Smooth surface

A smooth surface in ${\mathbb{R}}^3$ is a two-dimensional smooth embedded submanifold. Locally it is parametrized by two coordinates with linearly independent tangent vectors.

This regular surface has two connected components, $t=\pm \sqrt{1+x^2+y^2}$. The symmetric-matrix model of the two-sheeted hyperboloid realizes them as the positive-definite and negative-definite symmetric determinant-one $2\times 2$ matrices. Each component is diffeomorphic to ${\mathbb{R}}^2$; this differs from the connected one-sheet hyperboloid.

The displayed coordinates identify determinant-one real symmetric matrices of size two with the two-sheeted hyperboloid. The special linear congruence action on symmetric matrices is transitive on each sheet, with stabilizer $SO(2)$. In particular the positive sheet is the homogeneous space $SL(2,\mathbb{R})/SO(2)$, a model of the hyperbolic plane.

An embedded smooth surface $S\subset {\mathbb{R}}^3$ is inextendible when every connected smooth surface $\tilde{S}\subset {\mathbb{R}}^3$ containing $S$ is equal to $S$.