Yes, in 2 dimensions the Gaussian curvature (K) can be directly derived from the Riemann curvature (R) by K=-R.
Thus, there is no isometry between those two spaces. But conformal transformations are only angle preserving and do not preserve distances. So I believe that there must be a conformal...
i) Sorry, I posted the wrong metrics. Here are the correct two metrics:
http://tbf.me/a/Bu4jVZii) Here are the two penrose diagrams (I believe they are correct):
http://tbf.me/a/BntYLL
In both diagrams the red line is a horizon (chronology/ cauchy).
Could someone help me read/ compare the...
They are not asymptotically flat because of the identification of one coordinate; I call it asymptotically flat under identification. But without the identification they would be both asymptotically flat.
By 2-dimensions I mean a 2-dim manifold with Lorentz metric (-,+).
As a mathematician, I am not familiar with the Penrose diagrams, so I didn't draw them.
I thought there might be a necessary requirement/ premise or assumption so that we can conclude a "conformal relation".
But, what kind of...
I have two 2-dimensional space-times. One of them is flat the other one has not-vanishing curvature (Riemann tensor). But they seem to have a similar global and causal structure.
Of course, because of the 2-dimensional case they are local conformally flat.
I am looking for a relation between...
This means then that it doesn't make sense to calculate the shape operator at all?
Or is there a canonical embedding that could be picked (e.g. 3-dim Euclidean/ Minkowski space)?
The computation would depend on the embedding and thus the result would not deliver any general information that...
Thanks for your answer!
I am actually interested in the 2-dimensional (u,v)- "Schwarzschild" manifold (surface) defined by the metric above.
From the shape operator (e.g. the eigenvalues = principal curvatures) you can derive the Gaussian curvature and the mean curvature. I computed the Ricci...
Hello:
I would like to understand how to compute the shape operator (and eigenvalues etc) for a complex example like the Schwarzschild spacetime. It's easy for a submanifold in Euclidean space, but I don't know how to do it for the more advanced examples like the schwarzschild spacetime in...