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Why can a cylinder be flattened into a plane without distortion while a sphere cannot?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — geometry This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

In differential geometry, the ability to flatten a surface onto a plane without distortion depends on its Gaussian curvature—the product of the surface's two principal normal curvatures at a given point.12

A surface can be flattened into a plane (a process called being "developable") if and only if its Gaussian curvature is zero.2

Cylinder: At any point on a cylinder, one principal curvature is $0$ (along the vertical straight lines) and the other is $1 / r$ (along the horizontal circles). Because the product of these curvatures is $0 \times 1 / r = 0$ , the cylinder has a Gaussian curvature of zero. Consequently, a cylinder is locally isometric to a plane, meaning it can be unrolled or flattened without stretching or distorting the surface.2
Sphere: On a sphere of radius $r$ , both principal curvatures are $1 / r$ . The Gaussian curvature is therefore $(1 / r) \times (1 / r) = 1 / r^{2}$ . Because this value is not zero, a sphere cannot be flattened onto a plane without significant distortion, such as stretching or tearing.2

This principle is formalized in Gauss’s “Remarkable Theorem,” which states that if two surfaces are isometric, they must have the same Gaussian curvature.2