If curvature were an exact differential

In summary, the conversation discusses the concept of curvature being an exact differential and its relationship to the possibility of solving for the metric. The topic of Calabi-Yau manifolds and their properties is also brought up. The conversation concludes with a clarification on the different definitions and contexts of curvature.
  • #1
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?
 
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  • #2
I'm wondering what happens if you put the gaussian curvature in Stoke's theorem. So the curvature would have to be an exact differential. This might be physically important since the curvature of spacetime has a connection with energy and therefore mass, etc.
 
  • #3
Mike2 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?
the curvature is exact on a Calabi-Yau
 
  • #4
lethe said:
the curvature is exact on a Calabi-Yau
Interesting, I didn't know that. Thank you.

Are the 3 expanded dimensions part of the Calabi-Yau manifolds?
 
  • #5
Mike2 said:
Interesting, I didn't know that. Thank you.

Are the 3 expanded dimensions part of the Calabi-Yau manifolds?
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

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".
 
  • #6
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
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.


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".
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.
 
  • #7
Mike, I think you are confusing a few things here.

Curvature as a word is meaningless, unless you specify a connection and a context (is it extrinsic or intrinsic).

A metric space then may or may not be possible given your choice of connection on some topology.

I think what you are thinking of is the notion of gaussian curvature, and other related things like mean curvature, etc. This typically pressupposes a specific connection and an associated metric. Then you can play around with things.

Theres other ways to think about 'curvature' and the term is often used differently here, for instance you can specify a topological invariant, like the Chern class.. And this kinda tells you a little something about how your space acts independantly of the connection. Make sure that you keep the definitions in context.
 
  • #8
What curvature was it that uncurled as the universe expanded? Was it the curvature described by general relativity? Was it the gaussian curvature, what?

I do appeciate your responses, thanks.
 

1. What does it mean for curvature to be an exact differential?

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.

2. How is curvature related to differential geometry?

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.

3. Can you give an example of an exact differential in real life?

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.

4. What are the implications of curvature being an exact differential?

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.

5. Is it possible for curvature to be an exact differential?

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.

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