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What did Gauss prove about curvature and isometric surfaces?

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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 the field of differential geometry, Carl Friedrich Gauss made a fundamental breakthrough in 1827 regarding the relationship between curvature and the transformation of surfaces.1

His most significant contribution in this area is known as Gauss’s "Remarkable Theorem" (Theorema Egregium). Gauss proved that if two smooth surfaces are isometric—meaning one can be bent or transformed into the other without stretching or changing intrinsic distances—then they must possess the same Gaussian curvature at corresponding points.1

This theorem is considered "remarkable" because, although Gaussian curvature is initially defined using extrinsic measurements (the principal normal curvatures of a surface in space), Gauss demonstrated that it is actually an intrinsic property. Consequently, while a surface like a cylinder can be flattened into a plane without distortion because both have a Gaussian curvature of zero, a sphere cannot be flattened into a plane because their curvatures differ, making them non-isometric.12