**2**

votes

**0**answers

80 views

### Decay of the Fourier transform of a surface area measure

Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\...

**4**

votes

**1**answer

140 views

### Differential structures on compact Lie groups

Given a compact Lie group can there be a differential structure on it with respect to which one cannot define a smooth group operation?

**3**

votes

**0**answers

153 views

### How far can one reconstruct the boundary of a manifold M given its interior $int M$? [duplicate]

Suppose I keep in my pocket a manifold with boundary $M$ , and I provide you access to $int M := M \setminus \partial M$ up to homeomorphism/diffeomorphism. What can you deduce about $\partial M$? can ...

**2**

votes

**0**answers

426 views

### Space of derivations of holomorphic (analytic) functions

Let $M$ be a (real) smooth manifold, and $p \in M$.
The space of (linear) derivations $D:C^{\infty}(p) \to \mathbb{R}$ (i.e., maps satisfying $D(f+g) = D(f)+ D(g)$ and $D(fg)=D(f)g(p) + f(p)D(g)$) ...

**14**

votes

**1**answer

315 views

### Are homology spheres stably trivial?

A homology sphere is a closed smooth $n$-dimensional manifold with the same homology groups as $S^n$. Igor Belegradek's answer to a previous question of mine shows that the smoothness hypothesis is ...

**4**

votes

**1**answer

195 views

### On Johansson's Theorem on homotopy equivalences of 3-manifolds

Johansson's theorem states the following:
Given $f:M_1\rightarrow M_2$ (not a pair map) an homotopy equivalence between 3-manifolds with incompressible boundary.
Let $V_i$ be the components of the ...

**14**

votes

**5**answers

602 views

### Smoothness of the closest point on a submanifold

Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold.
Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...

**1**

vote

**2**answers

223 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 ...

**19**

votes

**2**answers

2k views

### Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked
Is it true, as rumours have it, that
you started to work on the embedding ...

**39**

votes

**7**answers

4k 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 ...

**5**

votes

**0**answers

84 views

### Relationship between Gaussian and bisectional curvature

Let $f:\mathbb CP^1\to X$ be an smooth embedding which its image is the curve $C$ where $X$ is a Kähler manifold. Do we have that $$\sup_{\mathbb CP^1}K(f^*\omega)<\sup_C K(\omega),$$where $K(f^*\...

**2**

votes

**1**answer

108 views

### homeomorphism type of punctured real projective spaces

Let $\mathbb{R}P^m$ be the $m$-dimensional real projective space and let $\mathbb{R}P^m\setminus\{*\}$ be the punctured space. I observe:
$\mathbb{R}P^2\setminus\{*\}$ is homeomorphic to a (open) ...

**2**

votes

**0**answers

105 views

### If $X$ is a compact smooth Riemannian manifold, why don't we integrate on a fundamental domain in the universal cover? [closed]

Let $X$ be a compact connected Riemannian manifold. The metric gives a local volume form. The universal cover is orientable, and has a precompact subspace locally isometric (with the covering metric) ...

**5**

votes

**3**answers

3k views

### Hessian as a tensor, multi-dimensional taylor series, and generalizations

The Hessian matrix $\{\partial_i \partial_j f \}$ of a function $f:\mathbb{R}^n \to \mathbb{R}$ depends on the coordinate system you choose. If $x_1,\cdots,x_n$ and $y_1,\cdots,y_n$ are two sets of ...

**3**

votes

**1**answer

136 views

### Is there a smooth polar decomposition for non-invertible matrices?

Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite.
$P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$,
and when $A$ ...

**10**

votes

**2**answers

180 views

### Triangulation with simplices of same volume

Let $M$ be a Riemannian smooth compact manifold.
It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ?
...

**5**

votes

**1**answer

74 views

### smoothing locally-finite (Borel-Moore chains)

Let $M$ be a smooth manifold. As is recorded in (for example) Lee's book, de Rham proved that one can calculated singular homology, $H_*(M)$ using smooth simplices. Does the result extend to Borel-...

**2**

votes

**1**answer

170 views

### Non-diffeomorphic smooth structures on the quotient of a manifold by an integrable distribution

In geometric quantization, one of the important ingredients is an integrable distribution $D$ (let's say real) on some manifold $M$ (symplectic, but this is not important). The resulting object is $M/...

**20**

votes

**2**answers

550 views

### Is there a smooth manifold which admits only rigid metrics?

Does there exist a (finite dimensional) smooth manifold $M$, such that every Riemannian metric on $M$ has no isometries except the identity?
Of course, such a manifold must not admit a diffeomorphism ...

**32**

votes

**7**answers

4k views

### Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{k!l!}...

**8**

votes

**1**answer

188 views

### Homotopy type of diffeomorphism which are the identity on and near the boundary

Let $M$ be a compact manifold with boundary. Denote by $Diff(M), Diff_\partial(M)$ and $Diff_{U\partial}(M)$ the groups of diffeomorphisms of $M$ and the subgroups of the ones that are the identity on ...

**3**

votes

**1**answer

128 views

### Constructing a “nice” cobordism

Denote by $\Sigma_g$ the closed, orientable surface of genus $g$. I want to construct a cobordism $M_g$ between $\Sigma_g$ and $\Sigma_{g+1}$ with the following two nice properties:
1) $M_g$ is an ...

**3**

votes

**1**answer

188 views

### Is there a degree one map from a product $B\times S^1 \to \#_n S^2 \times S^1$ for any n

For any $n \geq 1$, let $\Sigma_n$ denote the closed orientable surface of genus n. In http://arxiv.org/abs/1202.6302, the authors showed that for any $n$, there is a degree two, $\pi_1$-surjective, ...

**5**

votes

**0**answers

180 views

### Does $\#_n S^2×S^1$ really admit a map of non-zero degree from $B×S^1$?

In Proposition 4 on page 6 of this paper, http://arxiv.org/abs/1202.6302, the authors claim to produce a degree 2 $\pi_1$-surjective map $f$ between $M=S^1 \times \Sigma_2$ and $N=\#_2 S^2 \times S^1$ ...

**11**

votes

**1**answer

291 views

### Are there non-smoothable homotopy/homology spheres?

A homotopy sphere is a topological $n$-manifold $M$ which is homotopy equivalent to $S^n$.
A homology sphere is a topological $n$-manifold $M$ such that $H_i(M) \cong H_i(S^n)$ for all $i$.
Note, by ...

**9**

votes

**1**answer

109 views

### Harmonic function with injective boundary conditions is an immersion?

Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note ...

**3**

votes

**0**answers

50 views

### Transverse intersection in the $G$-orbit of paths

I know how to prove the following lemma but I assume that it is well-known. Can someone provide a reference for it?
Let $d>1$ and let $M$ be a $d$-dimensional connected smooth manifold with a ...

**19**

votes

**7**answers

3k views

### Why should I prefer bundles to (surjective) submersions?

I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also ...

**2**

votes

**0**answers

64 views

### A problem of defining addition in a Quotient space

Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Let the set of all re-parameterizations of curves is $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (...

**6**

votes

**0**answers

91 views

### $H(M)$ necessarily highly non-integrable, i.e. forms contact structure?

Let$$M^{2n - 1} = \{z \in \textbf{C}^n : \textbf{h}(\textbf{z}, \textbf{z}) \equiv \textbf{z} \cdot \overline{\textbf{z}} = \textbf{1}\}$$be the unit sphere in $\textbf{C}^n$. Consider the real-...

**4**

votes

**1**answer

80 views

### Locally nilpotent algebraic section of tangent bundle is complete?

Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that ...

**2**

votes

**1**answer

109 views

### Existence of non-negative extensions of smooth functions on axes

I am struggling to solve an extension problem of smooth functions, and I would like someone to help me.
The setting is as follows:
Let $X_1$, $X_2$, and $T$ be either the real lines $\mathbb R$ or ...

**29**

votes

**6**answers

3k views

### How can you tell if a space is homotopy equivalent to a manifold?

Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not ...

**7**

votes

**0**answers

251 views

### Can any smooth triangulation of a smooth manifold be blurred?

For the purposes of this question, let's say that a blurring
of a smooth triangulation $T$ of a smooth manifold $X$
is a smooth homotopy $h\colon [0,1] \times X \to X$ such that $h_0=\operatorname{id}...

**0**

votes

**1**answer

273 views

### Unique symplectic form in an adapted complex structure

I have the following question: Due to Stenzel, Lempert, Szöke etc. we know that a Riemannian manifold $(M,g)$ admits a complex structure on a neighbourhood of the zero section of the cotangent bundle. ...

**10**

votes

**1**answer

284 views

### A symmetric embedding of manifolds

Assume that $M$ is a manifold.
Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x_{1},x_{2},\ldots x_{i},\...

**0**

votes

**1**answer

101 views

### Variation of normals along loops

I've constructed a quotient space $M/\sim$ in $\mathbb{R}^d$ that must be a $2$-manifold. If $M/ \sim$ is a sphere then I know that its normal spaces must vary at least 90 degrees. That is $M / \sim$ ...

**1**

vote

**0**answers

82 views

### Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...

**11**

votes

**1**answer

253 views

### Whitehead products and Framed Manifolds

The attaching map for the top cell of the torus $S^n \times S^n$ is a map
$$
[x,y]: S^{2n-1} \to S^n \vee S^n
$$
where the notation is such that
$x,y : S^n \to S^n \vee S^n$ are the two inclusions–––...

**0**

votes

**1**answer

80 views

### full set of invariant functions on manifold

Let $M$ be a smooth manifold and $G$ a Lie group acting properly on $M$. Let $k$ be the codimension of a maximal dimensional $G$-orbit in $M$.
Is it always possible to construct $k$ functions $f_1, \...

**9**

votes

**2**answers

502 views

### Do smooth manifolds admit linear atlases? [duplicate]

There is a theorem of Whitney showing that a smooth manifold can be endowed with a compatible real-analytic atlas (later, it was proven that this analytic structure is essentially unique).
I am ...

**0**

votes

**0**answers

46 views

### Foliation by Umbilic Surfaces

Suppose $(M,g)$ is a simply connected 3 dimensional Riemannian Manifold which is a foliation by Umbilic surfaces.
Can I make the claim that there exists a coordinate system $(x_1,x_2,x_3)$ in which ...

**3**

votes

**0**answers

72 views

### How to visualize the dual objects of jets of functions?

I work with a smooth $f: M \to \Bbb C$ and I would like to have an object mimicking the concept of "$k$-th order differential" from multivariate calculus. For various reasons that are not important ...

**1**

vote

**0**answers

85 views

### “Nice” limits of sequences of smooth embeddings

Consider smooth embeddings of a manifold $M$ into some $\mathbb{R}^n$. If a sequence $f_k : M \to \mathbb{R}^n$ of such embeddings converges to some continuous function $f : M \to \mathbb{R}^n$, then ...

**0**

votes

**0**answers

50 views

### map of constant rank

Let $f_1, \dots, f_m \colon M^n \to \mathbb{R}$ be smooth functions on a manifold $M$, such that $F := (f_1, \dots , f_m) \colon M \to \mathbb{R}^m$ has constant rank $k <m,n $ on $M$.
I'm trying ...

**3**

votes

**0**answers

57 views

### Interpolating from a Hard Lefschetz class to a Kaehler class

Let $X$ be a compact smooth manifold that admits symplectic and Kaehler structures.
There is a paper by Ugarte, Rudyak, Tralle, and Ibanez, showing how the Lefschetz rank can vary along a path of ...

**23**

votes

**6**answers

2k views

### How to get convinced that there are a lot of 3-manifolds?

My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that ...

**3**

votes

**0**answers

81 views

### Using and understanding the Atiyah-Bott localization theorem/integration formula

I posted this on r/math, but was told I might have better success here given the level of the question.
Basically, I need to learn how to use the localization theorem to compute integrals on ...

**7**

votes

**2**answers

146 views

### Tangent bundle of smooth closed simply-connected $4$-manifold $w_1 = w_2 = 0$ can be trivialized in complement of point?

Let $M$ be a smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$. Can $TM$ be trivialized in the complement of a point?

**6**

votes

**0**answers

176 views

### A tangential fixed point property for manifolds embedded in Euclidean spaces

Assume that $M$ is a compact orientable manifold which is embedded in some Euclidean space $\mathbb{R}^{N}$
We say that $M\subset \mathbb{R}^{N}$ has the tangential fixed point property if for ...