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 $x_i,1\le i\le 4$. 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:

$x_1^2+x_2^2=1,x_3^2+x_4^2=1$

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