Toponogow's theorem

In geometry , Toponogow's theorem establishes the connection between Riemannian geometry and synthetic metric geometry. It clearly states that in a manifold with an upwardly restricted curvature, triangles are not thicker than in the comparison space of constant curvature.

It was proven in 1958 by Viktor Andreevich Toponogow .

Comparison rooms

For every number and every one there is an unambiguous, simply connected -dimensional Riemannian manifold of the sectional curvature constant . For this is the sphere of the radius , for the Euclidean space and for the hyperbolic space scaled with the factor . ${\displaystyle \delta \in \mathbb {R} }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n}$ ${\displaystyle E_{\delta }^{n}}$ ${\displaystyle \delta }$ ${\displaystyle \delta >0}$ ${\displaystyle {\frac {1}{\sqrt {\delta }}}}$ ${\displaystyle \delta =0}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \delta <0}$ ${\displaystyle {\frac {1}{\sqrt {-\delta }}}}$

Comparison triangle

A comparison triangle in . From follows . ${\displaystyle \mathbb {R} ^{2}=E_{\delta =0}^{2}}$ ${\displaystyle K_{M}\leq 0}$ ${\displaystyle d(x,y)\leq \|x^{\prime }-y^{\prime }\|}$

A geodesic triangle in a Riemannian manifold is a triangle with corners whose three sides are minimizing geodesics . ${\displaystyle \Delta (a,b,c)}$ ${\displaystyle M}$ ${\displaystyle a,b,c\in M}$

Let be an upper bound for the section curvatures in , thus . Then, for every geodesic triangle with sides (especially to any geodesic triangle if ) a comparison triangle in with ${\displaystyle \delta \in \mathbb {R} }$ ${\displaystyle M}$ ${\displaystyle K_{M}\leq \delta }$ ${\displaystyle \Delta (a,b,c)}$ ${\displaystyle d(a,b),d(a,c),d(b,c)\leq {\frac {\pi }{\sqrt {\delta }}}}$ ${\displaystyle \delta <0}$ ${\displaystyle \Delta (a^{\prime },b^{\prime },c^{\prime })}$ ${\displaystyle E_{\delta }^{2}\subset E_{\delta }^{n}}$

${\displaystyle d(a,b)=\lVert a^{\prime }-b^{\prime }\rVert ,\ d(a,c)=\lVert a^{\prime }-c^{\prime }\rVert ,\ d(b,c)=\lVert b^{\prime }-c^{\prime }\rVert }$.

This triangle is unique except for congruence if either or and all side lengths are less than . You then have a comparison image ${\displaystyle \delta \leq 0}$ ${\displaystyle \delta >0}$ ${\displaystyle {\frac {\pi }{\sqrt {\delta }}}}$

${\displaystyle \partial \Delta (a,b,c)\rightarrow \partial \Delta (a^{\prime },b^{\prime },c^{\prime })}$,

which (for example) assigns each point on the side the corresponding point on the side (i.e. the unique point with ), analogously for the other two sides. ${\displaystyle x}$ ${\displaystyle (a,b)}$ ${\displaystyle x^{\prime }}$ ${\displaystyle (a^{\prime },b^{\prime })}$ ${\displaystyle \lVert x^{\prime }-a^{\prime }\rVert =d(x,a)}$

Toponogow's theorem

Lower curvature bounds

Let it be a Riemannian manifold of the sectional curvature for a number . Be ${\displaystyle M}$ ${\displaystyle K_{M}\geq \delta }$ ${\displaystyle \delta \in \mathbb {R} }$

${\displaystyle \Delta (a^{\prime },b^{\prime },c^{\prime })\subset E_{\delta }^{2}}$

a comparison triangle to a geodetic triangle

${\displaystyle \Delta (a,b,c)\subset M}$.

Then applies

${\displaystyle d_{E_{\delta }^{2}}(x^{\prime },c^{\prime })\leq d_{M}(x,c)}$

for everyone . ${\displaystyle x\in \left[a,b\right]\subset \partial \Delta (a,b,c)\subset M}$

Upper curvature bounds

A corresponding sentence applies to the upper curvature bounds, whereby further prerequisites are required here.

Let be a Riemannian manifold of the section curvature . If is simply connected, and if the geodetic triangle has side lengths at most . ${\displaystyle M}$ ${\displaystyle K_{M}\leq \delta }$ ${\displaystyle \delta \leq 0}$ ${\displaystyle M}$ ${\displaystyle \delta >0}$ ${\displaystyle \Delta (a,b,c)\subset M}$ ${\displaystyle {\frac {\pi }{\sqrt {\delta }}}}$

Then applies to the comparison triangle ${\displaystyle \Delta (a^{\prime },b^{\prime },c^{\prime })\subset E_{\delta }^{2}}$

${\displaystyle d_{E_{\delta }^{2}}(x^{\prime },c^{\prime })\geq d_{M}(x,c)}$

for everyone . ${\displaystyle x\in \left[a,b\right]\subset \partial \Delta (a,b,c)\subset M}$

Inferences

From Toponogow's theorem it follows that Hadamard manifolds ( simply connected manifolds of non-positive intersection curvature) are CAT (0) -spaces and all have corresponding properties: they are contractible , two points can be connected by a unique geodesic and for geodesics is Function convex . ${\displaystyle \gamma _{1},\gamma _{2}}$ ${\displaystyle d(\gamma _{1}(t),\gamma _{2}(t))}$

literature

• Chavel, Isaac (2006), Riemannian Geometry; A Modern Introduction (second ed.), Cambridge University Press
• Berger, Marcel (2004), A Panoramic View of Riemannian Geometry, Springer-Verlag, ISBN 3-540-65317-1
• Cheeger, Jeff; Ebin, David G. (2008), Comparison theorems in Riemannian geometry, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4417-5

Web links

• W. Meyer: Toponogov's Theorem and Applications
• J. Eschenburg: Comparison Theorems in Riemannian Geometry
• U. Lang, V. Schroeder: On Toponogov's Comparison Theorem for Alexandrov Spaces