Let \(S\hookrightarrow M\) be an embedded submanifold. Then the tangent space \(T_pS\) may be considered as a subspace of \(T_pM\).

Since \(\iota\colon S\hookrightarrow M\) is a smooth immersion, the differential \(d\iota_p\) is injective and

\begin{equation*} d\iota_p(v)(f)=v(f\circ \iota)=v(f{\mid}_S). \end{equation*}

That is why we ask \(\iota\) to be an immersion.

Links Link to heading

• tangent space
• tangent space of a regular level set
• smooth immersion

Questions Link to heading

• Why \(\iota\) needs to be an immersion?