nLab Reeb sphere theorem





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id ⊣ id ∨ ∨ fermionic ⇉ ⊣ ⇝ bosonic βŠ₯ βŠ₯ bosonic ⇝ ⊣ Rh rheonomic ∨ ∨ reduced β„œ ⊣ β„‘ infinitesimal βŠ₯ βŠ₯ infinitesimal β„‘ ⊣ & Γ©tale ∨ ∨ cohesive Κƒ ⊣ β™­ discrete βŠ₯ βŠ₯ discrete β™­ ⊣ β™― continuous ∨ ∨ βˆ… ⊣ * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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The Reeb sphere theorem in differential topology says that:

If a compact differentiable manifold XX admits a differentiable function X→ℝX \to \mathbb{R} with exactly two critical points which are non-degenerate, then XX is homeomorphic to an n-sphere (with its Euclidean metric topology).

In fact this holds true even if the two critical points happen to be degenerate (Milnor 64, theorem 1’ on p. 166)


Let x minx_{min} be the critical point at which ff attains its minimum and f maxf_{max} the critical point at which it attains its maximum.

By the fact that XX is a differential manifold we find an closed neighbourhood BβŠ‚XB \subset X of x minx_{min} which is diffeomorphic to an closed ball.

By the assumption that there are no other critical points, ff is monotonically increasing along the flow lines Ξ³\gamma of βˆ‡f\nabla f on Xβˆ–{x min,x max}X \setminus \{x_{min}, x_{max}\}, in that the function βˆ‡ f(dΞ³ t)=βˆ‡f(βˆ‡f)=|βˆ‡f| 2 \nabla_f(d \gamma_t) = \nabla f(\nabla f) = {\vert \nabla f\vert^2 } is strictly positive:

|βˆ‡f| 2:Xβˆ–{x min,x max}β†’(0,∞)βŠ‚β„. \vert\nabla f\vert^2 \colon X \setminus \{x_{min}, x_{max}\} \to (0,\infty) \subset \mathbb{R} \,.

Hence for CβŠ‚XC \subset X a compact, the extreme value theorem implies that this function attains its minimum.

This implies that for every yβˆˆβ„y \in \mathbb{R} with f(x min)<y<f(x max)f(x_{min}) \lt y \lt f(x_{max}) there exists tβˆˆβ„t \in \mathbb{R} such that the flow Ο• βˆ‡f\phi_{\nabla f} along βˆ‡f\nabla f satisfies Ο• βˆ‡f(x,t)β‰₯y\phi_{\nabla f}(x,t) \geq y for all xβˆˆβˆ‚Bx \in \partial B. In particular this is the case for yy the maximum of f| Cf|_{C}. This implies that CC is contained in the image of BB under some flow of βˆ‡f\nabla f. But this image is, by the nature of the flow, diffeomorphic to a Euclidean ball.

In conclusion this shows that every compact subspace of XX is contained in an open subspace diffeomorphic to a Euclidean ball. With this the claim follows by the Brown-Stallings lemma.


Notice that the Reeb sphere theorem does not speak of diffeomorphism to an n-sphere, just about homeomorphism. Indeed, the theorem may be used as an ingredient in the construction of exotic smooth structures on some nn-spheres.


Due to:

  • Georges Reeb, Sur les points singuliers d’une forme de Pfaff completement integrable ou d’une fonction numerique, Comptes Rendus Acad. Sciences Paris 222 (1946) 847-849 [[crid:1571417125676878592]]


  • John Milnor, theorem 4.1 in Morse theory (pdf)

  • John Milnor, Differential topology, chapter 6 in T. L. Saaty (ed.) Lectures On Modern Mathematic II 1964 (web, pdf)

See also

Last revised on May 26, 2022 at 15:22:16. See the history of this page for a list of all contributions to it.