Bishop and O'Neil defined warped product manifolds. Space-times are good examples of such warped product manifolds. Is there a famous and important example of space-times I×M where M is itself a warped product manifold? One can construct many such examples, but I want an example from physics where the fiber of the space-times is a warped product manifold?
Kindly refer to the importance of the example in brief. A good reference is fine and enough.
Thanks in advance.
The standard space-times of string theory are of the form Minkowski\times Calabi-Yau manifold, where the Calabi-Yau is a Ricci flat Riemannian Manifold (with holonomy in SU(n) i.e. admitting a complex structure). Variations are de Sitter \times Calabi-Yau or Anti-de Sitter\times Calabi-Yau.
In particular, since de Sitter is a warped product R \times S^3 we get a warped product decomposition
de Sitter \times Calabi-Yau \iso R \times (3-Sphere \times Calabi-Yau)
with S^3 \times Calabi-Yau itself a warped product.
Thank you Rogier Brussee for your nice answer. Is it possible to give me a real one. As google answered me, Calabi-Yau manifold is a complex manifold and I wand a real one. Thanks again.
Every n dimensional complex manifold is in particular a 2n dimensional real differentiable manifold.
I have never really understood how string theory comes up with holonomy in SU(3). A quick google on Wikipedia mentions it leaves some of the original supersymmetry. What it does do however is provide a great supply of non trivial Ricci flat compact manifolds through Yau's resolution of the Calabi conjecture. Not every compact manifold admits a Ricci flat metric (e.g. it implies that the manifold is scalar flat and so every solution of the Dirac equation is parallel constant by the Weitzenbock formula. In dimension 4 this leads to obstructions on the differentiable structure from Seiberg Witten theory). Even if the manifold has a Ricci flat metric finding one is not easy. However, a complex manifolds admits a Ricci flat Kaehler metric iff it is Calabi Yau (Yau's resolution of the Calabi conjecture), i.e. has a trivial canonical line bundle. The latter is condition is something algebraic geometers have a good handle on and it is easy to produce many examples. For example, if follows from the so called adjunction formula (https://en.wikipedia.org/wiki/Adjunction_formula) that the simplest examples of a Calabi Yau's of dimension n is a generic hyper surface of degree n+1 in n dimensional projective space.
One of the standard models of space times are the Robertson-Walker spaces, (-I)X_f M, where I is a real interval , -I indicates it has the negative metric, and M is a constant-curvature real manifold. All of the complete simply-connected constant-curvature spaces are warped products: Euclidean spaces are direct products of Euclidean spaces; de-Sitter space is -R X_cosh S^3 and the constant curvature sphere can be written [-\pi/2,\pi/2] X_cos S^2, etc. In a paper by me and S. Alexander give a description of the big-bang Robertson-Walker spaces :
(with S. B. Alexander), Lorentz and semi-Riemannian spaces with Alexan-
drov curvature bounds, Comm. Anal. Geom. 16 (2008), 251-282.
Dear Richard L. Bishop
I am extremely happy to see your answer and I am very grateful for your help.
Would you kindly answer this question
https://www.researchgate.net/post/In_spacetimes_what_is_the_most_general_metric_of_the_existing_ones
Schwarzchild space-time is a warped product I x_f M such that M is again a warped product.
By the way, I shall also mention that the Reissner-Nordstrom space-time is also such an example.
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