**4**

votes

**1**answer

520 views

### On a remark in Foundations of mechanics, 2nd Edition, by Abraham and Marsden

I don't know if this question is appropriate to this site, but I posted here without an answer, so I tried this alternative.
Given a $2$-form $\omega$ on a manifold $M$, let us denote by $N$ the ...

**3**

votes

**1**answer

465 views

### Jet spaces between non Hausdorff manifolds

I found it very hard to find literature about smooth manifolds that are not required to be Hausdorff. In particular I'm interested in their local properties:
1.) Are the $r$-th order jet bundles ...

**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 ...

**6**

votes

**1**answer

268 views

### Does there exist a model of chains on oriented manifolds with both a strict intersection pairing and strict functoriality for closed embeddings?

Let $M$ be a smooth oriented $n$-dimensional manifold. My favorite model of $\operatorname{Chains}_\bullet(M) \otimes \mathbb R$ is the space of smooth compactly-supported de Rham forms on $M$, ...

**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 ...

**16**

votes

**1**answer

585 views

### How are these algebraic and geometric notions of homotopy of maps between manifolds related?

Let $M$ and $N$ be smooth manifolds, and $f,g: M \to N$ smooth maps. Denote by $(\Omega^\bullet M,\mathrm d_M)$ and $(\Omega^\bullet N, \mathrm d_N)$ the cdgas of de Rham forms in each manifold, and ...

**4**

votes

**2**answers

1k views

### On the smooth structure of the spaces of $k$-jets

I was asking myself, if the following list of conditions is sufficient to determine the usual smooth structure on the spaces of $k$-jets.
the map $j^k f:M\ni x\to j_x^k f\in J^k(M,N)$ is smooth, ...

**31**

votes

**4**answers

2k views

### Can cotangent bundles see exotic smooth structures?

I have two questions that are inspired by a couple of questions here on MO (referenced below), as well as by a conversation with some other grad students at a summer school.
Caveat: I'm not a ...

**1**

vote

**3**answers

442 views

### symplectic form with partition on unity

Assume $M$ is a $2n-$dimensional differentiable manifold. Let $(U_{i})$ be a open covering of $M$. With respect to this covering let $\rho_{i}$ be a partition of unity. Assume that on each $U_{i}$ we ...

**3**

votes

**2**answers

939 views

### The double of a smooth manifold with boundary?

$\def\mc#1{\mathcal#1}\def\seq#1{\langle#1\rangle}\def\bbR{\mathbb R}\def\gt{>}\def\dom{{\rm dom\ }}$In some instances, I have seen an appeal to the concept of "the double" of a smooth manifold ...

**0**

votes

**1**answer

407 views

### immersion: submanifold of complex manifold

Let $\alpha : \mathbb{C} \rightarrow M$ be an immersion and $M$ a $n$ dimensional complex manifold with complex structure $I$. Does then follow that $\alpha (\mathbb{C})$ is a one dimensional ...

**0**

votes

**0**answers

338 views

### covariant derivative complex manifold

Assume we have $X$ a complex manifold and $Y = Y^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ and $Z = Z^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ two vector fields on $X$. Let $\nabla$ be the ...

**2**

votes

**1**answer

206 views

### biholomorphism complex manifold induced structure

Let $X$ be a $n$ dimensional complex manifold with complex structure $I$ and assume one has a diffeomorphism $f : \mathbb{C} \rightarrow X$ of some open set $U$ in $\mathbb{C}$ into its image $f(U)$. ...

**9**

votes

**1**answer

476 views

### Is every closed embedded codimension-n submanifold cut out by a section of a rank-n vector bundle?

Let $M$ be a smooth manifold (over $\mathbb R$) and $N \hookrightarrow M$ a closed embedding. Locally near any point in $N$, I can find coordinates $x^1,\dots,x^{\dim M}$ on $M$ so that $N$ is the ...

**6**

votes

**1**answer

470 views

### fundamental domain of universal covering

Let $M$ be a connected compact manifold without boundary, $\pi:\widetilde{M}\to M$ be the universal covering map. A fundamental domain of $(\pi,\widetilde{M}, M)$ is a compact subset $D\subset ...

**1**

vote

**1**answer

622 views

### About the geometry of completely integrable systems

During a conversation I heard an assertion that I found at least dubious for the lack of adeguate hypothesis, but I am not able to imagine a counterexample, even if it is probably obvious to some of ...

**22**

votes

**2**answers

1k views

### How can we detect the existence of almost-complex structures?

Any smooth $k$-manifold $M$ comes with a well-defined map $f:M\rightarrow BGL_{k}(\mathbb{R})$ (up to homotopy) classifying its tangent bundle. Since $GL_{k}(\mathbb{R})$ deformation-retracts onto ...

**6**

votes

**0**answers

254 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 ...

**3**

votes

**1**answer

409 views

### Fixed points of the action of an algebraic group

Hello!
If a compact Lie group $K$ acts smoothly on a smooth manifold $M$, then the set $M^K$ of fixed points under this action is a smooth submanifold of $M$. This is proved for example in ...

**3**

votes

**2**answers

357 views

### uniqueness of regular/tubular neighborhood with equivariant boundary

Let $N$ and $N'$ be regular neighborhoods of a subpolyhedron $P$ in a closed PL manifold $M$, and suppose that $t$ is a free PL involution on $M$ such that each of $\partial N$, $\partial N'$ is ...

**11**

votes

**2**answers

561 views

### Geometric interpretation of the Pontryagin square

The Pontryagin square (at the prime 2) is a certain cohomology operation
$$
\mathfrak P_2: H^q(X;\Bbb Z_2) \to H^{2q}(X;\Bbb Z_4)
$$
which has the property that its reduction mod 2 coincides with ...

**23**

votes

**1**answer

1k views

### Can you flip the end of a large exotic $\mathbb{R}^4$

Can you flip the end of a large exotic $\mathbb{R}^4$
Background
Definition (Exotic $\mathbb{R}^4$):
An exotic $\mathbb{R}^4$ is a smooth manifold $R$ homeomorphic but not diffeomorphic to ...

**14**

votes

**13**answers

3k views

### What should be taught in a 1st course on smooth manifolds?

I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic ...

**9**

votes

**1**answer

803 views

### Looking for a simple proof that R^2 has only one smooth structure

So not so long ago, I asked for a simple proof that $\mathbf{R}$ has only one smooth structure. A proof that was communicated to me by Ryan Budney (link text) was the following:
So let me recall his ...

**3**

votes

**1**answer

428 views

### Do the focal points of a submanifold $M$ in $\mathbb{R}^k$ form a closed subset?

Let $M$ be a submanifold in an euclidean space $\mathbb{R}^k$, and $\nu(M)$ the normal bundle to $M$, let us denote $\phi$ the restriction to $\nu(M)$ of the exponential map for $\mathbb{R}^k$.
A ...

**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, ...

**10**

votes

**0**answers

374 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 ...

**55**

votes

**4**answers

2k views

### Parallelizability of the Milnor's exotic spheres in dimension 7

Are the Milnor's seven dimensional exotic spheres parallelizable?

**26**

votes

**3**answers

1k views

### Is it possible to improve the Whitney embedding theorem?

Edited to fix the example, as per Zack's suggestion.
Edit 2: So it turns out that when I think 'manifold' I tend to assume the nicest possible object. As I believe is standard, I would like to ...

**8**

votes

**1**answer

758 views

### Finite-dimensionality for de Rham cohomology

I was browsing through the litterature, hoping to find sufficient and necessary conditions for a smooth manifold to have finite-dimensional de Rham cohomology, but I can't find any satisfactory ...

**18**

votes

**2**answers

2k views

### Universal property of the tangent bundle

If $X$ is a scheme (over some base scheme, but which I will ignore) its tangent bundle $T(X)$ is defined as the relative spectrum of the symmetric algebra of its sheaf of differentials. Combining the ...

**13**

votes

**2**answers

786 views

### Nice application of generalized smooth spaces

I am a fan of category theory in general, and I appreciate that various brands of generalized smooth spaces (Diffeological spaces, Chen spaces, Frolicher spaces ...) form much nicer categories of ...

**6**

votes

**3**answers

521 views

### Can homologous submanifolds be connected by an immersed manifold with boundary?

Supposed I have an n-dimensional manifold M with a k-dimensional submanifold that is homologous to zero (or, equivalently, two homologous submanifolds). Can I always construct a k+1-dimensional ...

**2**

votes

**2**answers

2k views

### On a proof of the existence of tubular neighborhoods.

Studying analysis on manifolds, I have found, in the proof of the existence of tubular neighborhoods, a reference to theorem 3.1.2 in "Topologie algebrique et theorie des faisceaux" of Godement.
...

**5**

votes

**2**answers

809 views

### Is volume--preserving an intrinsic property?

Let $M$ be a compact smooth manifold without boundary. A Riemannian metric $g$ on $M$ induces a volume measure (or Lebesgue measure) $m_g$ on $M$.
A diffeomorphism $f:M\to M$ is said to be ...

**20**

votes

**6**answers

3k 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 ...

**17**

votes

**1**answer

815 views

### What's the Kirby Diagram of a universal $\mathbb{R}^4$?

What's the Kirby diagram of a universal $\mathbb{R}^4$?
Background
Define $\mathcal{R}$ as the set of smoothings of $\mathbb{R}^4$. For two oriented elements $R_1$, $R_2$ in $\mathcal{R}$ we can ...

**0**

votes

**0**answers

473 views

### A doubt on a problem in Manifolds, Tensor Analysis and Applications

Having tried to solve exercise 4.4-7 to have another proof of Frobenius Theorem, I would ask you a question.
This is what I have understood:
In Step 2 there is to prove, for any tangent subbundle ...

**1**

vote

**1**answer

234 views

### Commutativity of pullbacks and the exterior derivative as an unbounded operator on $L^2$

Let $d_c, \delta_c$ be operators with domains $D(d_c) = D(\delta_c) = C_{c}^\infty(\wedge T^\ast M)$. We let $d_c$ be the usual exterior derivative on compactly supported smooth forms, ie., $d_c\omega ...

**1**

vote

**1**answer

267 views

### why rotation group is a smooth 3-manifold [closed]

Do you need the implicit function theorem? Can you give an explicit charts?

**12**

votes

**1**answer

500 views

### smoothly varying smooth structures

Can one vary smooth structures on $\mathbb R^4$ smoothly/continuously?
This question popped out of Ben's answer here.

**7**

votes

**2**answers

649 views

### Fibrewise homotopy-equivalence of unit sphere bundles vs isomorphism of tangent bundles

Let $M$ be a smooth $m$-dimensional manifold, $TM$ its tangent bundle and $SM$ its unit sphere bundle.
Are there some simple examples where $SM$ is fibrewise homotopy-equivalent to the trivial ...

**3**

votes

**0**answers

218 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

**3**answers

672 views

### analytic structure on lie groups

I need a reference for a result I have heard only very vaguely "A lie group (smooth) has a compatible analytic manifold structure".
(Would even appreciate a concise way to refer to the result..)
I ...

**12**

votes

**2**answers

979 views

### Smooth structures on the connected sum of a manifold with an Exotic sphere

What can we say about the connected sum of a manifold $M^n$ with an Exotic sphere? Is is possible some of them are still diffemorphic to $M^n$. Is it possible to classifying all the exotic smooth ...

**4**

votes

**2**answers

2k views

### A metric for Grassmannians

Hello everybody! I'm reading an article by Ricardo Mane, Hausdorff dimension
is dipheomorphism. I'm having a technical problem. Sorry for my ignorance but Would you like a
references where I can find ...

**12**

votes

**3**answers

715 views

### orientations for zero-dimensional manifolds

I am teaching a course on manifolds, and soon I will have to prove the Stokes' theorem which, of course, involves defining oriented manifolds. There are many ways to define an oriented manifold. My ...

**5**

votes

**2**answers

496 views

### Definition of the Kervaire invariant for normal maps (as in Browder's book)

Browder's book "Surgery on simply-connected manifolds" defines the Kervaire invariant in a very general setting. My question is: how does one get the more usual definition of the invariant for a ...

**2**

votes

**2**answers

1k views

### When are diffeomorphisms from a manifold to itself homotopic to the identity?

I'm sure people in the field know this, but I'm not in the field. Under what conditions (be they on the manifold or the map) is a diffeomorphism from a differentiable manifold $M$ to itself homotopic ...

**9**

votes

**5**answers

1k views

### Ricci Curvature in Infinite Dimensions?

Is there a good notion of "Ricci curvature" in infinite dimensions?
My intuitive understanding of Ricci curvature is that it is some kind of an "average" of the curvature tensor over "different ...