**3**

votes

**0**answers

84 views

### Is $C^\infty(M,\mathbb{R})$ an ind-(smooth manifold)?

Let $\mathrm{Man}$ be the category of smooth manifolds (2nd countable, Hausdorff, no boundary, not necessarily compact) and smooth maps, and let $M$ be an object thereof. Is the presheaf $S \mapsto ...

**5**

votes

**1**answer

321 views

### Does pullback in the category of smooth manifolds always exists?

I am looking for an example where $f:Y\to X$ and $f':Y'\to X$, are both smooth maps of smooth manifolds, but the pullback does not exist.
Remarks:
1) A pullback in a certain category is defined as ...

**3**

votes

**1**answer

68 views

### Closed leaves on foliations of $\mathbb{R}^n$

I want to know if there exists a characterization of k-foliations of $\mathbb{R}^n$ which have all the leaves closed.
Do exists a $k$-foliation of $\mathbb{R}^n$ with a non-closed leaf?
In general, ...

**9**

votes

**3**answers

714 views

### Is each closed convex set a manifold with corners?

Assume that $C$ is a convex set in $\mathbb{R}^{n}$ with non empty interior.
Then consider its closure, is it a smooth manifold with corners?
Edit:
1) The closure of $C$ should be a smooth manifold ...

**10**

votes

**0**answers

90 views

### Which spherical space forms embed in $S^4$?

Is there any hope of getting a classification of which 3-dimensional spherical space forms are smoothly embeddable in $S^4$? I read that lens spaces cannot embed in $S^4$, but some other spherical ...

**6**

votes

**3**answers

300 views

### Examples of Stiefel-Whitney classes of manifolds

Let $M$ by an compact, connected $n$-dimensional manifold without boundary.
Are there any other computable examples of the Stiefel-Whitney class $w(M)$ except for $M=S^m, \mathbb{R}P^m,\mathbb{C}P^m, ...

**7**

votes

**0**answers

233 views

### Question about theorem in Arnold's book on action-angles variables

I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this book in the last part of this question)
If you don't have the book or ...

**11**

votes

**1**answer

213 views

### Homotopies with prescribed regular values

Let $M_1$ and $M_2$ be connected smooth manifolds and let $f_0,f_1:M_1 \rightarrow M_2$ be homotopic smooth maps such that some fixed point $p \in M_2$ is a regular value for both $f_0$ and $f_1$. ...

**1**

vote

**2**answers

86 views

### Retract embedding of $S^{n}$ in its unit tangent bundle

Acording to the comment of Mark Grant and the answer of Ryan Budney, I revise the question:
For what even $n$, there is a retract embedding of of $S^n$ in its unit tangent bundle?

**3**

votes

**0**answers

92 views

### Hodge de Rham operator and orientability

Let $(M,g)$ be a Riemannian manifold. One can consider the exterior algebra bundle $\Lambda(T^*M)$. The sections of this bundle are differential forms, to be noted by $\Omega^k(M)$. One can consider ...

**3**

votes

**1**answer

272 views

### characteristic classes of homotopy equivalent manifolds

Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal
$$
...

**3**

votes

**1**answer

74 views

### Set of density matrices

A density matrix is a matrix $\rho \in \mathscr{D}:=\{A \in \mathbb{C}^{n \times n}; A^*=A; \operatorname{tr}(A)=1; A \ge 0\}.$
In Quantum Mechanics it is natural to look at a group action
$\Phi: ...

**0**

votes

**2**answers

117 views

### Does an analytic tensorial Lie structure on $S^2$ gives a fiberwise Abelian Lie algebra structure?

Motivated by the answer to this question we ask:
Is it true to say that for every real analytic tensorial Lie algebra structure $\alpha$ on $\chi^{\infty}(S^2)$, all fibers are necessarily ...

**1**

vote

**1**answer

123 views

### Manifold_Lie algebra compatibility

In this question we try to improve some parts of this post as follows:
What is an example of a manifold $M$ and a Lie algebra $L$ (with the same dimension) such that $M$ does not admit ...

**3**

votes

**1**answer

92 views

### Existence of tubular neighborohoods of locally flat topological embeddings

Suppose $X$ is a topological manifold and $Y \subset X$ is a locally flat submanifold. We know that $Y$ doesn't necessarily have a tubular neighborhood. My definition of a tubular neighborhood of $Y$ ...

**5**

votes

**2**answers

313 views

### Riemannian metrics preserved by diffeomorphisms

Let $f \neq Id$ be a diffeomorphism (of a smooth manifold $M$) which admits some Riemannain metric on $M$ making it an isometry. How many different metrics are preserved by $f$?
Note that ...

**4**

votes

**0**answers

101 views

### No irreducible parallelizable manifold of given dimension

What is an example of a closed 4-manifold $M$ such that $M$ is parallelizable and $M$ is topologically (or at least smoothly) irreducible?
Topological irreducible: it is not homemorphic to ...

**3**

votes

**2**answers

141 views

### Vanishing of sheaf cohomology with compact support

Let $X$ be a smooth manifold. Let $F$ be a sheaf of $\mathbb{R}$-vector spaces on $X$. I have three closely related questions.
1) Under what sufficient conditions on $F$ for any compact subset ...

**3**

votes

**1**answer

95 views

### Calculating Homology of the Boundary of a Handlebody

Given a manifold $M$ with boundary $W = \partial M$, I know that having a handle decomposition of $M$ allows one to compute its homology, at least in nice cases, by - for example - using the Morse ...

**14**

votes

**1**answer

703 views

### Is it possible for a metric on a smooth manifold to be smooth?

Are there any smooth manifolds $M$ with the following property:
There exist a realizing metric $d$ (i.e $d$ induces the topology on $M$), and $d$ is smooth on all of $M \times M$?
If not, is it ...

**5**

votes

**0**answers

155 views

### Intuition behind the following theorem of Reeb?

What is the intuition behind the following theorem of Reeb?
If a compact manifold admits a function with only two critical points which are non degenerate, it is homeomorphic to the sphere.

**4**

votes

**1**answer

85 views

### Submersion theorem for smooth tame Frechet manifolds

If $M$ and $N$ are Banach manifolds, $f:M\rightarrow N$ is a smooth map, and $q\in N$ is a regular value, so $f$ is a submersion on $f^{-1}(q)$, it is well known that the level set $f^{-1}(q)$ is a ...

**3**

votes

**1**answer

117 views

### Stokes-like Theorem for Dolbeault Operator

I have a simple question regarding complex geometry: is there an analog for the Stokes Theorem for the Dolbeault Operator $\bar{\partial}$? For instance, suppose that $M$ is a closed complex manifold ...

**0**

votes

**0**answers

69 views

### Suitable reference on smooth manifolds for qualifying exam study?

Is there a single suitable reference to study for the smooth manifold (geometry) half of a typical Topology/Geometry PhD preliminary exam at an average AMS group I school? (Note: I know details about ...

**1**

vote

**0**answers

71 views

### Poisson algebra automorphisms of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. Let $V=\mathcal{C}^\infty(M)$ be the Lie algebra of smooth real valued functions. Suppose $f:\rightarrow V$ be an Lie-algebra isomorphism (an algebra ...

**14**

votes

**3**answers

595 views

### Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators

EDIT: According to some comments on this post I revise the title to remove the misunderestanding.
Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...

**3**

votes

**1**answer

145 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.
...

**1**

vote

**0**answers

89 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

36 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

101 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 ...

**0**

votes

**1**answer

311 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

251 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

197 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

116 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

120 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

171 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 ...

**3**

votes

**1**answer

161 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 ...

**2**

votes

**0**answers

359 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 ...

**4**

votes

**2**answers

189 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

183 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

98 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 ...

**1**

vote

**0**answers

63 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 ...

**6**

votes

**1**answer

483 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 ...

**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 $f \in C^{\infty}(M)$ ...

**17**

votes

**1**answer

504 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 ...

**2**

votes

**0**answers

44 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 ...

**-1**

votes

**1**answer

54 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 ...

**2**

votes

**1**answer

125 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 ...

**1**

vote

**1**answer

77 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 $V$ ...