Myers's theorem

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

Diameter

In order to formulate the sets of Bonnet and Myers, the term of the diameter of a Riemannian manifold is defined first. Let be a Riemannian manifold with distance function, see the definition here. Then called

Their diameter. It should be noted that the diameter of the sphere having a radius but is not.

Set of Bonnet

Be a complete, coherent Riemannian manifold. All sectional curvatures are bounded by a positive constant downward. Then a compact space with finite fundamental group and the diameter of the Riemannian manifold is at most.

Set of Myers

Be a complete, cohesive, n-dimensional Riemannian manifold, for which the Ricci tensor for all the inequality

Met. Then is compact, has a finite fundamental group and the diameter is at most.

Comments

• The set of Myers is a generalization of the theorem of Bonnet, because from a strictly positive sectional curvature follows the strictly positive Ricci curvature.

• The paraboloid is completely coherent with the scalar product induced by the Riemannian metric. In addition, it has a positive curvature section, however, is not compact. The paraboloid therefore does not meet the conditions of the theorem of Bonnet because its sectional curvature approximates arbitrarily to zero. Thus, this example shows that the requirement of a positive curvature section in the set of bonnet would not be sufficient.

• What is remarkable about the sets of Bonnet and Myers that they establish a link between local geometric and global topological properties. So you need the Riemann 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 independent of the Riemannian metric, or differentiable structure in their definitions and also do not depend on point of the manifold. Such sentences are therefore called local- global theorems. Other such local / global statements are the set of Cartan -Hadamard and the Gauss -Bonnet.

• One application of the theorem of Myers at Einstein manifolds. For an Einstein manifold with positive scalar curvature, the conditions of the theorem satisfied. It follows immediately that non-compact Einstein manifolds must have a negative or vanishing scalar curvature.