**18**

votes

**0**answers

470 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 ...

**16**

votes

**0**answers

418 views

### Monoid structure of oriented manifolds with connect sum

Take the class of all compact, connected, boundaryless, smooth oriented $n$-dimensional manifolds, each taken up to orientation-preserving diffeomorphism. This is a commutative monoid with operation ...

**15**

votes

**0**answers

193 views

### Finite limits in the category of smooth manifolds?

The category of smooth manifolds does not have all finite limits. However, it does have some limits: it has finite products, it has splittings of idempotents, and it has certain other limits if we ...

**14**

votes

**0**answers

617 views

### Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?

This is a followup to my previous question, which asked whether
the category of commutative or noncommutative C*-algebras or von Neumann algebras
is equivalent to the category of commutative or ...

**10**

votes

**0**answers

378 views

### Exotic smoothness and Parallelizability

Regarding the parallelizability of the Milnor's seven dimensional exotic spheres:
Parallelizability of the Milnor's exotic spheres in dimension 7
The following question naturally arises:
Suppose ...

**8**

votes

**0**answers

96 views

### When does a cobordism factorize the sphere?

Let $W: \emptyset \Rightarrow M$ be a smooth cobordism from $\emptyset$ to a smooth closed $n$-manifold $M$. Are there reasonably simple conditions on $W$ which guarantee the existence of another ...

**8**

votes

**0**answers

273 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 $M$. ...

**8**

votes

**0**answers

226 views

### Homology classes represented by $J$-holomorphic curves

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if
$$ du \circ j = J \circ ...

**6**

votes

**0**answers

255 views

### Universal property for complex blowup in smooth category

If $M$ is a smooth complex manifold and $N$ is a smooth complex surface, we may ask when a holomorphic map $f:M\rightarrow N$ lifts to a map $f:M\rightarrow [N:p]$, where $[N:p]$ denotes the blowup of ...

**5**

votes

**0**answers

217 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 ...

**5**

votes

**0**answers

114 views

### In cell-decomposed manifolds, how easy is it to arrange for the tubular neighborhood of a diagonal to contract onto the diagonal?

Suppose that you have decomposed a manifold $M$ into cells (I care most, if it matters, about compact oriented smooth manifolds; but if my question can be solved in the PL category, all the better). ...

**5**

votes

**0**answers

172 views

### How do metrics behave under joining along a manifold embedded in the boundary?

How do metrics behave under joining along a manifold embedded in the boundary?
This is, more-or-less, part of Problem 4.66 in Kirby's List:
Problem 4.66 How do metrics (e.g. Riemannian, Lorentz, ...

**5**

votes

**0**answers

141 views

### What is the name of the quotient of the Stiefel manifold of $k$-frames by the symmetric group of $k$ letters?

Let $V_k(\mathbb{R}^n)$ be the Stiefel manifold of $k$-frames in $\mathbb{R}^n$. The symmetric group of $k$ letters $\Sigma_k$ acts freely by permuting vectors in $k$-frames.
Does the quotient ...

**4**

votes

**0**answers

166 views

### $n$-Fold Framed Functions

Suppose that $M$ is a manifold. One can consider a suitably constructed space of generalized framed Morse functions on $M$, let's call it $\mathrm{Fun}^\mathrm{fr}(M)$. This space is known to be ...

**4**

votes

**0**answers

172 views

### Atlas of a manifold as a Sheaf

--Hopefully this question does not dublicate another--
In this question Tom Goodwillie pointed out, that the 'atlas part' of
the definition of a smooth manifold can be redefined in terms of
sheaves. ...

**4**

votes

**0**answers

173 views

### “Naïve”cobordism?

The naive view of (say, complex) cobordism of a space $X$ is that it should be the group generated by continuous families of closed manifolds parametrized by $X$. Acting even more naively, one may try ...

**4**

votes

**0**answers

127 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 ...

**4**

votes

**0**answers

194 views

### Ricci-flat metrics on Cotangent bundles in adapted complex structure

greetings,
Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...

**3**

votes

**0**answers

213 views

### Quotes from Connes

I found the following remark by Connes HERE:
"the main quality of the homotopy type of a manifold is to satisfy Poincare duality not only in ordinary homology but in K-homology with the Fredholm ...

**3**

votes

**0**answers

100 views

### Is a continuous map between smoothable manifolds of the same dimension always smoothable?

(My question is inspired by this math.SE question, whose
negative answer I showed by a dimension-increasing map.)
Is it the case that for all smoothable manifolds $M_0$ and $M_1$ with the same ...

**3**

votes

**0**answers

150 views

### smoothing a current

Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth ...

**3**

votes

**0**answers

127 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 ...

**3**

votes

**0**answers

72 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 ...

**3**

votes

**0**answers

219 views

### Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold?

Let $(M,g)$ be a geodesically complete Riemannian manifold that is not necessarily compact. Futhermore, assume that $M$ has at most exponential volume growth (ie., locally doubling property). Let ...

**2**

votes

**0**answers

145 views

### The Cauchy Problem in General Relativity: Existence of a Hausdorff Development

This is related to a problem that I posed about a year ago. I was given several references by a number of experts who were kind enough to entertain my rather arcane question. Those references were ...

**2**

votes

**0**answers

238 views

### Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed 4-manifolds. Are there any closed 4-manifolds with uncountably many ...

**2**

votes

**0**answers

76 views

### Smooth function over a manifold into an algebra

Let $M$ be a manifold and $A$ a $*$-algebra. Does is hold that
$$C^{\infty}(M,A) \cong C^{\infty}(M) \otimes A$$
where the RHS means that you take smooth functions which map into $A$. If this holds, ...

**2**

votes

**0**answers

72 views

### Strictly Convex Smoothing of a function defined on an affine manifold

A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex
if its restriction to each line is. An affine ...

**2**

votes

**0**answers

90 views

### Analytic version of the Cartan lemma

Assume that $\beta$ is a real analytic 2-form on an analytic manifold $M$ and $\alpha$ is an analytic non vanishing 1-form on $M$. Assume that $\beta \wedge \alpha=0$. Is there an analytic 1-form ...

**2**

votes

**0**answers

110 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 ...

**2**

votes

**0**answers

75 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

**0**answers

72 views

### Non-clean fiber products

Usually, the most general condition for fiber product of manifolds (or vector bundles) to exist is that we require the images cleanly intersects. See e.g.
When do fibre products of smooth manifolds ...

**2**

votes

**0**answers

108 views

### isomap and self intersections

I sample a 2D surface in $\mathbb{R}^3$ with $N$ points, and compute an isomap using pairwise weighted geodesic distances. I am thus able to embed this surface into a $M$ dimensional space in which ...

**2**

votes

**0**answers

369 views

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

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

**1**

vote

**0**answers

78 views

### Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here:
...

**1**

vote

**0**answers

58 views

### Cover a set with balls centered at smooth functions (Ascoli theorem)

Assume $M$ to be a compact $n$-dimensional manifold, endowed with a complete metric.
Let us consider the space $C^\infty(M)$ endowed with the standard $C^\infty$ topology, i.e. generated by the ...

**1**

vote

**0**answers

136 views

### Foliation of the tangent bundle of $n$-sphere

Is there a smooth $n$-dimensional foliation of $TS^{n}$,( here $n\neq1,3,7$) such that the zero section be a leaf of this foliation?

**1**

vote

**0**answers

99 views

### Riemann normal coordiantes and change of metric

Le $(M,g)$ be Riemannian manifold. Fix point $p\in M$. We can define the map
$$\exp: U \subset T_p M \rightarrow M$$
$$\exp(X) = \gamma_{p,X}(1)$$
where $t\mapsto \gamma_{p,X}(t)$ is geodesic such ...

**1**

vote

**0**answers

87 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 ...

**1**

vote

**0**answers

172 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 ...

**1**

vote

**0**answers

177 views

### HyperKaehler manifolds are Ricci-flat

Hi,
I have the following question: Let $M$ be a Hyperkaehler manifold with complex structures $I,J,K$ and Hyperkaehler metric $g$. Let $\omega_{I} = g(I *, *), \omega_{J} = g(J *, *), \omega_{K} = ...

**1**

vote

**0**answers

81 views

### Is there a reference for some notions relative to distributions of corank 1?

In a expository text on differential geometry I am reading about the geometry of distributions of a corank one.
Here the first properties are reported without proof, and no reference is given.
I ...

**0**

votes

**0**answers

75 views

### A consequenc of a Lie group act on a Riemannian manifold by isometry

I am learning differential geometry for using this topic in my research. I am stuck to prove following Result which I got in a article.
Formulation:
Let $ f: [0, 1]\rightarrow \mathbb{R}^2$ be a ...

**0**

votes

**0**answers

75 views

### A question on tangent bundle (and second tangent bundle)

Let $M$ be a $n$ dimensional manifold and $p:TM\to M$ be the projection map. Then $\ker Dp$ is a $n$ dimensional vector bundle on $TM$, as a sub bundle of $TT(M)$.
For what type of manifolds, ...

**0**

votes

**0**answers

97 views

### A special Lie subalgebra

Motivated by comments of the following post A question on involutions on the Lie algebra of vector fields we ask the following question:
Let $L$ be a Lie algebra. We consider the Lie subalgebra ...

**0**

votes

**0**answers

95 views

### Question about a particular estimate in Riemannian geometry

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...

**0**

votes

**0**answers

47 views

### Question on center-stable manifold

Assume that you have a gradient system smooth enough and a fixed point $x_{0}$. Is it true that if $x_{0} \in \omega(x)$ then $\gamma^{+}(x)$ must intersect the local center-stable manifold of ...

**0**

votes

**0**answers

66 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 ...

**0**

votes

**0**answers

77 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 ...

**0**

votes

**0**answers

105 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 ...