Difference between revisions of "Some retraction problems"

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(Created page with "Scale calculus was developed to make reparametrization actions on function spaces smooth. All the key issues and ideas can already be seen at the example of <math>S^1=\R/\Z</...")
 
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== A useful toy retraction ==
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1. State
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2. Investigate smooth paths
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3. Investigate Tangent space at strange point
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== The model lemmas ==
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== Retractions instead of sc-Banach manifolds ==
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1. Transversal constraint construction as retraction
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2. Target manifold as <math>\mathbb{R}^N</math>
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3. Retractions on retracts
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== The <math>\overline{\partial_J}</math> operator as sc-Fredholm ==
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1. Use the original definition of HWZ for sc-Fredholm to prove dbar is.
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2. Show that the linearized d-bar operator at a constant solution is an isomorphism
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Scale calculus was developed to make reparametrization actions on function spaces smooth.  
 
Scale calculus was developed to make reparametrization actions on function spaces smooth.  
 
All the key issues and ideas can already be seen at the example of <math>S^1=\R/\Z</math> acting on a space of nonconstant functions on <math>S^1</math>, for example the shift action
 
All the key issues and ideas can already be seen at the example of <math>S^1=\R/\Z</math> acting on a space of nonconstant functions on <math>S^1</math>, for example the shift action
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the structure of a scale-smooth manifold ... and to see that classical Banach space topologies fail at giving it a differentiable Banach manifold structure.  
 
the structure of a scale-smooth manifold ... and to see that classical Banach space topologies fail at giving it a differentiable Banach manifold structure.  
  
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== differentiability of shift map ==
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== A subtitle ==
  
 
Compute the directional derivatives of the shift map <math>\sigma</math>,  
 
Compute the directional derivatives of the shift map <math>\sigma</math>,  
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== local charts ==  
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== Another subtitle ==  
  
 
Verify that <math>\mathcal{B}</math> has local Banach manifold charts modeled on  
 
Verify that <math>\mathcal{B}</math> has local Banach manifold charts modeled on  

Revision as of 08:50, 12 June 2017


A useful toy retraction

1. State 2. Investigate smooth paths 3. Investigate Tangent space at strange point

The model lemmas

Retractions instead of sc-Banach manifolds

1. Transversal constraint construction as retraction 2. Target manifold as {\mathbb {R}}^{N} 3. Retractions on retracts

The \overline {\partial _{J}} operator as sc-Fredholm

1. Use the original definition of HWZ for sc-Fredholm to prove dbar is. 2. Show that the linearized d-bar operator at a constant solution is an isomorphism


Scale calculus was developed to make reparametrization actions on function spaces smooth. All the key issues and ideas can already be seen at the example of S^{1}=\mathbb{R} /\mathbb{Z } acting on a space of nonconstant functions on S^{1}, for example the shift action

\sigma :S^{1}\times {\mathcal {C}}_{{{\rm {id}}}}^{1}\;\to \;{\mathcal {C}}_{{{\rm {id}}}}^{1},\quad (\tau ,u)\mapsto u(\tau +\cdot )

{\text{on}}\qquad \qquad \qquad \qquad {\mathcal {C}}_{{{\rm {id}}}}^{1}:=\{u:S^{1}\to S^{1}\;|\;u\in {\mathcal {C}}^{1}\;{\text{and homotopic to the identity}}\}.\qquad \qquad \qquad \qquad .

The purpose of the following exercises is to give the quotient of S^1-valued functions homotopic to the identity, modulo reparametrization by shifts, {\mathcal {B}}:={\mathcal {C}}_{{{\rm {id}}}}^{1}/S^{1} the structure of a scale-smooth manifold ... and to see that classical Banach space topologies fail at giving it a differentiable Banach manifold structure.

A subtitle

Compute the directional derivatives of the shift map \sigma , first at (\tau ,u)=(0,0), then at (\tau ,u)\in S^{1}\times {\mathcal {C}}_{{{\rm {id}}}}^{2}.

Next, understand how to identify T{\mathcal {C}}_{{{\rm {id}}}}^{1}\simeq {\mathcal {C}}^{1}(S^{1},\mathbb{R} ), and write down a conjectural formula for the differential d\sigma :\mathbb{R} \times {\mathcal {C}}^{1}(S^{1},\mathbb{R} )\to {\mathcal {C}}^{1}(S^{1},\mathbb{R} ).

Then recall the definition of differentiability in terms of uniform linear approximation and see whether it can be satisfied at any base point (\tau ,u).


YOUR SOLUTION WANTS TO BE HERE


Another subtitle

Verify that {\mathcal {B}} has local Banach manifold charts modeled on E:=\{\xi \in {\mathcal {C}}^{1}(S^{1},\mathbb{R} )\,|\,\xi (0)=0\}. To set up a chart near a given point, pick a representative [u_{0}]\in {\mathcal {B}} with {\dot u}_{0}(0)\neq 0 and consider the map

\phi _{{u_{0}}}:\{\xi \in E\,|\,\|\xi \|_{{{\mathcal {C}}^{1}}}<\epsilon \}\to {\mathcal {B}},\qquad \xi \mapsto [u_{0}+\xi ].


YOUR SOLUTION WANTS TO BE HERE

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