Mathematicians of the 18th century studied surfaces in 3-space using unit normal fields, local coordinates, and differential calculus. The ebook General Investigations of Curved Surfaces of 1827 and 1825 , comprising translations of two papers by Carl Friedrich Gauss, establishes a now-famous Great Theorem: There is an “intrinsic” curvature that can be detected solely by making measurements within a surface. This may sound esoteric, but opens the possibility that our physical universe is curved in ways we can detect astronomically without making reference to a higher-dimensional space.
Informally, a plane is not curved, a cylinder is curved, and a sphere is curved. To a differential geometer, however, a cylinder and a sphere are “curved” in distinct senses: Two-dimensional beings living in a cylindrical universe cannot measure deviations from plane geometry by making local measurements, only by circumnavigating and detecting a “straight” closed path. By contrast, two-dimensional beings living in a spherical universe can measure deviations from plane geometry by surveying triangles: On a sphere, a triangle with “straight” (geodesic) sides has total interior angle greater than a straight angle. Precisely, on a unit sphere the total interior angle of a geodesic triangle is a straight angle plus the area enclosed by the triangle.
Nowadays in differential geometry (the branch of mathematics), we say a plane in 3-space is extrinsically flat because there exists a constant field of unit-length normal vectors, and is intrinsically flat because straight-sided triangles enclose a total interior angle equal to a straight angle. In the same senses, a cylinder in 3-space is extrinsically curved (no constant field of unit normal vectors) but intrinsically flat. A sphere in 3-space is curved (not flat) both extrinsically and intrinsically.
In a suitable universe, a sphere is extrinsically flat: Think of an equatorial 2-sphere in a round 3-sphere, or the product of a round 2-sphere with a number line. But thanks to Gauss's Great Theorem, in no universe is a round sphere intrinsically flat.