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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23
votes
8answers
3k views

Why should I prefer bundles to (surjective) submersions?

I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also ...
13
votes
1answer
939 views

What should I cite for the Poincaré conjecture?

I'm writing a paper that, rather unexpectedly, needs the Poincaré conjecture for one of the results. (The paper has almost nothing to do with differential geometry!) The conjecture was famously ...
3
votes
1answer
136 views

Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
6
votes
0answers
71 views

Lie subalgebras of $\chi^{\infty}(M)$ of codimension $n=dim M$

For a connected $n$ manifold $M$, the Lie algebra of all smooth vector fields is denoted by $\chi^{\infty}(M)$. For a pointe $p\in M$ we define $L_{p}=\{X\in \chi^{\infty}(M)\mid X(p)=0\} $....
6
votes
1answer
208 views

The minimum codimension of Lie subalgebra of $\chi^{\infty}(M)$

Assume that $M$ is an arbitrary manifold. Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one? If not, what is a counter ...
5
votes
0answers
122 views

Short and elegant definition of the $C^1$ topology

A friend told me that the $\mathbf{C^1}$-topology on the set $C^\infty(M,N)$ of smooth functions between two smooth manifolds $M$ and $N$ can be defined as the coarsest topology making the map $$ C^\...
15
votes
5answers
665 views

Smoothness of the closest point on a submanifold

Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold. Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...
3
votes
0answers
147 views

Manifolds and CW-complexes

Let us consider a category $C$ formed by topological spaces and continuous functions (or by smooth manifolds and smooth functions). We consider the morphism category $C_{2}$. An object of $C_{2}$ is a ...
2
votes
0answers
48 views

Thom form of holomorphic bundle over Kaehler manifolds/orbifolds

Consider a holomorphic vector bundle $\pi:E\rightarrow X$ of complex rank $m$ over a Kaehler manifold $X$. Can we find a Thom form $\Theta$ of $E$ such that as a form on the complex manifold $E$, it ...
21
votes
1answer
1k views

Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map $f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that $f^{n}(x_{0})...
2
votes
2answers
150 views

Infinite dimensional version of a simple fact on certain singular matrices

We consider the following simple fact about matrices. Then we try to generalize it in the context of smooth manifolds; Let $L$ be the collection of all $n \times n$ real matrices $A=(a_{ij})$ with ...
2
votes
0answers
107 views

Decay of the Fourier transform of a surface area measure

Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\...
4
votes
1answer
146 views

Differential structures on compact Lie groups

Given a compact Lie group can there be a differential structure on it with respect to which one cannot define a smooth group operation?
3
votes
0answers
163 views

How far can one reconstruct the boundary of a manifold M given its interior $int M$? [duplicate]

Suppose I keep in my pocket a manifold with boundary $M$ , and I provide you access to $int M := M \setminus \partial M$ up to homeomorphism/diffeomorphism. What can you deduce about $\partial M$? can ...
2
votes
0answers
426 views

Space of derivations of holomorphic (analytic) functions

Let $M$ be a (real) smooth manifold, and $p \in M$. The space of (linear) derivations $D:C^{\infty}(p) \to \mathbb{R}$ (i.e., maps satisfying $D(f+g) = D(f)+ D(g)$ and $D(fg)=D(f)g(p) + f(p)D(g)$) ...
15
votes
1answer
339 views

Are homology spheres stably trivial?

A homology sphere is a closed smooth $n$-dimensional manifold with the same homology groups as $S^n$. Igor Belegradek's answer to a previous question of mine shows that the smoothness hypothesis is ...
5
votes
1answer
212 views

On Johansson's Theorem on homotopy equivalences of 3-manifolds

Johansson's theorem states the following: Given $f:M_1\rightarrow M_2$ (not a pair map) an homotopy equivalence between 3-manifolds with incompressible boundary. Let $V_i$ be the components of the ...
1
vote
2answers
226 views

Reference request: $\mathcal{C}^\infty_c(M)$ is a topological vector space with the Whitney topology

Let $M$ be a smooth manifold and let $\mathcal{C}^\infty(M)$ be the space of all smooth real valued functions with the (strong) Whitney topology. This space is a topological vector space iff $M$ is ...
19
votes
2answers
2k views

Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
39
votes
7answers
4k views

What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$

Hello, I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ : For example : 1) Are all ...
5
votes
0answers
85 views

Relationship between Gaussian and bisectional curvature

Let $f:\mathbb CP^1\to X$ be an smooth embedding which its image is the curve $C$ where $X$ is a Kähler manifold. Do we have that $$\sup_{\mathbb CP^1}K(f^*\omega)<\sup_C K(\omega),$$where $K(f^*\...
2
votes
1answer
110 views

homeomorphism type of punctured real projective spaces

Let $\mathbb{R}P^m$ be the $m$-dimensional real projective space and let $\mathbb{R}P^m\setminus\{*\}$ be the punctured space. I observe: $\mathbb{R}P^2\setminus\{*\}$ is homeomorphic to a (open) ...
2
votes
0answers
108 views

If $X$ is a compact smooth Riemannian manifold, why don't we integrate on a fundamental domain in the universal cover? [closed]

Let $X$ be a compact connected Riemannian manifold. The metric gives a local volume form. The universal cover is orientable, and has a precompact subspace locally isometric (with the covering metric) ...
5
votes
3answers
3k views

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 ...
3
votes
1answer
141 views

Is there a smooth polar decomposition for non-invertible matrices?

Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite. $P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$, and when $A$ ...
10
votes
2answers
185 views

Triangulation with simplices of same volume

Let $M$ be a Riemannian smooth compact manifold. It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ? ...
5
votes
1answer
74 views

smoothing locally-finite (Borel-Moore chains)

Let $M$ be a smooth manifold. As is recorded in (for example) Lee's book, de Rham proved that one can calculated singular homology, $H_*(M)$ using smooth simplices. Does the result extend to Borel-...
2
votes
1answer
171 views

Non-diffeomorphic smooth structures on the quotient of a manifold by an integrable distribution

In geometric quantization, one of the important ingredients is an integrable distribution $D$ (let's say real) on some manifold $M$ (symplectic, but this is not important). The resulting object is $M/...
20
votes
2answers
555 views

Is there a smooth manifold which admits only rigid metrics?

Does there exist a (finite dimensional) smooth manifold $M$, such that every Riemannian metric on $M$ has no isometries except the identity? Of course, such a manifold must not admit a diffeomorphism ...
33
votes
7answers
4k 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 $\alpha\wedge\beta:=\frac{k!l!}...
8
votes
1answer
190 views

Homotopy type of diffeomorphism which are the identity on and near the boundary

Let $M$ be a compact manifold with boundary. Denote by $Diff(M), Diff_\partial(M)$ and $Diff_{U\partial}(M)$ the groups of diffeomorphisms of $M$ and the subgroups of the ones that are the identity on ...
3
votes
1answer
129 views

Constructing a “nice” cobordism

Denote by $\Sigma_g$ the closed, orientable surface of genus $g$. I want to construct a cobordism $M_g$ between $\Sigma_g$ and $\Sigma_{g+1}$ with the following two nice properties: 1) $M_g$ is an ...
3
votes
1answer
188 views

Is there a degree one map from a product $B\times S^1 \to \#_n S^2 \times S^1$ for any n

For any $n \geq 1$, let $\Sigma_n$ denote the closed orientable surface of genus n. In http://arxiv.org/abs/1202.6302, the authors showed that for any $n$, there is a degree two, $\pi_1$-surjective, ...
5
votes
0answers
181 views

Does $\#_n S^2×S^1$ really admit a map of non-zero degree from $B×S^1$?

In Proposition 4 on page 6 of this paper, http://arxiv.org/abs/1202.6302, the authors claim to produce a degree 2 $\pi_1$-surjective map $f$ between $M=S^1 \times \Sigma_2$ and $N=\#_2 S^2 \times S^1$ ...
11
votes
1answer
293 views

Are there non-smoothable homotopy/homology spheres?

A homotopy sphere is a topological $n$-manifold $M$ which is homotopy equivalent to $S^n$. A homology sphere is a topological $n$-manifold $M$ such that $H_i(M) \cong H_i(S^n)$ for all $i$. Note, by ...
9
votes
1answer
110 views

Harmonic function with injective boundary conditions is an immersion?

Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note ...
3
votes
0answers
50 views

Transverse intersection in the $G$-orbit of paths

I know how to prove the following lemma but I assume that it is well-known. Can someone provide a reference for it? Let $d>1$ and let $M$ be a $d$-dimensional connected smooth manifold with a ...
2
votes
0answers
64 views

A problem of defining addition in a Quotient space

Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Let the set of all re-parameterizations of curves is $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (...
7
votes
0answers
92 views

$H(M)$ necessarily highly non-integrable, i.e. forms contact structure?

Let$$M^{2n - 1} = \{z \in \textbf{C}^n : \textbf{h}(\textbf{z}, \textbf{z}) \equiv \textbf{z} \cdot \overline{\textbf{z}} = \textbf{1}\}$$be the unit sphere in $\textbf{C}^n$. Consider the real-...
4
votes
1answer
81 views

Locally nilpotent algebraic section of tangent bundle is complete?

Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that ...
2
votes
1answer
111 views

Existence of non-negative extensions of smooth functions on axes

I am struggling to solve an extension problem of smooth functions, and I would like someone to help me. The setting is as follows: Let $X_1$, $X_2$, and $T$ be either the real lines $\mathbb R$ or ...
30
votes
6answers
3k views

How can you tell if a space is homotopy equivalent to a manifold?

Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not ...
7
votes
0answers
255 views

Can any smooth triangulation of a smooth manifold be blurred?

For the purposes of this question, let's say that a blurring of a smooth triangulation $T$ of a smooth manifold $X$ is a smooth homotopy $h\colon [0,1] \times X \to X$ such that $h_0=\operatorname{id}...
0
votes
1answer
275 views

Unique symplectic form in an adapted complex structure

I have the following question: Due to Stenzel, Lempert, Szöke etc. we know that a Riemannian manifold $(M,g)$ admits a complex structure on a neighbourhood of the zero section of the cotangent bundle. ...
10
votes
1answer
287 views

A symmetric embedding of manifolds

Assume that $M$ is a manifold. Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x_{1},x_{2},\ldots x_{i},\...
0
votes
1answer
101 views

Variation of normals along loops

I've constructed a quotient space $M/\sim$ in $\mathbb{R}^d$ that must be a $2$-manifold. If $M/ \sim$ is a sphere then I know that its normal spaces must vary at least 90 degrees. That is $M / \sim$ ...
1
vote
0answers
82 views

Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
11
votes
1answer
254 views

Whitehead products and Framed Manifolds

The attaching map for the top cell of the torus $S^n \times S^n$ is a map $$ [x,y]: S^{2n-1} \to S^n \vee S^n $$ where the notation is such that $x,y : S^n \to S^n \vee S^n$ are the two inclusions–––...
0
votes
1answer
80 views

full set of invariant functions on manifold

Let $M$ be a smooth manifold and $G$ a Lie group acting properly on $M$. Let $k$ be the codimension of a maximal dimensional $G$-orbit in $M$. Is it always possible to construct $k$ functions $f_1, \...
9
votes
2answers
508 views

Do smooth manifolds admit linear atlases? [duplicate]

There is a theorem of Whitney showing that a smooth manifold can be endowed with a compatible real-analytic atlas (later, it was proven that this analytic structure is essentially unique). I am ...