At a regular point of a smooth surface, the product of the two principal curvatures k1 and k2; an intrinsic measure of how the surface bends that depends only on the first fundamental form and not on the particular embedding in space.

Principle

The intrinsic geometry of a surface determines Gaussian curvature: local metric data (lengths and angles) fix K, so K is invariant under isometries of the surface.

Demonstration

For a sphere of radius R, k1=k2=1/R so K=1/R^2; for a plane K=0; for a saddle point like a hyperbolic paraboloid k1>0, k2<0 and K<0.

Misapplication

Treating Gaussian curvature as the arithmetic mean of principal curvatures or as an extrinsic bending measure that changes under bending without stretching; confusing sign conventions without stating orientation.

Consequence

Gaussian curvature controls local geometric phenomena such as the behavior of geodesics, area distortion, and, via integral formulas, global topological invariants; it determines whether small neighborhoods are sphere-like, flat, or saddle-like.

Reversal

Exchange to an extrinsic viewpoint yields mean curvature or normal curvature information; flipping the concept gives quantities that depend on how the surface sits in space rather than only on its metric.

Boundary

Defined at regular (twice-differentiable) points of a two-dimensional surface; not defined at corners, edges, or on non-differentiable sets. For higher-dimensional manifolds one uses sectional or Ricci curvature instead.

Semantic Tension

Competing usages emphasize intrinsic versus extrinsic curvature; some texts conflate signs or define curvature via principal curvatures while others use metric formulas — the tension is between coordinate-free metric invariants and embedding-dependent measures.

Synthesis

Gaussian curvature is the intrinsic scalar formed by multiplying the principal curvatures at a smooth point; it encapsulates the surface's local metric bending and predicts whether neighborhoods are locally spherical, flat, or saddle-shaped.