The induced connection is Levi–Civita

Orthogonal projection is fibrewise linear. Projecting the affine-connection laws in [F2] and using [F1] therefore gives real bilinearity, ${\mathrm{\nabla }}_{fX}^{M}Y=f{\mathrm{\nabla }}_X^{M}Y$, and ${\mathrm{\nabla }}_X^M(fY)=(X(f)Y+f{\bar{\mathrm{\nabla }}}_{X}Y{)}^{\mathrm{\top }}=X(f)Y+f{\mathrm{\nabla }}_X^{M}Y,$ because $Y$ is tangent. Thus ${\mathrm{\nabla }}^M$ is an affine connection on $TM$.

The bracket of two fields tangent to $M$ is tangent: in a slice chart, their normal coordinate components vanish along the slice, and tangent derivatives of those zero restrictions vanish, so the coordinate formula gives zero normal components for $[X,Y]$. Ambient torsion freeness now yields ${\mathrm{\nabla }}_X^{M}Y-{\mathrm{\nabla }}_Y^{M}X=({\bar{\mathrm{\nabla }}}_{X}Y-{\bar{\mathrm{\nabla }}}_{Y}X{)}^{\mathrm{\top }}=[X,Y{]}^{\mathrm{\top }}=[X,Y]\mathrm{.}$ Hence the induced connection is torsion free.

For tangent fields $X,Y,Z$, ambient metric compatibility and the fact that tangent and normal vectors are orthogonal give $Xg(Y,Z)=X\bar{g}(Y,Z)=\bar{g}({\bar{\mathrm{\nabla }}}_{X}Y,Z)+\bar{g}(Y,{\bar{\mathrm{\nabla }}}_{X}Z)=g({\mathrm{\nabla }}_X^{M}Y,Z)+g(Y,{\mathrm{\nabla }}_X^{M}Z)\mathrm{.}$ Thus ${\mathrm{\nabla }}^M$ is compatible with the induced metric.

Steps 1.1–1.3 show that ${\mathrm{\nabla }}^M$ is a Levi–Civita connection. By [F3] it is the unique one for $g$.

On the empty or zero-dimensional submanifold all displayed identities are vacuous and the connection is the unique zero operator. In dimension one and at boundary points the same local calculations apply. Positive definiteness supplies the orthogonal splitting used in step 1.3. The hypothesis ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited exactly through [F1]'s smooth projection construction; the proof introduces no further choices.