The smooth-manifolds tag has no wiki summary.

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### 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 ...

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752 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 ...

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### 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 ...

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### 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.
...

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771 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 ...

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### 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 ...

**16**

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**1**answer

790 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 ...

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467 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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**1**answer

225 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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**1**answer

266 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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**1**answer

491 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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630 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 ...

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215 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 ...

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**3**answers

620 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 ...

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944 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 ...

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1k 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 ...

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**3**answers

690 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 ...

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488 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 ...

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964 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 ...

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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 ...

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**1**answer

381 views

### Understanding manifold GL+(3,R)/SO(3) ?

I'm trying to better understand the manifold GL+(3,R)/S0(3) which is diffeomorphic to positive definite symmetric matrices. My motivation is to understand U in F = RU where F in GL+(3,R) = deformation ...

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### Can all n-manifolds be obtained by gluing finitely many blocks?

Fix a dimension $n\geqslant 2$. Let $S= \{M_1,\ldots, M_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is generated by $S$ if it may ...

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### Slice knots and exotic $\mathbb R^4$

In the http://arxiv.org/abs/math/0606464v1 I read
"If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in
possession of a knot which is ...

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### Self homeomorphisms of $S^2\times S^2$

Every matrix $A\in SL_2(\mathbb{Z})$ induces a self homeomorphism of $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$. For different matrices these homeomorphisms are not homotopic, as the induced map on ...

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### 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 ...

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643 views

### In Diff, are the surjective submersions precisely the local-section-admitting maps?

Question as in title (Diff = category of smooth manifolds and smooth maps)
I thought I'd convinced myself this is true, so this is just a sanity check.
Also, what about for settings other than ...

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**1**answer

609 views

### Studying non-linear PDEs with manifolds

I'm sorry if this is an inappropriate forum to ask this question on, for I fear it is pretty undergraduate-level one :) I was contemplating on the study of non-linear PDEs. Is it possible to reduce a ...

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571 views

### Colimits of manifolds

This question tells us that in general colimits do not exist in the category of manifolds.
However, this negative answer is not very satisfying. A manifold can be considered as a colimit of its ...

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**1**answer

860 views

### Gluing two diffeomorphisms together

A fundamental construction in a first course on manifolds is to build a smooth function $\psi\colon \mathbb{R} \to \mathbb{R}$ with the property that for some $0<\delta<\epsilon$ we have
...

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### Exotic spheres detected in higher homotopy

Thinking about exotic 7-spheres, one can look at the maps $\cdots \rightarrow \Omega^2Diff(D^4, rel \space \partial) \rightarrow \Omega Diff(D^5, rel \space \partial) \rightarrow Diff(D^6, rel \space ...

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### Totally geodesic surfaces in fibered 3-manifolds

Is there an easy example of a (closed) hyperbolic 3-manifold that fibers over the circle but contains some totally geodesic surface?
(Of course such manifolds exist if the 'Virtually Fibered ...

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304 views

### Extending diffeomorphisms of Riemannian surfaces to the ambient space

Question 1: Given a smooth Riemannian surface $M$ in $R^3$ (i.e., a smooth Riemannian 2-manifold embedded in $R^3$) and a diffeomorphism $f: M\rightarrow M$ of class $C^{k\geq 2}$, does $f$ admit a ...

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### What is the Implicit Function Theorem good for?

What are some applications of the Implicit Function Theorem in calculus? The only applications I can think of are:
1) the result that the solution space of a non-degenerate system of equations ...

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### quasi-separated manifolds

Let $M$ be a smooth manifold. Let's call $M$ quasi-seperated if $M$ has the following property: If $B,C \subseteq M$ are open balls, then $B \cap C \subseteq M$ is a finite(!) union of open balls. By ...

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659 views

### On the $\mathbb R$-algebra structure on $C^\infty(M)$.

As it is more or less well-know, and as it has come up on MO a couple of times, the $\mathbb R$-algebra $C^\infty(M)$ of smooth functions on a (say) compact manifold contains essentially everything ...

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532 views

### The ring $C^{\infty}(M)$?

Let $M$ be a smooth paracompact manifold. I think that the ring $C^{\infty}(M)$ contains many (possibly almost all?) geometric or topological information about $M$.
(e.g. Let $E$ be a vector bundle ...

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### How well can we localize the “exoticness” in exotic R^4?

My question concerns whether there is a contradiction between two particular papers on exotic smoothness, arxiv:0807.4248v1 and arxiv:gr-qc/9404003v1. The former asserts:
"Let $M$ be a smooth closed ...

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### Is it true that exotic smooth R^4 cannot be diffeomorphic to RxN, where N is a 3-manifold?

Since $\mathbb{R}$ and any 3-manifold $N$ must be non-exotic, their product $\mathbb{R}\times N$ cannot possibly be diffeomorphic to exotic $\mathbb{R}^4$, correct?
Update: Andy Putman already ...

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### Is every graded manifold affine, and is this definition of graded manifold the right one?

The following definition is from:
Dmitry Roytenberg, "AKSZ-BV formalism and Courant algebroid-induced topological field theories", Letters in Mathematical Physics, 2007 vol. 79 (2) pp. 143-159, ...

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### exotic smooth structure clarification

Does the existence of exotic smooth structure in $\mathbb{R}^4$ imply the existence of an atlas which has a $C^0$ mapping to the Cartesian atlas, but not a $C^k$ mapping (for some finite $k$)? Does ...

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**1**answer

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### 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 ...

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### Level sets of Morse functions

Every compact two dimensional manifold admits a Morse function such that any its regular level set is at most two circles. I am interested in a generalization of that phenomenon.
Does there exist a ...

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### Which manifolds admit a diffeomorphism of order $n$?

Let $n>1$. Which smooth manifolds admit a diffeomorphism $f$ of order $n$?
For a closed orientable surface $S_g$ of genus $g$ and $n=2$ the answer is in the affirmative, since $S_g$ can be ...

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### When does a submersion have connected fibers?

Can we characterize when a submersion $F:M \to N$ between two smooth manifolds has connected fibers? If this is too hard, what are some sufficient conditions?

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### Checking whether the image of a smooth map is a manifold

I have a specific problem, but would also like to know how to tackle the general case. I will first state the genral question. Let $M$ be an embedded submanifold of $\mathbb{R}^n$ and let $F: ...

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### Do Lie algebroids pull back (along submersions)?

There are more general definitions, but for my purposes a Lie algebroid on a smooth manifold $X$ is a vector bundle $A \to X$, a map $\rho: A \to {\rm T}X$ of vector bundles over $X$, and a bracket ...

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### smooth sections of smooth fiber bundles

A maybe trivial question about fiber bundles (I'm not an expert, and I didn't find quickly an answer looking here and there). Suppose you are given fiber bundle $p\colon E\to M$, where
$E,M$ are ...

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### Exotic differentiable structures on R^4?

This was going to be a comment to Differentiable structures on R^3, but I thought it would be better asked as a separate question.
So, it's mentioned in the previous question that $\mathbb{R}^4$ has ...

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239 views

### A local transitivity property of the automorphism group of a foliated manifold

Let $(M,\mathcal F)$ be a smooth foliated manifold. An automorphism of $(M,\mathcal F)$ is a diffeomorphism of $M$ that takes leaves of $\mathcal F$ onto leaves. Let now $L$ be a leaf of $\mathcal F$. ...

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### Compactification theorem for differentiable manifolds ?

Just parallelling this question, that seemed not to admit an easy answer at all, let's "soft down" the category and ask the same thing in the case of $\mathcal{C}^{\infty}$-differentiable manifolds ...