Gauss equation for a Riemannian submanifold

Apply [F1] to the tangent field ${\mathrm{\nabla }}_Y^{M}Z$ and [F3] to the normal field $\mathrm{I}\mathrm{I}(Y,Z)$. Taking tangential components gives $({\bar{\mathrm{\nabla }}}_X{\bar{\mathrm{\nabla }}}_{Y}Z{)}^{\mathrm{\top }}={\mathrm{\nabla }}_X^M{\mathrm{\nabla }}_Y^{M}Z-S_{\mathrm{I}\mathrm{I}(Y,Z)}X\mathrm{.}$ Interchanging $X,Y$ gives the analogous formula, and [F1] gives $({\bar{\mathrm{\nabla }}}_{[X,Y]}Z{)}^{\mathrm{\top }}={\mathrm{\nabla }}_{[X,Y]}^{M}Z$.

Substitute the three identities of step 1.1 into the curvature commutator [F4]. Since [F2] identifies the intrinsic connection, $(\bar{R}(X,Y)Z{)}^{\mathrm{\top }}=R^M(X,Y)Z-S_{\mathrm{I}\mathrm{I}(Y,Z)}X+S_{\mathrm{I}\mathrm{I}(X,Z)}Y\mathrm{.}$

Pair step 2.1 with $W$. The normal component of $\bar{R}(X,Y)Z$ is orthogonal to $W$, and [F3] converts the two shape terms, so [F5] yields ${\mathrm{Rm}}^{\bar{M}}(X,Y,Z,W)={\mathrm{Rm}}^M(X,Y,Z,W)-\bar{g}(\mathrm{I}\mathrm{I}(X,W),\mathrm{I}\mathrm{I}(Y,Z))+\bar{g}(\mathrm{I}\mathrm{I}(Y,W),\mathrm{I}\mathrm{I}(X,Z))\mathrm{.}$ Metric symmetry and rearrangement give exactly the formula in the Statement.

The equation is vacuous on the empty submanifold. If tangent rank is zero every term vanishes. In tangent rank one the alternating curvature terms vanish, while the two quadratic terms are equal and therefore cancel; the second fundamental form itself need not vanish. If normal rank is zero the two II terms vanish and the identity reduces to equality of ambient and intrinsic tangent curvature. The pointwise calculation applies at boundary points, and positive definiteness supplies the orthogonal projections. The stated ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited through [F1]–[F3]; no new choice is made.