Difference between revisions of "Some retraction problems"

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== Independence of choice of retraction ==
 
== Independence of choice of retraction ==
  
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Let <math>\mathbb{E}</math> be some sc-Banach space, and suppose <math>r:\mathbb{E}\to \mathbb{E}</math> is an sc-smooth retraction, with <math>O = r(E_0)</math>.
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Let <math>\mathbb{E}</math> be some sc-Banach space, and suppose <math>r:\mathbb{E}\to \mathbb{E}</math> is an sc-smooth retraction, with <math>O = r(E_0)</math>. Recall that if <math>\mathbb{F}</math> is a sc-Banach space, then a function <math>f:O\to \mathbb{F}</math> is sc-smooth if and only if <math>f\circ r : \mathbb{E}\to \mathbb{F}</math> is sc-smooth.  Suppose <math>\rho:\mathbb{E}\to \mathbb{E}</math> is an sc-smooth retraction with the property that <math>\rho(E) = O</math>. 
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'''Question''': Is <math>f\circ \rho: \mathbb{E}\to \mathbb{F}</math> sc-smooth?
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'''Question''': The sc-differential structure
  
 
== The model lemmas ==
 
== The model lemmas ==

Revision as of 13:17, 13 June 2017


A useful toy retraction

Fix a non-negative function \beta \in {\mathcal {C}}_{0}^{\infty } for which \|\beta \|_{{E_{0}}}=\|\beta \|_{{L^{2}}}=1. We consider the sc-Banach space {\mathbb {E}}={\bigl (}H^{{k,\delta _{k}}}({\mathbb {R}},{\mathbb {R}}){\bigr )}_{{k\in {\mathbb {N}}_{0}}} with \delta _{0}=0. Define a family of linear projections \pi _{t}:E_{0}\to E_{0} for t\in \mathbb{R} by L^{2}-projection onto the subspace spanned by \beta _{t}:=\beta (e^{{1/t}}+\cdot ) for t>0 respectively \beta _{t}:=0 for t\leq 0. The corresponding retraction

{\mathbb {R}}\times {\mathbb {E}}\to {\mathbb {R}}\times {\mathbb {E}},\qquad (t,e)\mapsto (t,\pi _{t}(e))={\begin{cases}{\bigl (}t,\langle f,\beta _{t}\rangle _{{L^{2}}}\beta _{t}{\bigr )}&;t>0\\(t,0)&;t\leq 0\end{cases}}

is sc^{\infty } (see Lemma 1.23 in the HWZ sc-smoothness paper)) and a retraction (in fact, a splicing).

Question: What is the sc-retract of this sc-smooth retraction? Describe this space (somewhat) geometrically.

Question: Find a subset of {\mathbb {R}}^{2} homeomorphic to this sc-retract. Hint: It will contain the line {\mathbb {R}}\times \{0\}.

Question: Which paths of the form t\mapsto (t,\gamma _{t})\in {\mathbb {R}}\times {\mathbb {E}} are contained in the sc-retract and are sc-smooth? Can these paths be described via the above homeomorphism of the retract to {\mathbb {R}}^{2}?

Question: What is the tangent fiber of the sc-retract at the point (0,0)? Can all points in this fiber be achieved as tangent vectors to paths in the retract passing through (0,0)? Is the tangent vector at (0,0) of a path in the sc-retract always contained in the tangent fiber of the retract at (0,0)?

Independence of choice of retraction

Let {\mathbb {E}} be some sc-Banach space, and suppose r:{\mathbb {E}}\to {\mathbb {E}} is an sc-smooth retraction, with O=r(E_{0}). Recall that if {\mathbb {F}} is a sc-Banach space, then a function f:O\to {\mathbb {F}} is sc-smooth if and only if f\circ r:{\mathbb {E}}\to {\mathbb {F}} is sc-smooth. Suppose \rho :{\mathbb {E}}\to {\mathbb {E}} is an sc-smooth retraction with the property that \rho (E)=O.

Question: Is f\circ \rho :{\mathbb {E}}\to {\mathbb {F}} sc-smooth?

Question: The sc-differential structure

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 4. Does smoothness depend on choice of retraction?

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