$$\begin{align*} &\text{minimize }& f(\b{x}) \br &\text{subject to }& \b{h}(\b{x}) = \b{0}\br \end{align*}$$
where $\b{x} \in \R^n$, $f:\R^n \to \R$, $\b{h}:\R^n \to \R^m$, $\b{h} \in \mathcal{ C }^1$, and $m < n $.
The set of equality constraints $h(\b{x}) = 0, \b{h} : \R^n \to \R^m $ describe a surface
$$S = \set{ \b{x} \in \R^n : \b{h}(\b{x}) = 0} $$
Let $D \b{h}(\b{x})$ be the Jacobian matrix of $\b{h} = \transpose{ [ h_1, h_2, …, h_{ m } ] } $ at $\b{x}$, given by
$$D \b{h}(\b{x}) = {\Vce{ Dh_1(\b{x})}{ \vdots }{ Dh_m(\b{x})}} = {\Vce{ \transpose{ \nabla h_1(\b{x})}}{ \vdots }{ \transpose{ \nabla h_m(\b{x})}}} \qquad D \b{h}(\b{x}) \in \R^{m \times n} , m < n $$
Let point $\b{x}^* \in S$,
$$\b{x}^*\text{ is a regular point }\iff \rank{ D \b{h}(\b{x}^*)} = m .$$
Assuming that the points in $S $ are regular, the dimension of the surface $S $ is $n-m$.