The Mudcat Café TM
Thread #59418 Message #3321764
Posted By: Amos
12-Mar-12 - 12:09 PM
Thread Name: BS: The Mother of all BS threads
Subject: RE: BS: The Mother of all BS threads
Here's the latest dope on transcendental dimensionality.
"A Calabi–Yau manifold is a special type of manifold that shows up in certain branches of mathematics such as algebraic geometry, as well as in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold.
Calabi–Yau manifolds are complex manifolds that are higher-dimensional analogues of K3 surfaces. They are sometimes defined as compact Kähler manifolds whose canonical bundle is trivial, though many other similar but inequivalent definitions are sometimes used. They were named "Calabi–Yau spaces" by Candelas et al. (1985) after E. Calabi (1954, 1957) who first studied them, and S. T. Yau (1978) who proved the Calabi conjecture that they have Ricci flat metrics. In superstring theory the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry.
There are many different inequivalent definitions of a Calabi–Yau manifold used by different authors. This section summarizes some of the more common definitions and the relations between them.
A Calabi–Yau n-fold or Calabi–Yau manifold of dimension n is sometimes defined as a compact n-dimensional Kähler manifold M satisfying one of the following equivalent conditions:
The canonical bundle of M is trivial.
M has a holomorphic n-form that vanishes nowhere.
The structure group of M can be reduced from U(n) to SU(n).
M has a Kähler metric with global holonomy contained in SU(n).
These conditions imply that the first integral Chern class c1(M) of M vanishes, but the converse is not true. The simplest examples where this happens are hyperelliptic surfaces, finite quotients of a complex torus of complex dimension 2, which have vanishing first integral Chen class but the canonical bundle is not trivial.
For a compact n-dimensional Kähler manifold M the following conditions are equivalent to each other, but are weaker than the conditions above, and are sometimes used as the definition of a Calabi–Yau manifold:
M has vanishing first real Chern class.
M has a Kähler metric with vanishing Ricci curvature.
M has a Kähler metric with local holonomy contained in SU(n).
A positive power of the canonical bundle of M is trivial.
M has a finite cover that has trivial canonical bundle.
M has a finite cover that is a product of a torus and a simply connected manifold with trivial canonical bundle."
What I want to know, Mom, is what happens to people that makes 'em talk like that? Is it a parasite of some kind?