Totally geodesic submanifold

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. An embedded Riemannian submanifold $M\subseteq \bar{M}$ is totally geodesic when its normal-valued second fundamental form vanishes identically:

${\mathrm{I}\mathrm{I}}_p(u,v)=0\text{for every }p\in M\text{ and }u,v\in T_{p}M\mathrm{.}$

By The second fundamental form is a symmetric normal-bundle-valued two-tensor, this is an intrinsic pointwise condition on the embedding and ambient metric; it is independent of extensions, frames, and normal orientations. The choice hypothesis is inherited exactly through the smooth projection used to define $\mathrm{I}\mathrm{I}$ in Induced connection and second fundamental form, and the vanishing condition makes no additional choice.

The condition is vacuous for the empty submanifold and holds automatically when the tangent bundle or normal bundle has rank zero. It applies unchanged in rank one and at boundary points. Degenerate induced metrics are outside the Riemannian hypothesis. The next theorem proves the promised equivalence with ambient preservation of tangent derivatives and with the local geodesic condition; those are consequences, not part of this definition.