One-parameter family of metrics

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In summary, the two metrics, g and h, are associated with different curvature tensors, but the sum of them does not always lead to a non-degenerate metric.
  • #1
mach4
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I have a manifold M=S^4 which is endowed with a physical metric g.
I can define another metric on this manifold h (a pullback metric).

Does it make sense to define a one-parameter family of metrics G(u) on the manifold M in the form

G(u) = (1-u)*g + u*h , where u is a parameter in [0,1] ?

Are there any compatibility conditions?
Any help would be appreciated - Thx!
 
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  • #2
If you're talking about euclidean metrics, then that should work, since the sum of two symmetric positive definite matrices is symmetric and positive definite, and so qualifies as a metric. It won't work for minkoswkian metrics though, since, eg, diag(1,1,1,-1) and diag(1,1,-1,1) are both valid metrics, but their sum is not.
 
  • #3
Thanks for your help!
Both metrics are symmetric positive definite but non-Euclidean.

When I check G for
-symmetry
-bilinearity
-non-degeneracy
all criteria of a metric seemed to be satisfied.
I was just bothered by the fact that g and h are associated with different curvature tensors, but it seems that they simply add to define the new curvature tensor of G.

Did I understand correctly? In the case of the Minkowskian-metrics the 'non-degeneracy' is not satisfied and thus it does not define a metric.
 
  • #4
The curvature is not linear in the metric, so will not simply add. But it's true, you can get a continuous family of metrics with different curvatures (obviously the curvature will then vary continuously over this family). And yes, the problem is that the sum of Minkowski metrics is not necessarily non-degenerate.

By the way, by "Euclidean" I mean a positive definite metric, not a flat one. It's just to distinguish from "Minkowskian".
 
  • #5
uups - you are right. The Riemannian is clearly not linear in the metric. Bad mistake :(.
Thus, the operation of adding two positive definite metric is possible and does not lead to any inconsistencies. Great!
Thanks again for your help!
 

1. What is a one-parameter family of metrics?

A one-parameter family of metrics is a set of metrics or distance functions that are related to each other by a single parameter. This parameter can be continuously varied to obtain different metrics within the same family.

2. What is the purpose of studying one-parameter families of metrics?

One-parameter families of metrics are important in the field of geometry and topology as they allow us to compare and analyze different geometries within a single framework. They also have applications in physics, especially in studying the properties of space-time.

3. How are one-parameter families of metrics used in mathematical modeling?

One-parameter families of metrics are often used in mathematical modeling to represent a continuum of possible solutions to a problem. For example, in optimization problems, the parameter can represent different constraints that result in different optimal solutions.

4. Can one-parameter families of metrics be used to compare non-Euclidean geometries?

Yes, one-parameter families of metrics can be used to compare non-Euclidean geometries. By varying the parameter, we can obtain different metrics that can be used to analyze and compare different non-Euclidean geometries, such as hyperbolic or elliptic geometries.

5. Are there any real-world applications of one-parameter families of metrics?

Yes, there are many real-world applications of one-parameter families of metrics. For example, in economics, one-parameter families of metrics can be used to model and compare different market structures. In engineering, they can be used to optimize designs by varying different parameters. They also have applications in data analysis and image processing.

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