= Projected ambient connection
{title2=$D_XY=\pi\nabla_XY$}
For a <Riemannian metric> on an ambient manifold and an <embedded submanifold>, let $\pi:TN|_M\to TM$ be <orthogonal projection>. Project the <restriction of a connection to an embedded submanifold> to obtain $D_XY=\pi\nabla_XY$. Since $\pi Y=Y$, the <Leibniz rule> and linearity over <smooth functions> hold, making $D$ an <affine connection>. If the ambient connection is the <Levi-Civita connection>, pairing with tangent vectors shows that $D$ is a <metric connection>; projecting the zero-torsion identity shows it is a <torsion-free connection>. By uniqueness, $D$ is the induced <Levi-Civita connection>.
Back to article page
