Normal connection

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M\subseteq \bar{M}$ be an embedded Riemannian submanifold. For a tangent field $X\in \mathrm{\Gamma }(TM)$ and a normal field $\nu \in \mathrm{\Gamma }(\nu M)$, define the normal connection by

${\mathrm{\nabla }}_X^{\perp }\nu :=({\bar{\mathrm{\nabla }}}_X\nu {)}^{\perp }\mathrm{.}$

Here ${\bar{\mathrm{\nabla }}}_X\nu$ is well defined along $M$: the local extension-independence calculation in Induced connection and second fundamental form applies verbatim to any smooth field along $M$, including a normal one. Smoothness then follows locally before applying the smooth normal projection from Tangential and normal projections along a Riemannian submanifold.

This operation is a connection on $\nu M$. Indeed, the directional connection laws give

${\mathrm{\nabla }}_{fX}^{\perp }\nu =f{\mathrm{\nabla }}_X^{\perp }\nu ,{\mathrm{\nabla }}_X^{\perp }(f\nu )=X(f)\nu +f{\mathrm{\nabla }}_X^{\perp }\nu ,$

with real linearity in the other arguments. It is compatible with the metric induced on $\nu M$: for normal fields $\nu ,\mu$,

$X\bar{g}(\nu ,\mu )=\bar{g}({\mathrm{\nabla }}_X^{\perp }\nu ,\mu )+\bar{g}(\nu ,{\mathrm{\nabla }}_X^{\perp }\mu ),$

because tangent components of the two ambient derivatives are orthogonal to normal fields. This is metric compatibility in the sense of Metric compatible connection on a riemannian vector bundle.

The assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited exactly through the smooth restricted normal-bundle and projection construction; the displayed operation adds no choice. For the empty submanifold or a rank-zero normal bundle it is the unique zero connection. The same formulas apply in rank one, in tangent dimension zero, and at boundary points. Degenerate normal metrics are excluded by the Riemannian hypothesis.