Hilbert's theorem

Hilbert's theorem states that no complete regular surface of constant negative Gaussian curvature can be immersed in three-dimensional Euclidean space.

Gaussian curvature: A surface's intrinsic curvature, measured as the product of its principal curvatures. The theorem rules out immersions when this intrinsic quantity is constantly negative.
Differential geometry: The study of smooth spaces using calculus, geometry, and algebra. Hilbert's theorem is a global result in the geometry of smooth surfaces.
David Hilbert: German mathematician whose work shaped algebra, geometry, mathematical logic, and the foundations of mathematics. He proved the non-immersion result that bears his name.
Hilbert's theorem on positive definite forms: A theorem characterizing when a quadratic form is positive definite through positivity on nonzero vectors. This different result also bears Hilbert's name but concerns algebra, not immersed surfaces.
Complete metric space: A metric space in which every Cauchy sequence converges to a point in the space. Completeness prevents the surface from escaping the theorem through missing finite-distance boundaries.
Riemannian manifold: A smooth manifold equipped with a smoothly varying inner product on its tangent spaces. A surface's metric supplies the distances and completeness used in the theorem.
Non-Euclidean geometry: Geometries that differ from Euclidean geometry, including those with nonzero curvature. The theorem addresses whether a hyperbolic geometry can appear as a complete surface in Euclidean space.
Nash embedding theorem: A theorem stating that suitable Riemannian manifolds admit isometric embeddings into Euclidean spaces of sufficiently high dimension. Higher-dimensional ambient spaces can realize metrics that Hilbert's theorem excludes in three dimensions.
Geodesic: A curve that locally minimizes distance and has zero intrinsic acceleration on a Riemannian manifold. Geodesic behavior on a complete negatively curved surface underlies the global obstruction.
Isometric immersion: A smooth immersion that preserves the metric on every tangent space. The theorem is often framed as impossibility of realizing the intrinsic metric in three-dimensional space.

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