For an unconstrained optimization problem with objective function $F(x,y,z)$ the index of a stationary point is well-defined: If $(x^*, y^*, z^*)$ is a point where the gradient of $F(x,y,z)$ vanishes, then $(x^*,y^*,z^*)$ is a stationary point of $F(x,y,z)$ and its index is the number of negative eigenvalues of the hessian matrix of $F(x,y,z)$ with respect to $x$, $y$ and $z$.
However, if $F(x,y,z)$ is then subject to some other nonlinear constraint $g(x,y,z) = 0$, then how is the index of the stationary point is defined in this case? I would think that the bordered hessian in place of the hessian and then the number of negative eigenvalues of the bordered hessian at a stationary point is the index of the stationary point. But I do not see any reference which clearly states this definition. In the wiki article, as well as this one, it only gives a condition for a stationary point not being a minimum nor a maximum. But they do not clearly define the index of such a stationary point. Is there any reference where I could find a precise definition?