For an N-dimensional hypersurface $f(\pmb{x})=0$ embedded in N+1 flat dimensional space in which the coordinates $x^1,x^2,\dots,x^{N+1}$ are N+1 Cartesian ones, we can introduce $N$ local parameters $u^1,u^2,\dots,u^N$ such that $\pmb{x}=\pmb{x}(u^1,u^2,\dots,u^N)$. Then, a simple question is, what is zweibein for this hypersurface $f(\pmb{x})=0$? Please give some textbooks for references.
Question 1: The terminology zweibein may not be so commonly used in mathematics textbooks. In physics textbooks, the termiology dyad is also used instead.
Question 2: The zweibein may be referred to as the tetrad defined by $\pmb{e}^a_μ=∂\pmb{x}(u^1,u^2,\dots,u^N)/∂u^μ$ See: https://en.wikipedia.org/wiki/Tetrad_formalism
Most good books on general relativity and supergravity have reviews on tangent spaces and Vielbeine/Vierbeine, as these are needed to express fermions on the tangent spaces to a curved space. Consider Weinberg's classic Gravity text Ch 12.5.
Here, I'll give you a simplest example of Zweibeine on a beach-ball, $S^2$ which might serve you when you learn the general theory.
The constraint is, of course, $f=x^2+y^2+z^2-1=0$, so $df/2=0=zdz+ydy+xdx$, hence $$ dz=-(x dx +y dy)/z, $$ hence $$ dz^2=(xdx+ydy)^2/(1-x^2-y^2)=(xdx+ydy)^2/w, \qquad w\equiv 1-x^2-y^2 . $$
Consequently, $$ ds^2=g_{\mu\nu}dx^\mu dx^\nu \\ =dx^2+dy^2+(x^2 dx ^2 +2xydx dy +y^2dy)/w, $$ so that $$ g_{\mu\nu}= \delta_{\mu\nu} +x^\mu x^\nu /w, $$ and its inverse $$ g^{\mu\nu}= \delta_{\mu\nu} -x^\mu x^\nu , $$ $$ \det g_{\mu\nu}=1/w . $$
The Zweibeine are basically the "square roots" of the metric, $$g_{\mu\nu}=\delta_{ab} V_\mu^a V_\nu^b , \\ \delta^{ab}=g^{\mu\nu} V_\mu^a V_\nu^b , $$ $$ V_\mu^a=\delta_{\mu a} -\frac{x^\mu x^a}{1-w}\left (1+\frac{1}{\sqrt {w}}\right ) $$ $$V^{\mu a}=\delta_{\mu a} -\frac{x^\mu x^a}{1-w}(1+{\sqrt {w}}). $$ The latter convert commuting gradients $\partial_\mu$ to non-commuting gradients on the tangent space, $V_a^\mu \partial_\mu$. The positive square root of w may be equivalently supplanted by its negative square root, if desired.
Check them.
In matrix form, $$ g_{\mu\nu}= \begin{pmatrix} 1+x^2/w&xy/w\\ xy/w&1+y^2/w \end{pmatrix}, $$ $$ g^{\mu\nu}= \begin{pmatrix} 1-x^2&-xy\\ -xy&1-y^2 \end{pmatrix}, $$ and check the square root of the metric is $$ V_{\mu}^{a}= \begin{pmatrix} 1-x^2v&-xyv\\ -xyv&1-y^2v \end{pmatrix}, \qquad v\equiv (1+1/\sqrt{w})/(1-w). $$