ReebStabilitätssatz

The Reeb stability theorem, named after Georges Reeb, is a fundamental result in differential geometry concerning the structure of smooth manifolds. It states that if a smooth manifold M admits a Morse function f with only one critical point, then M must be diffeomorphic to a sphere. A Morse function is a smooth function whose critical points are all non-degenerate. Non-degenerate means that the Hessian matrix at each critical point is invertible. The "index" of a critical point is the number of negative eigenvalues of the Hessian.

The theorem is particularly significant because it implies that any manifold that can be "built" by thickening a single point, in a specific geometric sense, must be a sphere. This single critical point is necessarily a global minimum or maximum, meaning it has an index of 0 or n, where n is the dimension of the manifold. The proof typically involves constructing a gradient flow from the single critical point and demonstrating that this flow leads to a sphere-like structure.

The Reeb stability theorem has important implications for understanding the topology of manifolds. It provides a strong topological constraint on the existence of Morse functions with a minimal number of critical points. In essence, it suggests that if a manifold is topologically "simple" enough to be described by a Morse function with only one critical point, then its overall shape must be that of a sphere. This theorem has been influential in various areas of mathematics, including topology, dynamical systems, and mathematical physics.