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Mike2
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If curvature were an exact differential so that the derivative of the curvature were zero, then is it possible to solve for the metric?
the curvature is exact on a Calabi-YauMike2 said:If curvature were an exact differential so that the derivative of the curvature were zero, then is it possible to solve for the metric?
Interesting, I didn't know that. Thank you.lethe said:the curvature is exact on a Calabi-Yau
the expanded 3 dimensions? huh? what expanded 3 dimensions? I have no idea what you are asking, so i guess i will just say some stuff about Calabi-YausMike2 said:Interesting, I didn't know that. Thank you.
Are the 3 expanded dimensions part of the Calabi-Yau manifolds?
If I remember right, space-time is suppose to be divided between 3 expanding spatial dimensions, 1 time dimension, and 6 compactified spatial dimensions. I wasn't sure whether the expanding 3 dimension were part of the Calabi-Yau manifold or not. It sounds to me from your answers below that it does not.lethe said:the expanded 3 dimensions? huh? what expanded 3 dimensions? I have no idea what you are asking, so i guess i will just say some stuff about Calabi-Yaus
I was talking in theory can we find the metric given an exact curvature. Or whether there was some math that tells us that there is not sufficient information to get the metric given an exact differential. Perhaps we also need some boundary conditions.A Calabi-Yau manifold is a Kähler manifold whose first Chern class vanishes. by a theorem of Yau, this implies that the Ricci tensor also vanishes. Calabi-Yaus are compact.
In general, we do not know how to find the metric for a Calabi-Yau, so the answer to the original question "can we solve for the metric if the curvature is exact" is "no".
If curvature were an exact differential, it would mean that it is a mathematical concept that can be measured and described precisely. This would allow for accurate calculations and predictions.
Curvature is a fundamental concept in differential geometry, which is the branch of mathematics that studies the properties of curves and surfaces in space. It is used to quantify the amount of bending or twisting of a curve or surface at a given point.
One example of an exact differential in real life is the change in temperature over distance. Temperature is an exact differential because it can be measured precisely at any point in space and time.
If curvature were an exact differential, it would have significant implications in fields such as physics and engineering. It would allow for more accurate and precise calculations, leading to improved designs and predictions.
Yes, it is possible for curvature to be an exact differential in certain situations. For example, in a perfectly symmetrical and uniform space, the curvature may be an exact differential. However, in most real-world scenarios, curvature is not an exact differential and is instead described as a non-exact differential.