Difference between revisions of "Geodesic manifold"
From Encyclopedia of Mathematics
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| − | ''at a point | + | {{TEX|done}} |
| + | ''at a point $x$'' | ||
| − | A submanifold | + | A submanifold $M^k$ of a smooth manifold $M^n$ (Riemannian or with an affine connection) such that the geodesic lines (cf. [[Geodesic line|Geodesic line]]) of $M^n$ that are tangent to $M^k$ at $x$ have a contact of at least the second order with $M^k$. This requirement is fulfilled at all points if any geodesic in $M^k$ is also a geodesic in $M^n$. Such geodesic manifolds $M^k$ are called totally geodesic manifolds. |
Latest revision as of 13:57, 29 April 2014
at a point $x$
A submanifold $M^k$ of a smooth manifold $M^n$ (Riemannian or with an affine connection) such that the geodesic lines (cf. Geodesic line) of $M^n$ that are tangent to $M^k$ at $x$ have a contact of at least the second order with $M^k$. This requirement is fulfilled at all points if any geodesic in $M^k$ is also a geodesic in $M^n$. Such geodesic manifolds $M^k$ are called totally geodesic manifolds.
Comments
Also called geodesic submanifold and totally geodesic submanifold, respectively.
References
| [a1] | W. Klingenberg, "Riemannian geometry" , Springer (1982) (Translated from German) |
How to Cite This Entry:
Geodesic manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_manifold&oldid=16613
Geodesic manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_manifold&oldid=16613
This article was adapted from an original article by Yu.A. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article