Flat torus

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Definition

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

Equational description

Consider \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_1x_2-plane and the unit circle in the x_3x_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 \R^{m+n} obtained as an isometric direct product of a curve in \R^m and a curve in \R^n, has zero sectional curvature. Template:Justify