- #1
ivl
- 27
- 0
Hi all,
First of all I would like to apologize, as my knowledge of differential geometry is not as good as it should be (I only took an introductory course for physicists).
Anyhow, here is a question for you. Any help is greatly appreciated!
-I have a four dimensional manifold, whose properties are rather banal (for instance, all closed differential forms are also exact).
-On this manifold, I am given a nowhere-vanishing 1-form (I call it "dt").
-I would like to know if this nowhere-vanishing exact 1-form "dt" determines a foliation (the folia are t=const surfaces).
An attempt to an answer: dt is an exact 1-form, and t is a function (which maps points in the manifold to real numbers). Now, if dt is nowhere zero, t is a monotonic function. The surfaces of t=const are thus folia which are monotonically labeled.
Related topics: Frobenius theorem, space+time foliation
First of all I would like to apologize, as my knowledge of differential geometry is not as good as it should be (I only took an introductory course for physicists).
Anyhow, here is a question for you. Any help is greatly appreciated!
-I have a four dimensional manifold, whose properties are rather banal (for instance, all closed differential forms are also exact).
-On this manifold, I am given a nowhere-vanishing 1-form (I call it "dt").
-I would like to know if this nowhere-vanishing exact 1-form "dt" determines a foliation (the folia are t=const surfaces).
An attempt to an answer: dt is an exact 1-form, and t is a function (which maps points in the manifold to real numbers). Now, if dt is nowhere zero, t is a monotonic function. The surfaces of t=const are thus folia which are monotonically labeled.
Related topics: Frobenius theorem, space+time foliation