**2**

votes

**1**answer

34 views

### Can the Laplace operator be represented as a sum of second order derivational operators

Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated to the metric is denoted by $\Delta$.
Are there $n$ vector fields $X_{1},X_{2}, \ldots, X_{n}$ ...

**3**

votes

**1**answer

118 views

### Smooth manifolds for which every metric is geodesically convex

Are there non compact smooth manifolds which have the property that every Riemannian metric is geodesically convex?
Note that a manifold for which every Riemannian metric is complete must be compact.
...

**32**

votes

**7**answers

3k views

### What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$

Hello,
I would like to know if there is a known necessary and sufficient
property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ :
For example :
1) Are all ...

**1**

vote

**0**answers

74 views

### How close (Homology-wise) can we approximate a topological manifold with a PL or smooth one?

Sorry if this question is to naive or badly phrased. I am curious about the following problem, given a manifold $M$, how "close" can we find a smooth or PL manifold, $N$, with a map $f:N\to M$. The ...

**1**

vote

**1**answer

32 views

### Tube formula for r-neighbourhood of a manifold

Let $P$ be a topologically embedded submanifold in a Riemannian manifold $M$. Then the tube $T(P, r)$ of radius $r \geq 0$ about $P$ is the set of all points $m \in M$ such that there exists a ...

**3**

votes

**1**answer

89 views

### Generalized functions on a product of two manifolds

Let $X,Y$ be smooth compact manifolds. Let $C^\infty(X)$ and $C^{-\infty}(X)$ denote the spaces of smooth and generalized functions on $X$ respectively. We have the obvious canonical linear map ...

**8**

votes

**2**answers

292 views

### Homeo-Fixed point property

Edit: According to comment of Michał Kukieła I revised the question
A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point.
...

**0**

votes

**1**answer

293 views

### Dimension of two homotopy equivalent manifolds [closed]

Let $M,N$ be a closed (connected, without boundary, say smooth) manifolds which are homotopy equivalent. Does it follows that they are of the same dimension? One should be aware of examples of ...

**3**

votes

**1**answer

240 views

### What is the (co-)homology of $K(\mathbb{R}_\delta,n)$?

To elaborate on the question from the title, $\mathbb{R}_\delta$ is the additive group of real numbers (without any topology) and $K(\mathbb{R}_\delta,n)$ is an Eilenberg-MacLane space. I would like ...

**2**

votes

**1**answer

188 views

### Construction of appropriate Morse functions

I am interested in the properties of connectedness of level sets of Morse functions. Let $M$ a compact smooth $n$-manifold, and $1\leq k<n$. Is it possible to construct $k$ Morse functions ...

**1**

vote

**1**answer

113 views

### Reference request: $\mathcal{C}^\infty_c(M)$ is a topological vector space with the Whitney topology

Let $M$ be a smooth manifold and let $\mathcal{C}^\infty(M)$ be the space of all smooth real valued functions with the (strong) Whitney topology. This space is a topological vector space iff $M$ is ...

**4**

votes

**1**answer

103 views

### Differential Operators On A Curve And On Osculating Circle

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...

**5**

votes

**1**answer

166 views

### Automorphism group of a fiber bundle surjects onto diffeomorphism group?

This should surely be well-known by I have not been able to find a good reference to the following question: Given a smooth fiber bundle $\pi\colon P \longrightarrow M$ over a smooth manifold $M$ with ...

**2**

votes

**0**answers

325 views

### Differential and pre-differential of Jacobi identity

Let M be a manifold.
To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied?
That is a Lie algebra structure for which $[X,fY]=f[X,Y]$.
(For ...

**3**

votes

**1**answer

156 views

### Reference request: Intrinsic definition of the strong Whitney topology on $\mathcal{C}^{\infty}(M,\mathbb{R})$ without using charts or jets

Let $M$ and $N$ be smooth manifolds. There is a description of the strong Whitney topology on $\mathcal{C}^{\infty}(M,N)$ in terms of partial derivative in charts (using locally finite sets of charts ...

**4**

votes

**2**answers

180 views

### Exponential rule for Whitney-$\mathcal{C}^{\infty}$-topology

Let $M,N,X$ be smooth manifolds. Equip the space of smooth functions between two manifolds with the (strong) Whitney- $\mathcal{C}^\infty$-topology.
The evaluation map $$ev\colon ...

**6**

votes

**1**answer

180 views

### Closed orientable 4-manifold with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1\times H^1\to H^2$

I am looking for an example of a closed orientable 4-manifold $M$ with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1(M;\Bbb Z_2)\times H^1(M;\Bbb Z_2)\to H^2(M;\Bbb Z_2)$.
A non-orientable ...

**0**

votes

**0**answers

96 views

### Certain principal bundle structure on $\mathbb{R}^{n} \setminus \{0\}$

I ask this question in MSE and I received no answer, so I repeat it here:
Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a ...

**6**

votes

**1**answer

404 views

### Symplectic reversing diffeomorphisms on a compact symplectic manifold

I Ask this question in MSE and I received interesting comments and ideas. I repeat the question here for more discussion:
Let $(M,\omega)$ be a compact symplectic manifold.
Is there a ...

**1**

vote

**0**answers

51 views

### Taylor expansion in Riemannian foliations

Take:
$M$ a Riemannian manifold, ${X_0}\in M$,
$N_{X_0}$ a submanifold of $M$ going through ${X_0}$,
and $Z \in N_{X_0}$ in a neighborhood of ${X_0}$.
At ${X_0} \in N_{X_0}$, we consider the ...

**17**

votes

**1**answer

475 views

### Differential Topology over $\mathbb{Q}$

I have a specific question in mind, but it requires some explanation and context before it can be formally stated. To summarize it in a sentence, this is it:
Are every two rational manifolds of the ...

**0**

votes

**0**answers

79 views

### a construction on Stiefel manifolds

Are there any references concerning the following space $V(k,N,X)$ and $U(k,N,X)$? And the cohomology of these spaces? Thanks.

**-2**

votes

**1**answer

80 views

### Exponential map and convergence [closed]

I posted this question on Math Stack Exchange, but nobody answered so I decided to ask this question here.
Suppose that $M$ is smooth compact manifold and let $y \to x$. Let also $f \in C^{\infty}(M)$ ...

**19**

votes

**1**answer

924 views

### Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map
$f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that ...

**2**

votes

**1**answer

116 views

### Nowhere vanishing, normalized vector field with bounded derivatives

It is well-known that any non-compact manifold admits a nowhere vanishing vector field. If we have a Riemannian metric we may pick such a vector field and normalize it so that at every point it has ...

**2**

votes

**0**answers

40 views

### Second derivative in auto-parallel equation

I am trying to derive the auto-parallel equation for a curve. Consider a smooth manifold $M$ with chart map $x$, and let $\gamma$ be a smooth curve in $M$. The action of its tangent vector ...

**11**

votes

**1**answer

1k views

### How well can we localize the “exoticness” in exotic R^4?

My question concerns whether there is a contradiction between two particular papers on exotic smoothness, Exotic Structures on smooth 4-manifolds by Selman Akbulut and Localized Exotic Smoothness by ...

**4**

votes

**0**answers

168 views

### manifold branched covering space for orbifolds

An orbifold structure on some topological space $X$ is a covering of $X$ with local quotient charts $V/G$, where $V$ is some connected manifold and $G$ effectively acts on $V$ via a finite group of ...

**-1**

votes

**1**answer

52 views

### Glueing smooth functions give a smooth function if reparametrized [closed]

Given $\mathbf{r}:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{R}^2$ be a $C^{1}$ application, with $\mathbf{r}'(t)=(0,0)\Longleftrightarrow\ t=t_0$, and
$$\mathbf{r}(t)=\begin{cases} \mathbf{r}_1 ...

**1**

vote

**1**answer

76 views

### Cayley Subspaces in a Calibrated 8-Space

Suppose we are given $(\mathbb{R}^8,\Phi)$, where $\Phi$ is the self-dual 4-form that defines $Spin(7)\subset SO(8)$ (Cayley calibration, see Notes on the Octonians, page 23). Now some 4-subspaces $V$ ...

**-1**

votes

**1**answer

118 views

### Uniqueness of a smooth function [closed]

Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in ...

**1**

vote

**1**answer

225 views

### symplectic reduction for pair $(M,D)$

Let $M$ be a symplectic manifold with divisor $D$. Then how can we define symplectic reduction for pair $(M,D)$?

**8**

votes

**1**answer

239 views

### is there a diffeomorphism with only finite orbits but of infinite order?

I asked this in stackexchange, but got no answer, so I am trying here.
Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have the following properties:
All its orbits are ...

**1**

vote

**0**answers

66 views

### Real algebraic surface

Let $f(u,v,z)\in \mathbb{Q}[u,v,z]$ be a polynomial in three variables such that $X_{\mathbb{R}}\subset \mathbb{R}^{3} $ (the associated surface of real solution) is smooth. Suppose that the set of ...

**7**

votes

**3**answers

384 views

### When a homeomorphism is a diffeomorphism w.r.t to a suitable smooth structure?

Assume we have a homeomoprhism $\phi:M\rightarrow M$, where $M$ is a topological manifold which admits at least one smooth structure.
Is it always possible to construct a smooth structure on $M$ ...

**5**

votes

**0**answers

298 views

### Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?

Let $M$ be a smooth compact manifold, $\mathcal{S}=\Gamma(\odot^2T^*M)$ the smooth tame Frechet space of smooth symmetric $2$-covariant tensors, and $\mathcal{M}=\Gamma(\odot^2_+T^*M)$ the smooth tame ...

**2**

votes

**0**answers

37 views

### References to study Weak and Strong Topologies and aproximations on function spaces of manifolds

I´m studing weak and strong topologies and aproximations on the function space $C^{\infty}(M,N)$ of two manifolds $M$ and $N$. I´m using the book Differential Topology of Morris Hirsch but it is a ...

**6**

votes

**2**answers

291 views

### Charts needed for an atlas

I just read this question link and asked myself, if there is any easy way to decide how many charts you actually need to cover a given compact manifold in $\mathbb{R}^3$, maybe at least in this ...

**20**

votes

**0**answers

600 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 ...

**3**

votes

**1**answer

85 views

### Polygons with centroid at origin and vertices on compact codimension one submanifold of $\mathbb{R}^{n}-\{0\}$

Let $M$ be a compact codimension one submanifold of $\mathbb{R}^{n}$ which does not contaion $0$ and the origin lies in the bounded component of$\mathbb{R}^{n}-\{0\}$.
Is it true to say that:
...

**2**

votes

**0**answers

134 views

### (co)limits in the category of diffeological spaces vs. category of smooth manifolds

I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...

**9**

votes

**0**answers

213 views

### Reference for a proof of the fiberwise Stokes theorem

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary,
the difference between the fiberwise integral of the differential and the ...

**3**

votes

**0**answers

77 views

### Reference request: linearly independent cycles in a manifold

The following seems to be well known to experts, but I would be happy if there is a paper or textbook that I can cite.
Note: all of the manifolds are assumed to be without boundary.
Suppose that $C$ ...

**0**

votes

**0**answers

61 views

### Cohomology operators inducing local basis of $1-$forms

Suppose that $\partial$ is a non-trivial ($\partial \neq 0$) cohomology operator on an $m-$dimensional manifold $M$ (that is: $\partial:\Omega(M)\to\Omega(M)$ is a degree $1$ derivation such that ...

**3**

votes

**1**answer

222 views

### Ehresmann fibration theorem for manifolds with boundary

All manifolds in consideration may have nonempty boundary and may be disconnected.
Let me fix a definition first. A map between smooth manifolds $M\rightarrow N$ is a fiber bundle, iff it's locally ...

**6**

votes

**2**answers

388 views

### Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question.
I can think of at least four topologies to put on $C_c(M)$:
Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney ...

**1**

vote

**1**answer

108 views

### Are normal deformations of an embedding open in the $C^{\infty}$-space of embeddings of a compact smooth manifold`

Let $M$ be a compact smooth manifold (closed, for simplicity), $n\in\mathbb{N}$ and equip the space of embeddings $Emb(M,\mathbb{R^n})$ wih the Whitney-$C^{\infty}$-topology. (The weak and the strong ...

**6**

votes

**0**answers

210 views

### Schoenflies and symplectic topology

The final report from a workshop on Morse theory in low-dimensional and symplectic topology includes the following question, posed by Michael Hutchings: Can we apply symplectic geometry to solve the ...

**5**

votes

**1**answer

218 views

### Strange problem about triplets of differential forms

Suppose we have the following map:
$$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$
...

**7**

votes

**2**answers

668 views

### Stokes theorem with corners

I've found the following version of Stokes' theorem in G. Stolzenberg's lecture notes 19:
Notation:
for $1 \le n \le m$
$\Lambda(m, n) = \{ \lambda: \{1,...,n\} \rightarrow \{1,...,m\} \ | \ ...