Equivalent characterizations of a totally geodesic submanifold

By [F2], the normal component of ${\bar{\mathrm{\nabla }}}_{X}Y$ is exactly $\mathrm{I}\mathrm{I}(X,Y)$. Thus condition 1 holds if and only if condition 2 holds.

Pulling [F2] back along a smooth curve $\gamma$ in $M$ and evaluating on its velocity gives the curvewise Gauss formula ${\bar{D}}_t{\gamma }^{\mathrm{\prime }}=D_t^M{\gamma }^{\mathrm{\prime }}+\mathrm{I}\mathrm{I}({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})\mathrm{.}$ This follows in a local frame directly from the pullback derivative in [F5], so it is independent of field extensions. If condition 1 holds and $\gamma$ is an intrinsic geodesic, [F3] and [F5] make the first term zero and [F1] makes the second zero. Hence $\gamma$ is an ambient geodesic, proving condition 3.

Conversely assume condition 3. Fix $p\in M$ and $v\in T_{p}M$. By [F6] there is an intrinsic geodesic $\gamma$ with $(\gamma (0),{\gamma }^{\mathrm{\prime }}(0))=(p,v)$. Condition 3 and [F5] make both covariant accelerations in step 1.2 zero, so ${\mathrm{I}\mathrm{I}}_p(v,v)=0$. Since $p,v$ were arbitrary, this holds for every tangent vector.

By symmetry and bilinearity [F4], polarization gives $2{\mathrm{I}\mathrm{I}}_p(u,v)={\mathrm{I}\mathrm{I}}_p(u+v,u+v)-{\mathrm{I}\mathrm{I}}_p(u,u)-{\mathrm{I}\mathrm{I}}_p(v,v)=0.$ Thus $\mathrm{I}\mathrm{I}=0$, proving condition 1 and completing the equivalence.

On the empty manifold all three universal conditions hold. In dimension zero all geodesics are constant and $\mathrm{I}\mathrm{I}$ is zero; the proof applies unchanged in dimension one and in codimension zero. Both manifolds are explicitly boundaryless because [F5] uses that convention; parameter intervals have nonempty interior and any included endpoints use their stated one-sided derivative. Positive definiteness supplies the orthogonal decomposition. The only choice is the declared ${\mathrm{A}\mathrm{C}}_{\omega }$ inherited through [F1]–[F3] and used by [F6]; step 2.1 invokes existence for one supplied $(p,v)$ at a time.