Difference between revisions of "Totally-geodesic manifold"
From Encyclopedia of Mathematics
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''totally-geodesic submanifold'' | ''totally-geodesic submanifold'' | ||
| − | A submanifold | + | A submanifold $M^n$ of a [[Riemannian space|Riemannian space]] $V^N$ such that the geodesic lines (cf. [[Geodesic line|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|second fundamental form]] vanishes; this is equivalent to the vanishing of all normal curvatures of $M^n$. |
Latest revision as of 16:07, 10 July 2014
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