In
mathematics, the
differential geometry of surfaces deals with the
differential geometry of
smooth surfaces with various additional structures, most often, a
Riemannian metric. Surfaces have been extensively studied from various perspectives:
extrinsically, relating to their embedding in
Euclidean space and
intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the
Gaussian curvature, first studied in depth by
Carl Friedrich Gauss (articles of 1825 and 1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.