# Geodesic manifold

From Encyclopedia of Mathematics

*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://www.encyclopediaofmath.org/index.php?title=Geodesic_manifold&oldid=31980

This article was adapted from an original article by Yu.A. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article