66_a2/