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Science Forum Index » Physics - Research Forum » Simple geometry question...
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| Stathis... |
 Posted: Sun Jun 01, 2008 5:41 pm |
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Guest
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Hello group.
I have two questions concerning surfaces.
If you have a surface, let's say a sphere, and you know that it has
certain "external" geometric properties such as global symmetries (in
the case of the sphere it is obvious), what is the mathematical
technique (if there is a fairly "standard" one) to form a differential
equation or other relation that will help me derive an ordinary
representation of this surface, or a class of ordinary
representations?
The second one is about metrics. When the off-diagonal terms of a
metric are zero
(i.e g_i_j=0), how does this relate to the symmetries of the surface
or in general any global properties it might have? The sphere is an
example, but I'm asking about the more general case.
Thank you very much!
Stathis Artis |
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| Igor Khavkine... |
| Posted: Wed Jun 04, 2008 12:13 pm |
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On Jun 1, 11:41 pm, Stathis <stathis.ar... at (no spam) gmail.com> wrote:
Quote: I have two questions concerning surfaces.
If you have a surface, let's say a sphere, and you know that it has
certain "external" geometric properties such as global symmetries (in
the case of the sphere it is obvious), what is the mathematical
technique (if there is a fairly "standard" one) to form a differential
equation or other relation that will help me derive an ordinary
representation of this surface, or a class of ordinary
representations?
What exactly do you mean by an "ordinary representation"? The answer
to your question hinges on when precisely you are looking for. Let me
make a guess here and suppose that you are interested in so-called
symmetric spaces (the sphere is one of them), which are defined by the
property of being as symmetric "as possible". Here's a book that
covers this subject in much detail:
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces
(AMS, 2001)
Quote: The second one is about metrics. When the off-diagonal terms of a
metric are zero
(i.e g_i_j=0), how does this relate to the symmetries of the surface
or in general any global properties it might have? The sphere is an
example, but I'm asking about the more general case.
When a space-time has certain symmetries (Killing vectors), it is
sometimes possible to find coordinate systems in which some of the off-
diagonal components of the metric vanish. You may wan to look up
Wald's or Misner, Thorn & Wheeler's books on General Relativity for
details (look up Killing vector in the index).
On the other hand, the vanishing of off-diagonal components of the
metric does not necessarily imply the presence of symmetries. For
example, at any given point, we can choose a coordinate system such
that the metric takes on Minkowski form at that point (but not
necessarily anywhere else). Another example, is the proper time gauge
in the Hamiltonian formulation of GR. This gauge requires a choice of
coordinate such that the time-space, g_0i, i=1,2,3, components of the
metric vanish. In principle, it is possible to find such a coordinate
system in any small open set (and perhaps even globally, although
there might be global obstructions).
If this doesn't answer your question, you may want to follow up and
clarify it.
Hope this helps.
Igor |
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