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)|
Geodesic manifold. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Geodesic_manifold&oldid=31980