#18175: Implement categories for topological and metric spaces and related
categories
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Reporter: tscrim | Owner: tscrim
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.8
Component: categories | Resolution:
Keywords: geometry, | Merged in:
topology, sd67 | Reviewers:
Authors: Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | 95a30aa57fc62f23a884790b57835d107d8bdeef
public/categories/topological_metric_spaces-18175| Stopgaps:
Dependencies: #18174 #17160 |
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Comment (by bpillet):
Hi,
I believe Almost Complex manifolds are real manifolds with ''added
structure'' (namely a [1,1]-tensor), meanwhile complex manifolds can
either be seen as
* (a) Almost Complex manifold ''satisfying further axioms''
(integrability)
* (b) Manifolds (over '''C''') ''satisfying axioms'' (holomorphic
transition)
The definitions (a) and (b) are equivalent, but the transition from (b) to
(a) is easy meanwhile the transition (a) to (b) is given by the Newlander
Nirenberg theorem and involves solving complicated PDEs which cannot be
made automatic. Therefore, it should be implemented as different objects,
because we cannot get complex coordinates from the knowledge of a
vanishing Nijenhuis tensor.
Another consequence : Complex manifolds must have complex holomorphic
coordinates hence it inherits from the analytic category. But should its
coordinates seen as real analytic or complex analytic ?
* It cannot be complex analytic since complex analytic manifolds are
tautologically complex manifolds.
* But, it should not be real analytic because from the same almost complex
manifold with real analytic coordinates it may be impossible to get our
complex manifold back. Moreover it makes no sense to consider complex
coordinates which are real analytic (for example $\bar{z}$ as a coordinate
on '''C''') because it does not preserve the complex structure on '''C'''
but only preserves the structure of '''R^2'''.
I don't really know anything about manifolds over '''Q''', but I fear
transitions maps should also satisfy some ''rationality conditions'', and
more generally over any field, one have to take the good functions. Hence
making the category looks like `Algebra` might be more difficult.
Another example : over the "ring" of quaternions, what would be the good
functions ? Actually, there are several definitions possible each yielding
different theories.
Basile
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Ticket URL: <http://trac.sagemath.org/ticket/18175#comment:24>
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