Reduction To Generalized Crystallography
The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form with curvature is isometric to, hyperbolic space, with curvature is isometric to, Euclidean n-space, and with curvature is isometric to, the n-dimensional sphere of points distance 1 from the origin in .
By rescaling the Riemannian metric on, we may create a space of constant curvature for any . Similarly, by rescaling the Riemannian metric on, we may create a space of constant curvature for any . Thus the universal cover of a space form with constant curvature is isometric to .
This reduces the problem of studying space form to studying discrete groups of isometries of which act properly discontinuously. Note that the fundamental group of, will be isomorphic to . Groups acting in this manner on are called crystallographic groups. Groups acting in this manner on and are called Fuchsian groups and Kleinian groups, respectively.
Read more about this topic: Space Form
Famous quotes containing the words reduction and/or generalized:
“The reduction of nuclear arsenals and the removal of the threat of worldwide nuclear destruction is a measure, in my judgment, of the power and strength of a great nation.”
—Jimmy Carter (James Earl Carter, Jr.)
“One is conscious of no brave and noble earnestness in it, of no generalized passion for intellectual and spiritual adventure, of no organized determination to think things out. What is there is a highly self-conscious and insipid correctness, a bloodless respectability submergence of matter in manner—in brief, what is there is the feeble, uninspiring quality of German painting and English music.”
—H.L. (Henry Lewis)