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Results tagged with differential-topology
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user 148161
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
44
votes
4
answers
3k
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Do rings of smooth functions differ from rings of continuous functions?
Let $M$, $N$ be connected nondiscrete compact smooth manifolds. Can the ring of continuous functions on $M$ be isomorphic to the ring of smooth functions on $N$?
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13
votes
1
answer
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Are there textbooks on differential geometry in the language of smooth sets or smooth derive...
In differential geometry it is often natural to speak of infinite-dimensional manifolds (e.g., the manifold of mappings between two smooth manifolds). Different versions of generalized smooth spaces a …
8
votes
1
answer
385
views
Is it possible to define contact manifolds as manifolds with a G-structure?
Many geometries (Riemannian, symplectic, complex, Kähler, Calabi-Yau) can be defined as categories of G-structures on manifolds with the first integrability condition (zeroing of torsion of G-structur …
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9
votes
2
answers
370
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Cartesian-closed full subcategory of locally ringed spaces containing smooth manifolds
This coming fall, I will be teaching a course on differential topology to a small group of strong students. In preparation for it, I'm trying to find a category $\mathrm{GDiff}$ with the following pro …
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6
votes
0
answers
172
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Is the category of diffeological spaces a full subcategory of locally ringed spaces?
It is known that the natural functor of smooth functions from the category of smooth manifolds into the category of locally ringed spaces is a full embedding (see, for example, here).
Is a similar fu …
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11
votes
1
answer
961
views
Intuition/meaning behind/physical content of the concept of a smooth structure
Some mathematical structures are visualized very well. I imagine how a shapeless bunch of points (a set; the only property of which is quantity) is collected in one or another soft form (topological s …
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10
votes
2
answers
650
views
Homotopy properties of Lie groups
Let $G$ be a real connected Lie group. I am interested in its special homotopy properties, which distinguish it from other smooth manifolds
For example
$G$ is homotopy equivalent to a smooth compact …
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2
votes
0
answers
74
views
Is the reversibility of inflation of a subset equivalent to its smoothness?
$D_r(x)$ denotes a closed ball of radius $r$ centered at $x$.
Definition. Let $M \subset \mathbb{R}^n$.
$D_r (M): = \bigcup\limits_{x \in M} D_r (x)$
$Int_r (M): = \{x ~|~ D_r(x) \subset M\}$
Questi …
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4
votes
1
answer
291
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Classification of functorial smooth vector fiber bundles
Let $\mathrm{Bundle}$ be the category whose objects are smooth vector fiber bundles over $\mathbb{R}$, and morphisms are fiberwise smooth linear map (that is, the base is not assumed to be fixed).
Let …
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