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For a submanifold with nondegenerate induced metric, being a totally geodesic submanifold means that its second fundamental form vanishes. The Gauss formula then identifies intrinsic covariant acceleration with ambient covariant acceleration, so every intrinsic geodesic is an ambient geodesic throughout its domain.

Equivalently, an ambient geodesic initially tangent to the submanifold stays in it locally, by uniqueness of the geodesic equation. This is a local statement. An open interval on a straight line in the Euclidean plane has zero second fundamental form, but its extended ambient geodesics eventually leave the interval. A closed embedded submanifold avoids that exit obstruction: local preservation and closedness extend containment along the ambient geodesic.

This is the arbitrary-codimension version of a totally geodesic hypersurface. A great circle on the round sphere supplies a curved-ambient example.