Hyperboloid ($x^2/a^2+y^2/b^2-z^2/c^2=\pm 1$)

The displayed surfaces, with positive semiaxes, are hyperboloids. The plus sign gives a one-sheet hyperboloid, while the minus sign gives a two-sheeted hyperboloid.

A one-sheet hyperboloid is a connected, doubly ruled surface, diffeomorphic to a cylinder.

On $x^2+y^2=z^2+1$, an axial plane such as $y=0$ cuts out two disjoint meridian geodesics. The tangent plane $x=1$ cuts out the two straight ruling lines $y=\pm z$, which are geodesics meeting orthogonally at $(1,0,0)$.

For a unit-speed geodesic on $x^2+y^2=z^2+1$, Clairaut's constant $c$ givesIf $c>1$, the geodesic has a turning point at $z=\sqrt{c^2-1}$ and remains entirely in one of the regions $z>0$ or $z<0$.

The waist circle $z=0$ is a geodesic. Its Gaussian curvature is $-1$, whereasoff the waist. Since isometries preserve Gaussian curvature, every isometry preserves this circle setwise.

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.