Bonnet-Myers theorem

The set of Myers (after Sumner Byron Myers ) is a mathematical statement of the field of Riemann geometry , a branch of differential geometry . This statement can be understood as a generalization of Bonnet's theorem and is therefore also called Bonnet-Myers ' theorem . For the sake of completeness, Bonnet's theorem is first formulated here, which is named after the mathematician Pierre Ossian Bonnet .

diameter

In order to be able to formulate the theorems of Bonnet and Myers, the concept of the diameter of a Riemann manifold is defined first. Let be a Riemannian manifold with the distance function , for a definition see here . Then you call ${\ displaystyle (M, g)}$ ${\ displaystyle d}$

{\ displaystyle \ operatorname {diam} (M): = \ sup \ {d (p, q) \ mid p, q \ in M ​​\}}

their diameter. It should be noted that the diameter of the sphere with radius is not but . ${\ displaystyle R}$ ${\ displaystyle 2R}$ ${\ displaystyle \ pi R}$

Theorem of Bonnet

Let be a complete , connected Riemannian manifold . All section curvatures are bounded downwards by a positive constant . Then there is a compact space with a finite fundamental group and the diameter of the Riemann manifold is at most . ${\ displaystyle M}$ ${\ displaystyle \ textstyle {\ frac {1} {R ^ {2}}}}$ ${\ displaystyle M}$ ${\ displaystyle \ pi R}$

Myers' theorem

Let be a complete, connected, n-dimensional Riemannian manifold, for which the Ricci tensor for all the inequality ${\ displaystyle (M, g)}$ ${\ displaystyle V \ in TM}$

{\ displaystyle \ operatorname {Ric} (V, V) \ geq {\ frac {n-1} {R ^ {2}}} g (V, V)}

Fulfills. Then is compact, has a finite fundamental group and the diameter is at most . ${\ displaystyle M}$ ${\ displaystyle \ pi R}$

Remarks

Myers' theorem is a generalization of Bonnet's theorem, since a strictly positive cutting curvature results in a strictly positive Ricci curvature.

The paraboloid is connected and complete with that induced by the scalar product of the Riemannian metric. It also has positive section curvature, but is not compact. The paraboloid does not meet the requirements of Bonnet's theorem because its section curvature approaches zero at will. So this example shows that the requirement of a positive section curvature in Bonnet's theorem would not be sufficient. ${\ displaystyle \ {(x, y, z) \ in \ mathbb {R} ^ {3} \ mid z = x ^ {2} + y ^ {2} \}}$ ${\ displaystyle \ mathbb {R} ^ {3}}$

What is remarkable about the theorems of Bonnet and Myers is that they establish a connection between local geometric and global topological properties. So one needs the Riemannian metric for the definition of the corresponding curvature tensor. The global, topological properties here are the compactness of the manifold and the finiteness of the fundamental group. These topological properties are in their definition independent of the Riemannian metric or the differentiable structure and also do not depend on the point of the manifold. Such sentences are therefore called local-global theorems . Other such local / global statements are the Cartan-Hadamard theorem and the Gauss-Bonnet theorem .

Myers' theorem is applied to Einstein's manifolds . For an Einstein manifold with positive scalar curvature , the requirements of the theorem are met. From this it immediately follows that non-compact Einstein manifolds must have negative or vanishing scalar curvatures. ${\ displaystyle (M, g)}$ ${\ displaystyle S}$ ${\ displaystyle \ textstyle \ operatorname {Ric} (X, Y) = {\ frac {1} {n}} Sg (X, Y)}$

literature

John M. Lee: Riemannian Manifolds. An Introduction to Curvature. Springer, New York 1997, ISBN 0387983228
Manfredo Perdigão do Carmo: Riemannian Geometry , Birkhäuser, Boston 1992, ISBN 0-8176-3490-8