Geodesic manifold

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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.


Also called geodesic submanifold and totally geodesic submanifold, respectively.


[a1] W. Klingenberg, "Riemannian geometry" , Springer (1982) (Translated from German)
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Geodesic manifold. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Yu.A. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article