Fourmurface

Fourmurface is a term used in differential geometry to denote a two-dimensional smooth manifold that is embedded in four-dimensional Euclidean space or, more generally, in a four-dimensional ambient manifold. As a codimension-two submanifold, a fourmurface inherits a Riemannian metric from the ambient space and possesses a two-dimensional normal bundle. Its extrinsic geometry is described by the second fundamental form, together with the Gauss and Codazzi equations, which relate intrinsic curvature to how the surface bends in the ambient space.

Examples include the standard unit sphere S^2 in R^4 defined by x1^2+x2^2+x3^2+x4^2=1 and the Clifford torus S^1(1)×S^1(1) embedded in R^4 as (cos t, sin t, cos s, sin s). More generally, any smooth embedding of a closed or noncompact two-dimensional manifold into R^4 qualifies as a fourmurface.

Characteristics of a fourmurface include that its topology matches the abstract surface, while the ambient embedding determines extrinsic invariants such as the normal bundle structure and linking with other submanifolds. In oriented cases, the normal bundle has rank two; the Whitney sum formula and characteristic classes constrain possible embeddings. Locally, every point has coordinates in which the surface appears as the graph of two functions from R^2 to R^2.

Applications and relevance: fourm surfaces arise in the study of four-manifolds, knotting phenomena in higher dimensions, and mathematical physics, including models where a two-dimensional worldsheet is embedded in a higher-dimensional space. They provide a setting to examine how curvature and topology interact under higher-codimension embeddings.