Let \(S\hookrightarrow M\) be an embedded submanifold of a Riemannian manifold \((M,g)\). Then \(S\) the pullback metric \(\iota^*g\) is called induced metric and \((S,\iota^*g)\) is referred as Riemannian submanifold.
Remark
- The induced metric has a simple form. Let \(v,w\in T_pS\). Then \begin{equation*} (\iota^*g)(v,w)=g(d\iota_p(v), d\iota_p(w))=g(v,w). \end{equation*}