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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2
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340 views

Conformally-flat

Assume given a smooth manifold $(\mathbb{R}^n, g)$, where the metric is a scaled identity $g = e^{2f}I$. Is there a way to know if this is always a non-positive (sectional) curvature manifold? Note ...
0
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0answers
248 views

Sets that are diffeomorphic to $(0,1)^k$

Let $W\subset \mathbf R^{k}$ be an open set. Are there conditions on $W$ that guarantee the existence of a map $T:(0,1)^k \rightarrow W$ such that: (i) $T$ is surjective, (ii) $T$ is continuously ...
10
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1answer
513 views

Smooth four-manifolds with contractible universal cover

Let $X$ be a smooth compact four-manifold with definite non-trivial intersection form. Can the universal cover of $X$ be contractible? It semms to me that the answer is negative when $X$ is simply ...
0
votes
1answer
225 views

local kählerforms on complex manifold

hallo, Let $M$ be a complex manifold. Assume we have a covering of $M$ by complex charts $\{U_{i}\}$. Furthermore assume that we have on each $U_{i}$ a Kählerform $\omega_{i}$ (i.e. $d\omega_{i} = ...
5
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2answers
1k views

one-parameter subgroup and geodesics on Lie group

Hi, Given a Matrix Lie Group, I would like to know if the one-parameter subgroups (which can be written as $\exp^{tX}$) are the same as the geodesics (locally distance minimizing curves). Geodesics ...
3
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3answers
979 views

looking for a book on banach manifolds

Hi, I am looking for a book on Banach manifolds. Can somebody recommend me something. Thanks in advance. leo
3
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2answers
713 views

book on calabi yau manifolds

hi, does anybody know a good book on calabi yau manifolds (i am a beginner) ? thanks in advance lois
6
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4answers
871 views

The Schwartz Space on a Manifold

I asked this question a couple of days ago on math.stackexchange, but have yet to receive a response, so I have decided to post this here. This question is also vaguely related (both questions arose ...
12
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1answer
509 views

What are the smooth manifolds in the topos of sheaves on a smooth manifold?

The category of internal locales in the Grothendieck topos of sheaves on a locale X is equivalent to the slice category over X. In other words, internal locales over X are precisely morphisms of ...
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3answers
646 views

When is the union of embedded smooth manifolds a smooth manifold?

Suppose we have k embeddings of one single smooth manifold into one other, such that the intersections are manifolds,too. What are sufficient conditions, such that the union of those embeddings is a ...
28
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3answers
2k views

When is a submanifold of $\mathbf R^n$ given by global equations?

Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...
6
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2answers
295 views

Cohomology of fixed point subspaces

Suppose $M$ is a smooth manifold and $\phi : M \to M$ is a homeomorphism whose fixed point set is a smooth submanifold $M_{\phi}$. Is there any relation between the cohomology ring of $M_{\phi}$ and ...
13
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3answers
832 views

How to disjoint two cycles with zero intersection?

Suppose that $M^n$ is an orientable connected (thanks to Greg) manifold and $Z^k$ with $Z^{n-k}$ are two real cycles in $M^n$ with zero index of intersection $Z^k\cdot Z^{n-k}=0$ (for me these cylces ...
14
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0answers
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 ...
0
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2answers
426 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
2k 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
1k 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
588 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
523 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
470 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
272 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 $M$, ...
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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
595 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, ...
32
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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 ...
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3answers
449 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
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2answers
1k 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
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1answer
419 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
360 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
209 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
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1answer
484 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
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1answer
489 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
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1answer
630 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 ...
23
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2answers
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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
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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
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1answer
421 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
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2answers
363 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
586 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 ...
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13answers
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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
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1answer
814 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
446 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, ...
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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 ...
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?
27
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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
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1answer
779 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 ...
19
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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 ...