Whitney Embedding Theorem

The Whitney embedding theorem is a foundational result in differential topology that establishes conditions under which a smooth manifold can be embedded into a higher-dimensional Euclidean space without self-intersections. Proven by Hassler Whitney in 1936, the theorem states that any smooth d-dimensional manifold can be embedded in Euclidean space of dimension at most 2d+1.

The theorem is the topological prerequisite for Takens' theorem. Where Whitney's result concerns abstract manifolds, Takens' theorem concerns dynamical systems: it shows that the delay-coordinate map constructed from a time series is not merely an embedding in the abstract sense, but a specific embedding that preserves the dynamical structure of the system — its trajectories, attractors, and bifurcations.

Whitney's theorem tells us that space is sufficient; Takens' theorem tells us that observation is sufficient. The gap between them — between the abstract existence of an embedding and the concrete construction of one from a single time series — is precisely the domain of empirical nonlinear dynamics.