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How are surfaces with zero Gaussian curvature related to planes?

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differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

In differential geometry, surfaces with zero Gaussian curvature are fundamentally related to planes through the concept of isometry.

A surface with constant zero Gaussian curvature is known as a developable surface. According to Minding’s theorem, such surfaces are locally isometric to a plane, meaning they can be flattened onto a plane without stretching or distorting intrinsic distances.1

While a plane itself has zero Gaussian curvature because all its normal curvatures are zero, other surfaces—such as a cylinder—also possess zero Gaussian curvature. This occurs because the product of their principal curvatures is zero (for a cylinder, one principal curvature is zero along its straight lines, while the other is non-zero). Consequently, these surfaces share the same intrinsic geometry as a plane, allowing them to be "unrolled" or mapped onto a flat surface.1