The smooth-manifolds tag has no wiki summary.
11
votes
1answer
330 views
Derivation on real analytic manifolds
Let $M$ be a real analytic manifold. By $C^{\omega}(M)$ we mean the algebra of all analytic functions from $M$ to $\mathbb{R}$. Assume that $D$ is a derivation on $C^{\omega}(M)$ .
Is there a ...
1
vote
1answer
63 views
Markov Partitions for toral automorphisms
I know that my question is more practical than theoretical. But, I do know where to look for the theoretical sources.
I want to find a program in the case that it exists (does it?), or to program it. ...
8
votes
2answers
179 views
The boundary of a domain whose interior is diffeomorphic to the ball
We know that there are compact manifolds with diffeomorphic interiors but their boundaries are not homeomorphic (see the question Manifolds with homeomorphic interiors).
My question is about a very ...
1
vote
1answer
71 views
Integration from vector bundles
Let $(E,M,p)$ be a smooth n dimensional vector bundle. Then $(TE,TM,Dp)$ is a 2n dimensional vector bundle. We restrict this bundle to $M\subset TM$. We denote this restricted bundle by $F$, as a ...
1
vote
1answer
74 views
Are these two bundles, stably equivalent?
Let $(E,M,p)$ be a n dimensinal smooth vector bundle where $M$ is a k dimensional manifold. We assign to $M$, two different vector bundles $F_{1}$ and $F_{2}$ over $M$ as follows:
1)$TE$ is a ...
6
votes
1answer
250 views
Can the monoidal structure on manifolds be strictified?
I'm asking this question purely out of curiosity.
Let $\{M_\alpha\}_{\alpha\in A}$ be a collection of closed smooth manifolds, with exactly one in every diffeomorphism class of closed smooth ...
3
votes
1answer
183 views
Exponential mapping versus flow
In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet ...
7
votes
2answers
208 views
Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a ...
-1
votes
1answer
145 views
A closed manifold with a subset with the same ring cohomology
Is there an example of a closed manifold $M$ with a proper subset $A\subset M$ such the inclusion $i:A \to M$ gives a ring isomorphism $i^{*}$ between $\mathbb{Z}$-cohomologies?
In this question ...
3
votes
1answer
178 views
Is any smooth homeomorphism isotopic to a smooth embedding?
Let $f:D^m\to \mathbb{R}^m$ be a smooth map ($D^m$ is the unit ball).
We call $f$ embedding if it is a homeomorphism on the image and the
derivative $D_xf$ is nonsingular at each point $x\in D^m$ ...
0
votes
0answers
63 views
Foliation values of Exotic spheres
In the following question, we defined the foliation values of an smooth manifold;
Foliation values of a manifold
Let $S_{i}$'s, $i\in \{0,1,\ldots,27\}$, be the smooth structures of topological ...
2
votes
0answers
104 views
Foliation values of a manifold
Let $M$ be a smooth n dimensional manifold. The foliation values of $M$, denoted by $F(M)$, is defined as
\begin{equation} F(M)=\{ 1\leq k\leq n\mid \text{there exist an smooth $k$ dimensional ...
1
vote
0answers
79 views
Frobenius rank of a manifold
The rank of an smooth manifold M is defined by Milnor, as follows:
"The maximum number of independent commuting vector fields on M"
For example it is well known that the rank of $S^{3}$ is 1 (Lima, ...
3
votes
0answers
115 views
A symplectic version of critical points
According to the interesting comment of Mohammad F Tehrani, I revise the question as follows:
Assume $n>2$. For what type of compact n dimensional manifolds $M$ we can say:
For every smooth ...
6
votes
1answer
152 views
A lagrangian version of the Withney theorem
Let $M$ be a smooth n dimensional manifold. Is there an smooth embedding $f:M \to \mathbb{R}^{2n}$ which image is a Lagrangian submanifold of $\mathbb{R}^{2n}$?
3
votes
1answer
159 views
Fundamental proof of the baby case of Hofer's theorem about displacement energy
In 1990, Hofer proved that the displacement energy of a standard ball in $C^{n}$ equals it's Gromov area.
Here is the baby case: Consider a smooth bounded function $f:R^{2}\rightarrow R$. Consider ...
0
votes
0answers
46 views
Interpolating between two points on Stiefel manifold
I'm looking for a formula to interpolate between two given matrices from the Stiefel manifold (orthogonal n by k matrices).
I do not know the tangent direction, I only know the start and end points ...
-1
votes
1answer
136 views
$S^n$ admit a real polarization $D\subset TS^n$?
When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$
-2
votes
1answer
97 views
A question on parallelizable manifolds [closed]
Let $M$ be a manifold with the property that $f^{*}(TM)$ is isomorphic to TM, for every diffeomorphism $f$ on $M$. Does this imply that $M$ is parallelizable?
3
votes
1answer
226 views
Totally non parallelizable manifold
Does there exist a manifold M which all iterated tangent bundles are non parallelizable manifolds? That is$ M, TM , T^2(M), \ldots ,T^n(M)\ldots$ are non parallelizable manifold?
What is ...
8
votes
1answer
453 views
Pontryagin numbers on manifolds with an $S^1$-action
Let $M$ is a smooth compact manifold with an $S^1$-action with isolated fixed points. Suppose the representation of $S^1$ at tangent spaces at all fixed points is known. Can one then find all ...
4
votes
1answer
315 views
Quantization of symplectic vector space and choice of lagrangian subspaces
My question is related to Geometric Quantization.
I don't undrestand the philosophy of following assertion
If $(V,\omega)$ be a symplectic vector space then the quantizations of
$V$ ...
6
votes
1answer
245 views
how to obtain a generalized Morse function out of a fiber bundle?
I already asked this question in MSE but did not get any answer/comment yet.
Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, ...
6
votes
0answers
211 views
Space of embeddings of an $n$-ball into an $n$-manifold
Let $M$ be a smooth $n$-manifold without boundary, and let $B$ be the open unit ball in $\mathbb{R}^n$. I am trying to understand the space $\text{Emb}(B,M)$ of smooth embeddings of $B$ into $M$. ...
3
votes
0answers
61 views
Analytic stuctures on $\mathbb R^n$ and the nilpotent ideal of supermanifolds
I have two questions which are somewhat related:
(a) It is a well known result (of Freedman?) in differential topology that $\mathbb R^4$ has exotic smooth structures. Apparently, it is known that ...
11
votes
3answers
589 views
Is the class of n-dimensional manifolds essentially small?
Question: Consider the proper class of all $n$-dimensional smooth manifolds. If we take the equivalence classes where two manifolds are identified if there exists a diffeomorphism between them, is ...
3
votes
1answer
439 views
Is it possible to define a structure of differentiable manifold on the Hilbert cube $[0,1]^\mathbb{N}$?
Is it possible to define a structure of differentiable (smooth) manifold on the Hilbert cube $[0,1]^\mathbb{N}$ ?
Has it been done in the literature?
In textbooks, only the Banach case is treated, ...
12
votes
1answer
190 views
Hyperbolic Manifolds which fiber over the circle
If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by ...
2
votes
1answer
136 views
isotopy classes of embeddings of the torus
Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$?
For each free homotopy classes $\gamma$ of mappings of the circle ...
2
votes
0answers
79 views
Proving that two given functionally structured spaces are isomorphic
The relevant definitions are listed below. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups; and Section 2, Chapter II of Bredon's Topology and ...
9
votes
1answer
340 views
Do partitions of unity exist if we impose additional conditions on the derivatives?
Let $ ~~\cup_{k=-1}^{\infty} U_k = \mathbb{R} $ be an open covering of
$\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to
the cover exists, i.e. there exists smooth
...
6
votes
2answers
212 views
Pseudofree $T^2$ actions on spheres
Is it possible to construct a smooth action of $S^1\times S^1$ on $S^{2n+1}$ ($n\ge 2$) such that no point on $S^{2n+1}$ has an infinite stabilizer?
Note that if such an action exists, it can not be ...
6
votes
2answers
208 views
Fixed component of an $S^1$ action on $S^n$
Suppose $S^1$ is acting smoothly on $S^n$ and $M$ is a connected component of the set of fixed points of the action. What can be said about $M$?
Is it true that $\pi_1(M)=0$? (sorry this first bit ...
8
votes
3answers
333 views
Configuration spaces of the torus
I would like a reference that calculates the rational homology of the unordered configuration spaces of the torus.
2
votes
3answers
211 views
Fatou sets and topological entropy
Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from ...
1
vote
1answer
244 views
The space of generalized complex structures in sense of N.Hitchin is contractible?
Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...
4
votes
1answer
136 views
Lipschitz Approximation to a PW Smooth Map
Suppose I have a triangulated smooth manifold, $\tau : |K| \rightarrow M$ (so that $\tau | _{\sigma}$ is smooth for each $\sigma \in K$), and a piecewise smooth map, $f: M \rightarrow \mathbb{R}^n$. ...
3
votes
2answers
105 views
intersection of Whitney stratifications
Let $X$ be an oriented smooth manifold with dimension $n$. If $U$ and $V$ are two oriented closed submanifolds of $X$ and $U$ is transverse to $V$ in $X$. Then $U\cap V$ (suppose the intersection is ...
0
votes
0answers
94 views
transverse intersection of Whitney stratifications
Let $M$ be a smooth manifold. If $X$ and $Y$ are two Whitney objects, i.e. subsets with a given Whitney stratification, then $X$ and $Y$ are transverse if each stratum of $X$ is transverse to each ...
10
votes
2answers
432 views
Does a smooth homeomorphism of closed manifolds preserve cobordism fundamental class?
Let $f:M\to N$ be a smooth map of closed oriented smooth manifolds which is also a homeomorphism. Let $[M]\in H_\bullet(M;\mathbb Z)$ denote the fundamental class (and similarly for $N$). It is ...
4
votes
2answers
259 views
Is it impossible for the dimension of a topological space to increase under a smooth map?
First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X_1 \cup X_2 \ldots ...
4
votes
1answer
109 views
Hilbert Manifolds and embedding
In the Wikipedia article on Hilber manifolds, it is claimed that every Hilbert manifold can be smoothly embedding onto an open subset of the model Hilbert space. However, no explicit reference is ...
5
votes
0answers
103 views
formal smooth morphism with a formal smooth source
Let $f:X\rightarrow Y$ a morpism between $k$-schemes ( $k$ a field).
We suppose that X is formally smooth and f is formally smooth and surjective.
Do we have that $Y$ is formally smooth?
Or if it's ...
10
votes
1answer
550 views
Does the gluing procedure in Robert Wald’s book *General Relativity* yield a Hausdorff spacetime?
Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity.
Definition 1: A triplet $ (\Sigma,h,k) $ is called an initial data set if $ ...
14
votes
0answers
328 views
Do there exist exotic 4-tori?
More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds?
Related: if such a ...
0
votes
1answer
99 views
A version of implicit function theorem when sections are not everywhere smooth?
Let $V_1, V_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$
a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section
...
1
vote
0answers
109 views
Can one always extend a smooth section defined on a non compact submanifold to the whole manifold, provided it extends continuously to the closure?
Let $V \rightarrow M$ be a smooth vector bundle over a smooth compact manifold $M$
(without boundary) and $X \subset M$ a smooth submanifold of $M$, that is not necessarily closed.
Suppose $s: X ...
5
votes
0answers
204 views
Embedding tower in low codimension
If $F$ is a suitably nice functor from manifolds to spaces, it has a degree $k$ "polynomial" approximation $T_k F$ in the sense of embedding calculus. We set $T_\infty F := \mathrm{holim} T_k F$.
The ...
2
votes
0answers
72 views
Squared displacement function on Lorentzian manifolds
Hi,
Let $\varphi$ be an isometry of a simply connected pseudo Riemannian manifold $M$. The squared displacement function of $\varphi$ is $d^2_{\varphi}(p):=d^2(\varphi (p),p)$, $p\in M$, where $d$ is ...
2
votes
1answer
329 views
Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space?
Suppose that $X$ is a compact, finite dimensional manifold and $Y$ is an infinite dimensional, second countable ($C^\infty$-)Banach manifold. Let $\nu \in \mathbb{N}$.
Question: Is the space ...
