Recent content by Reverie

1.) “There are three broad aspects to our debt crisis.” America is not in a debt crisis. Public debt as a percentage of GDP indicates how heavily a country is indebted in comparison to its ability to pay as measured by its GDP. Here is a Wikipedia article with the computed ratios for many...

I would start by examining an introductory book on Riemannian Geometry. One possibility is Riemannian Geometry written by Gallot, Hulin, and Lafontaine. I recommend doing some calculations and understanding some examples. The field of geometry is motivated by visual "intuitions", but not...

R^(n+1)\{0} is a subset of R^(n+1). You can explicitly write down a global trivialization for R^(n+1) that restricts to a global frame for R^(n+1)\{0}. A global frame for R^(n+1) is... (1,0,...,0) (0,1,0,...,0) . . . (0,...,0,1)

Let f:S^1-->R be any smooth map and consider Im(f) subset of R. Let m=max Im(f). At any point x in the inverse image of m, the map f will not be a submersion. Take local coordinates c around the point x. Then, f compose c:(-e,e)-->R is not a submersion. Hence, f is not a submersion. The...

What you wrote is wrong and a bit difficult to read. 1.)A submersion from S^1 -> R does not exist. 2.)A submersion from R -> S^1 does exist. 3.)Your definition of submersion: "By a submersion from M to N, I mean a map f:M-->N whose tangent map is surjective." is missing a key element...

Do any theories of gravity exist other than general relativity that are capable of explaining the perihelion of mercury's orbit? In particular, I would like to know if a theory of gravity exists that does not impose the fact that nothing can exceed the speed of light in a vacuum. Newton's theory...

This is the multivariate version of the second derivative test from calculus. If the second derivative is positive you are at a minimum. If the second der is Negative you are at a maximum. Let D be the discriminant matrix, and h a 2x1 column vector. At a minimum h^T*D*h>0 for all small h...

The operations are linear on each fiber. So, if you solve w(Y)=0 and find one X such that w(X)=f, then X+Y is such that w(X+Y)=f. The question is not optimally formulated, and it is a little unclear why you are asking this question. Do you have an application in mind? Are you reading a proof...

a differential 1-form on a manifold acting on a vector field on a manifold yields a function.

Or none. Let w=0(the 0 1-form). Let f be a non-zero function.

When one index is raised and one index is lowered, the summation symbol is omitted. There is nothing more to it. The main difficulty is accepting that such a convention is actually useful. It is useful because Einstein cleverly chose to make some indices raised and some indices lowered.

As was mentioned previously, the physical significance depends on the application. Maybe this explanation will help. Let V be a three dimensional vector space with basis {e1,e2,e3}, and let W be a four dimensional vector space with basis {f1,f2,f3,f4}. Then V tensor W is a 12 dimensional...

It appears to be essentially worked out in Scott's paper. I'll post a bit more about it when I've worked it out.

Note to moderators... this is actually a serious question... The reference to Adams' book was for amusement... but the rest is serious... after all... The Answer to The Ultimate Question Of Life, the Universe and Everything...and... I'm really interested in the answer... o:)

Hi, Every Thurston Geometry (X,Isom(X)) with the exception of the geometry modeled on S^2xR can be achieved as a 3D Lie group with a left-invariant metric. That is, the space X=G, where G is a 3D Lie group with a left-invariant metric. After picking a left-invariant frame field consisting of...