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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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a counter-example of Lie Group

Is there an example of a $n$-dimensional Lie group whose left invariant $n$-form is not right invariant? As far as I know, the Lie group can't be compact. But I don't know how to construct an example.
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75 views

Lower bound on $\epsilon$-covers of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every point $...
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1answer
96 views

Special spheres: principal curvatures with different signs

For $\varepsilon > 0$, we say that a closed, connected and oriented immersed hypersurface $M^n$ of a riemannian manifold $(N^{n+1},g)$ is $\varepsilon$-convex whenever all principal curvatures of $...
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25 views

Cut on hypersurfaces and angular defects

I like very much the elementary property that if one cuts a geodesic triangle onto a sphere (one can use 3 plans that contain $0$). The cut surface of the sphere is given by the sum of the angles of ...
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1answer
232 views

Handlebody decomposition of 4-spheres without 3-handles

There used to be many candidates for an exotic 4-sphere, but a lot of them are now known to be the standard smooth $S^4$. The ones of Cappell-Shaneson (maybe not all of them?) were described in terms ...
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73 views

Trying to understand why this local coordinates parametrizes a manifold

First of all, I would like to say that I think this question fits better on Math Overflow than on Math Stack Exchange, in view of the proposal of the two sites. However, if my analysis of the ...
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0answers
81 views

Classification of Principal $G$ bundles and vector bundles in smooth sense

Suppose $G$ is a Topological group then classification theorem of Principal $G$ bundles says that there is a Principal $G$ bundle $EG\rightarrow BG$ such that any principal $G$ bundle over a ...
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76 views

Cobordant of 5d manifolds, and the generalization of bordisms

Some of the 5-dimensional manifolds are (co)bordant via oriented cobordism. For example, if I understand correctly, 5-dimensional Dold manifold and Wu manifold are manifolds which are cobordant to ...
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157 views

Quotient space, a fundamental group, and higher homotopy groups 2

Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...
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225 views

Non-Abelian fundamental group? — a bizarre example

For the quotient space $G=G_0/G_1$, knowing the homotopy groups of $G_0$ and $G_1$, one can determine homotopy groups from the long exact sequence $$ ... \to \pi_n(G_1) \to \pi_n(G_0) \to \...
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135 views

Are there non-cuspy triangulations of smooth manifolds?

In (as it turned out my misunderstanding of) the literature, a "smooth triangulation" seems to mean: a homeomorphism from a simplicial complex, such that on each simplex the map can be extended to a ...
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143 views

Cell structures of Dold manifold and Wu manifold

In Dold's 1956 paper Erzeugende der Thomschen Algebra N, Dold studied the Dold manifold $P(m,n)=(S^m\times\mathbb{CP}^n)/\tau$ where $\tau$ acts as $-1$ on $S^m$ and a complex conjugation on $\mathbb{...
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54 views

Construction of homogeneous space

Given a Hilbertspace $\mathcal{H}$ of dimension $n<\infty$ and all hermitian matricies $Symm(\mathcal{H})$. I'd guess that the set $M_{2,2} \subset Symm(\mathcal{H})$ of all matricies of rank 4 and ...
4
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1answer
129 views

Surjections on generalized homology theory

In Davis-Januszkiewica´s paper Hyperbolization of polyhedra , the authors hyperbolized every closed n-manifolds K to get a new manifold, say M(K),together with a map $f_K$ from M(K) to K, then they ...
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112 views

Completing the proof of that the set of points where $f(x) = 0$ is a $k$-manifold [closed]

[I have asked this question with the previous versions of my answer in math.SE; however, I did not get any comment / answer, so I thought I might asked this in here with the improved version of my ...
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69 views

Injective differential of linear operators on a Hilbertspace

Given a complex Hilbertspace $\mathcal{H}$ of dimension $\dim(\mathcal{H}) = d$ and the set $$\mathcal{F} := \{q\in L(\mathcal{H})\vert\quad \text{rank}(q) = 4 \quad \wedge \lambda^q_{1,2} < 0\ \ \...
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684 views

What are the possible Stiefel-Whitney numbers of a five-manifold?

On a compact five-manifold, the Stiefel-Whitney number $w_2w_3$ can be nonzero. An example is the manifold $SU(3)/SO(3)$, and also another example is a $\mathbb{CP}^2$ bundle over a circle where the ...
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3answers
277 views

Classification of line bundles by second cohomology of a manifold

In the book Loop spaces, Characteristic classes and geometric quantization by Brylinski I see following result when trying to motivate geometric description of $H^3(M,\mathbb{Z})$. $H^2(M,\mathbb{Z}...
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90 views

$\ell_p$ geodesic distance on smooth Riemannian manifold and Logarithmic Sobolev Inequalities

Bear with me, I'm not a professional geometer. Recently, I've been studying Logarithmic Sobolev Inequalities (LSI) for probability distributions on manifolds (e.g as done in works of Bobkvo et al. ...
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2answers
364 views

Is the boundary of a manifold topologically unique? [duplicate]

Let $X$ be a manifold without boundary and let $Y$ and $Z$ be two manifolds with boundary such that $X$ is homeomorphic to their interiors: $X \cong Y^\circ \cong Z^\circ$. Does it follow that $Y \...
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103 views

Diffeomorphism group action on the space of embeddings

Let $S$ and $M$ be two finite-dimensional smooth manifolds with $\dim S\le \dim M$. Then it is known (e.g.Kriegl-Michor's book) that the set $\mathrm{Emb}(S, M)$ of all smooth embeddings $S\to M$ is ...
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92 views

Quotient by a non-free action of a Lie group and manifolds with corners

The quotient manifold theorem says that If $G$ is a Lie group acting freely and properly on a smooth manifold $M$ then $M/G$ has a (unique) smooth structure such that the projection $\pi:M\to M/G$...
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234 views

A cohomology associated to a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. Let $$\Omega_{\omega}^k(M)=\{\alpha \in \Omega^{k}(M)\mid \alpha \wedge \omega \;\;\text{is an exact form}\}$$ Then we have a chain comlex$$\...
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74 views

Fenchel conjugate on a Hadamard manifold

Let $M$ be a Hadamard manifold and let $F:M\to\mathbb{R}$ be a real-valued convex function on $M$. What would be the Fenchel-Young conjugate of $F$? In general for a real locally convex vector space $...
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1answer
109 views

Lie brackets of automorphisms

Let $F$ be the vector fields of a differential manifold $M$, let $[X,Y]$ be the Lie brackets of $F$, now let $a$ be an automorphism of $F$ for the structure of real vector space of $F$. I consider now ...
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177 views

Thom's first isotopy lemma

Thom's first isotopy lemma says that given $f:M\to P$ a smooth map between smooth manifolds and a closed Whitney stratified subset $S$ of $M$, such that $f|_S:S\to P$ is proper and $f|_X:X\to P$ is a ...
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1answer
200 views

Dimension (manifold) of matrices with exact $r$ positive and $r$ negative eigenvalues

For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset $$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid \mbox{rank} (A) = 2r \}$$ is a manifold of dimension $2n(2r)-(...
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1answer
330 views

Which topology for $C^\infty(X)$ works?

Let $X$ be a smooth manifold. What is the appropriate topology on $C^\infty(X)$ such that a linear functional $\lambda$ on $C^\infty(X)$ is continuous iff it can be represented as a limit of the form $...
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1answer
89 views

Parallel transport with minimum length curve

I would know if it is possible to create a natural parallel transport on a Riemannian manifold $M$ with the following procedure. Define a minimum length curve (m.l.c.) $\gamma$ from $P$ to $Q$ ...
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1answer
112 views

Can every De Rham cohomology class be represented by a closed form $\alpha$ with $L_X \alpha=0$

Assume that $M$ is a manifold and $X$ is a vector field on $M$. Is it true to say that every closed form is De Rham-cohomologue to a closed form $\alpha$ with $L_X \alpha =0$?
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107 views

Conditions to the existence of periodic orbits of non vanishing vector fields on $\mathbb{T}^2$

I'm doing a research about Filippov systems on $\mathbb{S}^3$ with discontinuities on $\displaystyle\frac{1}{\sqrt{2}}\cdot\mathbb{T^2} =\left\{\displaystyle\frac{1}{\sqrt{2}}x ; \ x \in \mathbb{S}^1\...
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2answers
501 views

Topological obstructions to existence of immersion

Let $M$ be a smooth, non-compact manifold. a) Can one always find a smooth, compact manifold $N$ with $\dim(N) = \dim(M)$ and a smooth embedding $i: M \to N$ ? b) If not, are there some concrete ...
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Small perimeter-minimizing disks on curved surfaces

Suppose that I have a smooth curved surface, and I choose an arbitrary point $Q$ on that surface. Say the Gaussian curvature at that point is $K$. What I am wondering is, is there an expression for ...
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Is the image of the map $A \to \bigwedge^{k}A $ a weakly embedded submanifold?

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd $2 \le k \le d-2$. Define $H_{>k}=\{ A \in \End(...
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53 views

Smooth functions with values in bornological vector space

Let $U$ be an open set in $\mathbb{R}^n$ (or more generally, a manifold) and let $V$ be a separated bornological vector space. Do we have $$C^\infty(U, V) \cong C^\infty(U) \,\hat{\otimes}\, V,$$ as ...
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131 views

Smale's relative h-cobordism theorem

In Smale's On the structure of manifolds paper there is his relative version of the h-cobordism theorem, specifically Theorem 3.1 (and 1.4). Roughly speaking this concerns the situation where one has ...
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65 views

Bounding the dimension of the euclidean space in which any $n$-manifold embeds “$k$-uniquely” in it

(The question will be interesting for topological/Pl as well but in order to not be too vague I will restrict the meaning of manifold to smooth manifold without boundary). I'm interested in the ...
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2answers
252 views

A map of spaces implementing the Pontryagin Thom collapse map? (collapse maps in families)

Let $M$ be an $n$ dimensional smooth manifold and let $j: M \to \mathbb{R}^{m}$ be an embedding. Associated to this embedding we can form the "collapse map" which is a pointed map from a sphere to the ...
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1answer
314 views

Twisting bordism classes

Let $X$ be a reasonable topological space (I'd be happy to assume that $X$ is a smooth closed manifold) and let $f\colon M^n \rightarrow X$ be a continuous map from a smooth oriented $n$-manifold $M^n$...
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1answer
53 views

The continuity of the the stable and unstable in definition of hyperbolic sets for flows

I would like to know whether the continuity of the stable and unstable subbundles $E^{s}$ and $E^{u}$ follows from the growth conditions as in the discrete case, or must be hypothesized, in the ...
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71 views

Pull back group cohomology onto handle decomposition

A construction encountered in the Dijkgraaf-Witten invariant uses the following ingredients: An oriented, (assumed here to be smooth) manifold $M^n$ A finite group $G$ (and a field, chosen to be $\...
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Extension of a function from a subset of a manifold to the unit sphere

There is a line in this following paper (page no- 221, the paragraph before the Lemma 2.2) Otsu, Yukio; Shiohama, Katsuhiro; Yamaguchi, Takao, A new version of differentiable sphere theorem, Invent. ...
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147 views

Smooth extension of maps on Manifolds [closed]

Currently I'm studying differentiable manifolds using the books of Boothby and Lee. I encounter the following problem: Suppose $M$ and $N$ are smooth manifolds, $U$ an open subset of $M$, and $F:U\...
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Is Yetter's invariant multiplicative under connected sum?

Classical formulation Consider the (untwisted) Dijkgraaf-Witten invariant, defined for an oriented, connected, closed manifold $M$ and a finite group $G$: $$DW_G(M) := \lvert \operatorname{Hom}(\...
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2answers
213 views

Map between manifolds and open dense subsets

Let $X$,$Y$ be compact, connected, smooth manifolds of the same dimension. Suppose you have a surjective smooth map $f : X \rightarrow Y$, such that $|f^{-1}(p) | \leq k$ for all $p \in Y$. Let $U \...
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1answer
61 views

What are examples of a second order operational tangent vector on an infinite dimensional Hilbert space

In the book "a convenient setting for global analysis" they describe the order of an operational tangent vector on a convenient vector space. http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf ...
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1answer
50 views

Finite pre-images implies (local) branch cover?

Let $M_{1},M_{2}$ be (possibly non-compact) 2-dimensional, connected, smooth, orientable manifolds of finite topological type. Suppose you have smooth, surjective map $F:M_{1} \rightarrow M_{2}$, and ...
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2answers
685 views

Topological Classification of Four-Manifolds

It is known that the topological classification of a closed Riemann surface is determined by its genus. Similar statements are proven for other compact Riemann surfaces with boundaries/marked points. ...
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1answer
91 views

On sections into Banach bundles over a compact manifold

Let $M$ be a smooth, compact manifold and $\xi: \mathcal B \to M$ a smooth complex Banach bundle over $M$. Here, smooth is understood to be in the Fréchet-sense. Further, let $p: V \to M$ be an ...
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286 views

An orientable non-spin${}^c$ manifold with a spin${}^c$ covering space

Is there a closed, smooth, orientable manifold which is not spin${}^c$ but has a finite cover which is spin${}^c$? Such examples exist when spin${}^c$ is replaced by spin: an Enriques surface is not ...