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Totally-geodesic manifold

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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://www.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