Totally-geodesic manifold
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
totally-geodesic submanifold
A submanifold $M^n$ of a Riemannian space $V^N$ such that the geodesic lines (cf. Geodesic line) of $M^n$ are also geodesic lines in $V^N$. A totally-geodesic submanifold is characterized by the fact that for every normal vector of $M^n$ the corresponding second fundamental form vanishes; this is equivalent to the vanishing of all normal curvatures of $M^n$.
Comments
The existence of totally-geodesic submanifolds in a general Riemannian manifold is exceptional. Conversely, the existence of many such totally-geodesic submanifolds is used in various recent work to characterize some special manifolds, e.g. symmetric spaces. See [a1].
References
[a1] | W. Ballmann, M. Gromov, V. Schroeder, "Manifolds of non-positive curvature" , Birkhäuser (1985) |
[a2] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963–1969) |
How to Cite This Entry:
Totally-geodesic manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Totally-geodesic_manifold&oldid=32412
Totally-geodesic manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Totally-geodesic_manifold&oldid=32412
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article