Liebmann's theorem

The set of Liebmann is a classical result of the differential geometry , which according to the German mathematician Heinrich Liebmann is named. It deals with the characterization of spherical surfaces in three-dimensional Euclidean space .

The sentence

First formulation

In modern terms, Liebmann's theorem says the following:

Let be a coherent and compact area of the class in three-dimensional Euclidean space and let the Gaussian curvature of be a constant . ${\displaystyle F\subset {\mathbb {E} }^{3}}$ ${\displaystyle C^{4}}$ ${\displaystyle F}$ ${\displaystyle K}$

Then a positive number of the form is for a real number and coincides with the surface of a three-dimensional full sphere of radius , is therefore a sphere of the form for a . ${\displaystyle K}$ ${\displaystyle K=r^{2}}$ ${\displaystyle r>0}$ ${\displaystyle F}$ ${\displaystyle R={\tfrac {1}{r}}}$ ${\displaystyle F=S_{R}^{2}(p)}$ ${\displaystyle p\in {\mathbb {E} }^{3}}$

Second formulation

In three-dimensional space a sphere of radius is always a coherent and compact surface and always has the constant Gaussian curvature . Therefore, Liebmann's theorem can also be formulated as follows: ${\displaystyle R}$ ${\displaystyle C^{4}}$ ${\displaystyle K={\tfrac {1}{R^{2}}}}$

In three-dimensional space, only the spheres are cohesive and compact surfaces with constant Gaussian curvature. ${\displaystyle C^{4}}$

Third formulation

In terms of its topological properties, a surface is a 2-manifold . Since a coherent and compact 2-manifold is also called a closed surface in topology , Liebmann's theorem can also be given in the following way, very abbreviated: ${\displaystyle F\subset {\mathbb {E} }^{3}}$

In three-dimensional space, the spheres are the only closed surfaces with constant Gaussian curvature. ${\displaystyle C^{4}}$

Relation to other results

Liebmann's second sentence

In 1900 Heinrich Liebmann put forward another sentence that is closely related to the above. This second Liebmann sentence is in modern formulation as follows:

Let be a coherent and compact surface of the class in three-dimensional Euclidean space and let the Gaussian curvature of consistently positive and the mean curvature of a constant . ${\displaystyle F\subset {\mathbb {E} }^{3}}$ ${\displaystyle C^{4}}$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle H}$

Then it coincides with the surface of a sphere of radius . ${\displaystyle F}$ ${\displaystyle R={\frac {1}{|H|}}}$

In other words and in short:

In three-dimensional space, the spheres are the only closed surfaces with a consistently positive Gaussian curvature and a constant mean curvature. ${\displaystyle C^{4}}$

The theorem of Cohn-Vossen and Herglotz

Liebmann's theorem can be related to the question of how a coherent compact surface of three-dimensional Euclidean space must be constituted in order to be isometrically - in the sense of the isometry of Riemannian manifolds  - to be a unitary sphere or a general sphere. The theorem by Cohn-Vossen and Herglotz provides information about this, which goes back to Stefan Cohn-Vossen and Gustav Herglotz and which is also a classic result of differential geometry:

Standing in three-dimensional Euclidean space , two closed surfaces of the class , each positive Gaussian curvature to each other in isometric, then there exists a rigid motion with , so that in transferred. ${\displaystyle {\mathbb {E} }^{3}}$ ${\displaystyle F,F^{*}\subset {\mathbb {E} }^{3}}$ ${\displaystyle C^{\infty }}$ ${\displaystyle \phi }$ ${\displaystyle \phi (F)=F^{*}}$ ${\displaystyle F}$ ${\displaystyle F^{*}}$

This phrase refers to the Austrian surveyor Charles Strub Ecker in his differential geometry as identity set for Eiflächen and calls him a basic for a metric theory of Eiflächen set . In differential geometry, an egg surface is understood to mean any closed surface of three-dimensional Euclidean space which is at least of the class and has positive Gaussian curvature throughout . The theorem of Cohn-Vossen and Herglotz can therefore also be formulated as follows: ${\displaystyle C^{3}}$ ${\displaystyle K>0}$

In three-dimensional Euclidean space, two isometric egg surfaces of the class are always congruent . ${\displaystyle C^{\infty }}$

Hilbert's theorem

The first partial statement of Liebmann's theorem is closely related to a general result presented by David Hilbert in 1900. It represents the central result of Appendix V ( On Areas of Constant Gaussian Curvature ) of its Fundamentals of Geometry and can be stated as follows:

In three-dimensional Euclidean space there is no surface of constant Gaussian curvature . ${\displaystyle C^{\infty }}$ ${\displaystyle F\subset {\mathbb {E} }^{3}}$ ${\displaystyle K<0}$

Hilbert was led to this result by asking whether the construction of a non-Euclidean plane ( pseudosphere ) provided by Eugenio Beltrami is conceivable as a whole embedded in three-dimensional space . He comes to a negative to this question and writes explicitly:

", D. H. we recognize that there is no singularity-free and everywhere regular analytical surface of constant negative curvature. In particular, the question raised at the beginning must therefore also be answered in the negative, whether the WHOLE LOBATSCHEFSKIJsche level can be realized in the BELTRAMI way by a regular analytical surface in space. "

Remarks

1. When it comes to the question of which class of differentiability the surfaces have to be in order for the above sentences to be valid, different information can be found in the literature. If one always assumes -surfaces, then the propositions are valid throughout. In many cases, the statements of the sentences can still be proven under weakened conditions. For example, the Russian mathematician AW Pogorelow showed in 1952 that the statement of the theorem by Cohn-Vossen and Herglotz is also valid for a considerably more general class of egg surfaces with limited Gaussian curvature. ${\displaystyle C^{\infty }}$
2. The concept of the egg surface goes back to Wilhelm Blaschke . According to an important theorem by Jacques Hadamard , an egg surface is always orientable , diffeomorphic to the 2-sphere and strictly convex , whereby strict convexity is understood to mean that the egg surface is located completely on one side of the tangential plane belonging to this point at each of its points , i.e. always lies completely within one of the two closed half-spaces which are formed by the tangential plane. ${\displaystyle {\mathbb {E} }^{3}}$   ${\displaystyle S^{2}}$  

literature

• M. Berger , B. Gostiaux : Differential Geometry: Manifolds, Curves and Surfaces (=  Graduate Texts in Mathematics . Volume 115 ). Springer Verlag, New York (et al.) 1988, ISBN 0-387-96626-9 ( MR0917479 ).
• David Hilbert : Fundamentals of Geometry. With supplements from Dr. Paul Bernays (=  Teubner Study Books: Mathematics ). 11th edition. Teubner Verlag, Stuttgart 1972, ISBN 3-519-12020-8 ( MR1109913 ).
• David Hilbert: On surfaces of constant Gaussian curvature . In: Trans. Amer. Math. Soc . tape 2 , 1901, p. 87-99 ( ams.org [PDF]). MR1500557
• Wilhelm Klingenberg : A lecture on differential geometry (=  Heidelberg pocket books . Volume 107 ). Springer Verlag, Berlin (among others) 1973, ISBN 3-540-06253-X ( MR0415512 ).
• E. Kreyszig : Differential Geometry. Curves - surfaces - manifolds (=  Mathematics and its applications in physics and engineering: A series . Band 25 ). 2nd, revised edition. Academic publishing company Geest & Portig, Leipzig 1968 ( MR0229174 ).
• Wolfgang Kühnel : Differential Geometry. Curves - surfaces - manifolds (=  STUDIUM ). 5th updated edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-1233-9 .
• Detlef Laugwitz : Differential Geometry (=  mathematical guidelines ). 2nd revised edition. Teubner Verlag, Stuttgart 1968 ( MR0243432 ).
• H. Liebmann : A new property of the sphere . In: Nachr. Kgl. Ges. Wiss. Göttingen, Math.-Phys. Class . tape 44 , 1899, pp. 44-55 .
• Heinrich Liebmann: About the bending of closed surfaces with positive curvature . In: Math. Ann . tape 53 , 1900, pp. 81-112 , doi : 10.1007 / BF01456030 . MR1511083
• Horst Schubert : Topology. An introduction (=  mathematical guidelines ). 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).
• Karl Strubecker: Differential Geometry III. Theory of surface curvature (=  Göschen Collection . 1180 / 1180A). Walter de Gruyter Verlag, Berlin 1959.
• Rolf Walter: differential geometry . 2nd, revised and expanded edition. BI-Wissenschaftsverlag, Mannheim ( inter alia) 1989, ISBN 3-411-03216-2 .

References and footnotes

1. ↑ Klingenberg, p. 106.
2. ↑ Kühnel, p. 133.
3. ↑ Berger-Gostiaux, p. 382.
4. ↑ Kreyszig, p. 243.
5. ↑ The consideration here are manifolds without boundary , so no manifolds with boundary and lokaleuklidisch ; d. That is, every point in it has an open environment that is homeomorphic to a Euclidean space , and thus a system of surroundings in the relative topology with all the topological properties that are also given for the points of Euclidean space. So every area point within the areas considered here has an open environment that is homeomorphic . See Schubert, p. 210 / Kühnel, p. 143. ${\displaystyle {\mathbb {E} }^{2}}$
6. ↑ Laugwitz, p. 162.
7. ^ Kühnel, p. 134.
8. ↑ Kreyszig, p. 243.
9. ↑ An isometry of Riemannian manifolds thus leaves the entire Riemannian structure and (in this sense) the entire internal geometry of the manifolds involved invariant; see. Walter, pp. 123,156.
10. ↑ Klingenberg, p. 106.
11. ↑ Berger-Gostiaux, p. 427.
12. ↑ Strubecker, p. 202.
13. ↑ Strubecker, p. 202. / Klingenberg, p. 100.
14. ↑ Walter, p. 195.
15. ↑ Hilbert, p. 231 ff.
16. ^ Hilbert also published this work in the Transactions of the American Mathematical Society of 1901; see. Hilbert: Over areas of constant Gaussian curvature . In: Trans. Amer. Math. Soc . 1901, p. 87 ff . ; see. also Berger-Gostiaux, p. 428.
17. ↑ Hilbert, p. 237.
18. ↑ Hilbert: On surfaces of constant Gaussian curvature . In: Trans. Amer. Math. Soc . 1901, p. 97 .
19. ↑ See Walter, p. 310. Here the author writes in Appendix II of his book, in which the properties of differentiable manifolds are summarized: For the sake of uniformity, only the case is dealt with here, but everything is framed in such a way that it is also based on weaker differentiability assumptions (mostly or ) remains valid. ${\displaystyle C^{\infty }}$ ${\displaystyle C^{1}}$ ${\displaystyle C^{2}}$
20. ↑ Strubecker, pp. 202-203.
21. ↑ Klingenberg, pp. 100-102.
22. ↑ See also section “Related Results” in the article “Tietze's Theorem (convex geometry)” .