Conditions for Solution to Pullback Equation on Forms?

In summary, the conversation discusses a quotient map given by the mapping torus of a compact surface with nonempty boundary and a homeomorphism, as well as the pullback equation and conditions for solving it. The problem is that the induced map on cohomology is not necessarily onto. The question asks for potential conditions on a 1-form ω for the existence of a form β that pulls back to ω.
  • #1
WWGD
Science Advisor
Gold Member
6,910
10,292
Hi, All:

I have a quotient map given by the mapping torus (S,h) , where S is a compact surface with nonempty boundary, and h: S→S is a homeomorphism. Let I=[0,1].
The mapping torus ## S_h## of the pair (S,h) is defined as the quotient q: $$ q:S \times I/~$$ , where (x,0)~(h(x),1), i.e., we glue ## S \times I## along h.

Now, let ω be a 1-form on S . I'm trying to see under what conditions we can solve the pullback equation:
$$ q* β =ω $$ ,

i.e., I want to find a form β on ## S_f## which pulls-back to ω . Now, the problem is that the induced map ## q*: Hm( S \times I/~)→ S \times I##on cohomology is not necessarily onto -- q is not a homeomorphism, for one, tho the gluing map is almost as nice as can be, since it is a homeomorphism.

Can anyone think of conditions on ω under which there is a form β with ω= q* β ?

Thanks.
 
Last edited:
Physics news on Phys.org
  • #2
Never mind, thanks, I got it.
 

1. What is a pullback equation on forms?

A pullback equation on forms is a type of differential equation that involves pulling back differential forms from a target manifold to a source manifold. It is used in the study of differential geometry and plays a crucial role in understanding geometric structures and transformations.

2. What are the conditions for a solution to a pullback equation on forms?

The main conditions for a solution to a pullback equation on forms are the existence of a smooth map between the source and target manifolds, and the compatibility of the pullback operation with the differential structure of the manifolds. Additionally, certain regularity conditions may also be required for the solution to be well-defined.

3. Can a pullback equation on forms have multiple solutions?

Yes, a pullback equation on forms can have multiple solutions. This is because the pullback operation is not always unique and different choices of pullback maps can lead to different solutions. However, under certain conditions, the solution may be unique.

4. How are pullback equations on forms used in practical applications?

Pullback equations on forms have various applications in mathematics and physics, particularly in the study of differential geometry and geometric structures. They are used to describe transformations between manifolds and are also relevant in topics such as gauge theories, general relativity, and symplectic geometry.

5. What are some common techniques for solving pullback equations on forms?

Some common techniques for solving pullback equations on forms include using integral equations, variational methods, and numerical methods such as finite difference or finite element methods. The choice of technique depends on the specific form of the equation and the desired level of accuracy.

Similar threads

  • Differential Geometry
Replies
7
Views
4K
Replies
6
Views
857
Replies
21
Views
1K
  • Differential Geometry
Replies
9
Views
3K
  • Differential Geometry
Replies
4
Views
3K
  • Differential Geometry
Replies
7
Views
9K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
3
Views
3K
  • Classical Physics
Replies
21
Views
1K
  • Math Proof Training and Practice
Replies
16
Views
5K
  • Differential Geometry
Replies
20
Views
5K
Back
Top