Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

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14
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613 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 ...
0
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2answers
425 views

$\infty$-forms and $\infty$-plectic geometry

Can you have $\infty$-forms on infinite-dimensional manifolds or elsewhere and what are they used for?
2
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2answers
1k views

Inner products on differential forms

Given a Riemannian metric $g$ on a smooth manifold $M$, one defines an $L^2$-inner product on the space $\bigwedge^\ast(M)$ of differential forms by $$ \langle \alpha, \beta \rangle_g = \int_M ...
20
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2answers
979 views

When is a closed differential form harmonic relative to some metric?

Let $\omega$ be a closed non-exact differential $k$-form ($k \geq 1$) on a closed orientable manifold $M$. Question: Is there always a Riemannian metric $g$ on $M$ such that $\omega$ is ...
3
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1answer
570 views

What are elementary applications of the Frobenius'Theorem in the Classical Differential Geometry?

Usually in a first course on differential geometry we learn some classical results on the geometry of curves and surfaces in the ordinary euclidean space, and just later in more advanced courses we ...
1
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1answer
198 views

Is anything known about flow along vector fields over complete normed fields?

In Bourbaki "Variétés différentielles et analytiques" there is a statement (without proof) that for a vector field on smooth manifold over complete normed field (characteristic zero) there is a local ...
4
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1answer
522 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
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1answer
468 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
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0answers
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
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1answer
270 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
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0answers
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
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1answer
589 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
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2answers
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
4answers
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
3answers
443 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
2answers
963 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
1answer
411 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
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0answers
343 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
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1answer
207 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
1answer
479 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
1answer
473 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
1answer
628 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
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2answers
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
0answers
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 ...
3
votes
1answer
411 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
2answers
359 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
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2answers
573 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
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1answer
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
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13answers
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
1answer
809 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
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1answer
434 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
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0answers
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
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0answers
377 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
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4answers
3k views

Parallelizability of the Milnor's exotic spheres in dimension 7

Are the Milnor's seven dimensional exotic spheres parallelizable?
26
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3answers
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
1answer
767 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
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2answers
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
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2answers
791 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
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3answers
530 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
2answers
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
2answers
821 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 ...
21
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6answers
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
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1answer
817 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
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0answers
476 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
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1answer
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 ...
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1answer
268 views

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

Do you need the implicit function theorem? Can you give an explicit charts?
12
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1answer
502 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
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2answers
659 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
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0answers
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
3answers
692 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 ...