Flat torus

Definition

The flat torus in $\mathbb {R} ^{4}$ is defined as the isometric direct product of two circles of equal radius, embedded in orthogonal $\mathbb {R} ^{2}$s.

Equational description

Consider $\mathbb {R} ^{4}$ with coordinates $x_{i},1\leq i\leq 4$. The flat torus obtained by taking the direct product of the unit circle is the $x_{1}x_{2}$-plane and the unit circle in the $x_{3}x_{4}$ plane is defined as the set of points satisfying the following two equations:

$x_{1}^{2}+x_{2}^{2}=1,\qquad x_{3}^{2}+x_{4}^{2}=1$

Curvature

The flat torus has zero sectional curvature, on account of being a direct product of two curves. In fact, any surface in $\mathbb {R} ^{m+n}$ obtained as an isometric direct product of a curve in $\mathbb {R} ^{m}$ and a curve in $\mathbb {R} ^{n}$, has zero sectional curvature. Template:Justify