A point $p\in M$ is a critical point of $f$(1) if for any smooth curve$\gamma : (-\epsilon, \epsilon)\to M$ with $\gamma(0)=p$, the tangent vector

$\frac{d(f\circ\gamma)}{dt} |_{t=0} = 0
\,.$

The critical point is regular if for one (or equivalently any) chart$\phi : U^{\open}\to \mathbb{R}^n$, where $p\in U$ and $\phi(p) = 0\in \mathbb{R}^n$, the Hessian matrix