Consider a embedded submanifold \(S\) on \(\mathbb{R}^n\). The tangential connection for \(X,Y\in \mathfrak{X}(S)\) is defined by

\begin{equation*} \nabla^{\perp}_XY=\pi^{\perp}(\bar{\nabla}_X(\widetilde{Y})), \end{equation*}

where \(\widetilde{Y}\) is an extension of \(Y\) on \(\mathbb{R}^n\), \(\hat{\nabla}\) is the euclidean connection and \(\pi^{\perp}\colon T_p\mathbb{R}^n\to T_pS\) the projection along the normal outward unit. (For more infos take a look at [@lee2018riemannian_manifolds, pp.86].)

Remarks