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# Myers ' theorem states that if the Ricci curvature is bounded from below on a complete Riemannian manifold by, then the manifold has diameter, with equality only if the manifold is isometric to a sphere of a constant curvature k. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite fundamental group.
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# and Myers
# Peng H., Ruan, Z, Long, F, Simpson, JH, Myers, EW: V3D enables real-time 3D visualization and quantitative analysis of large-scale biological image data sets.
# and theorem
# Yoga of the Grothendieck – Riemann – Roch theorem ( K-theory, relation with intersection theory ).
# X has a sub-base such that every cover of the space by members of the sub-base has a finite subcover ( Alexander's sub-base theorem )
# Reducing the theorem to sentences ( formulas with no free variables ) in prenex form, i. e. with all quantifiers ( and ) at the beginning.
# If A is a Lebesgue measurable set, then it is " approximately open " and " approximately closed " in the sense of Lebesgue measure ( see the regularity theorem for Lebesgue measure ).
There is no formal distinction between a lemma and a theorem, only one of intention – see Theorem # Terminology.
One consequence of Toda's theorem is that a polynomial-time machine with a # P oracle ( P < sup ># P </ sup >) can solve all problems in PH, the entire polynomial hierarchy.