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Questions tagged [smooth-manifolds]

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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Some details about Kirillov-Kostant Poisson bracket

Let $G$ be a finite dimensional Lie group with Lie algebra $\mathfrak{g}$. The Kirillov-Kostant Poisson bracket on $\mathfrak{g}^*$ is defined as $$\{\cdot ,\cdot \} :C^{\infty}(\mathfrak{g}^*)\times ...
Mahtab's user avatar
  • 257
14 votes
1 answer
401 views

Where is the Steenrod Realization problem at?

I'm wondering if there is a more modern reference out there for the Steenrod Realization Problem than the book of Connor and Floyd? Realizing homology classes in a manifold via embedded submanifolds, ...
Ryan Budney's user avatar
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11 votes
1 answer
382 views

$\infty$-categorical description of $n$-manifolds

$\newcommand{\Mfld}{\mathsf{Mfld}} \newcommand{\Space}{\mathsf{Space}} \newcommand{\Sh}{\operatorname{Sh}} \newcommand{\PSh}{\operatorname{PSh}}$ I am wondering there is (or is expected to be) an $\...
user39598's user avatar
  • 325
-2 votes
0 answers
50 views

Tangent space of Lie group SO(n) [migrated]

I have a potentially simple question here, about the tangent space of the Lie group SO(n), the group of orthogonal $n\times n$ real matrices (I'm sure this can be asked more generally). For any $R\in \...
CComp's user avatar
  • 119
5 votes
2 answers
330 views

Is every codimension-one homology class of a closed manifold represented by a $\pi_1$-injective embedded submanifold?

Let $M$ be a connected closed orientable smooth $n$-manifold and $\nu \in H_{n-1}(M, \mathbb{Z})$ a non-trivial codimension-one homology class. It is known that $\nu$ can be represented by an embedded ...
24601's user avatar
  • 302
8 votes
1 answer
209 views

The closure of the space of Riemannian metrics with a fixed isometry class

Let $M$ be a closed manifold, and let $\mathscr{M}$ be the space of all Riemannian metrics over $M$. It is known that this is a Fréchet manifold. Consider also $\mathscr{D}$ the diffeomorphisms group ...
MyShepherd's user avatar
6 votes
1 answer
210 views

Ideals of functions whose zero locus is a submanifold

Let $M$ be a smooth $m$ dimensional manifold. Suppose that $f_1,\dots,f_k\in C^\infty(M)$ are smooth functions such that the zero locus $$N:=Z_f=\lbrace p\in M:\ f_i(p)=0,\ \forall i=1,2,\dots,k\...
Bence Racskó's user avatar
1 vote
1 answer
139 views

Lower bound on injectivity radius at one point implies lower bound on injectivity radius for a closed manifold

I’m interested in a closed Riemannian manifold $(M^n,g)$ with $sec<0$ and $diam(M)\leq D$. My question is: If at some point $p\in M^n$, the injectivity radius $injrad(p)\geq1$, then can we get $...
Xin Qian's user avatar
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3 votes
2 answers
150 views

First examples of Lie-Rinehart algebras that are not coming from Lie algebroids

I heard the idea of a Lie-Rinehart algebra first time from an algebraist. I noticed there is a similarity between description of Lie algebroid on a manifold and the algebraic notion of Lie-Rinehart ...
Praphulla Koushik's user avatar
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0 answers
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Charecterizing (Riemannian) submanifolds of the Bures-Wasserstein manifold

I'm still learning Riemannian geometry, so please correct any mistakes. I am interested in the Bures-Wasserstein manifold of centered Gaussians of dimension $d$. In this case, the manifold $\mathcal{M}...
Afham's user avatar
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6 votes
0 answers
119 views

Is there a canonical smooth structure on tame Fréchet orbit type stratifications?

In finite dimension orbit type stratifications, it is known that the orbit space $M/G$ resulting from an action of a proper Lie Group $G$ on a smooth manifold $M$, satisfying a set of certain ...
MyShepherd's user avatar
16 votes
2 answers
957 views

Homotopy equivalent but non-homeomorphic high-dimensional manifolds

I have a question motivated by the classification theory of simply-connected closed $4$-manifolds. Questions: Given any $n\geq 5$, is it possible to find two simply-connected closed $n$-manifolds $M$ ...
Random's user avatar
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14 votes
0 answers
169 views

Are Lie groupoids just groupoids internal to smooth manifolds?

It seems to be common to say "no" - but is this true? Two weeks ago I asked for a counterexample, but received no replies. To give some background, let's recall that the difference between ...
Konrad Waldorf's user avatar
12 votes
1 answer
353 views

Approximate classifying space by boundaryless manifolds?

As pointed out by Achim Krause, any finite CW complex is homotopy equivalent to a manifold with boundary (by embedding into $\mathbb R^n$ and thickening), and so every finite type CW complex can be ...
0207's user avatar
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0 answers
59 views

Condition to show $\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is (is not) a manifold

Consider $\mathscr{A}: S^{n\times n} \to \mathbb{R}^{m}$, $b \in \mathbb{R}^{m}$, I would like to know when $\mathscr{M}:=\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is a manifold. ...
wsz_fantasy's user avatar
13 votes
1 answer
545 views

Identifying two definitions of orientation on a vector space

Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$: A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive ...
Jean's user avatar
  • 133
5 votes
3 answers
857 views

Naturality of Lie bracket — alternate proof

Let $M$ and $N$ be smooth manifolds, and let $F: M \to N$ be a smooth map. Let $X$ and $Y$ be vector fields on $M$, and let $\tilde{X}$ and $\tilde{Y}$ be vector fields on $N$. We say that $X$ and $\...
Zhang Yuhan's user avatar
6 votes
0 answers
109 views

Example of a groupoid internal to the category of smooth manifolds that is not a Lie groupoid

This questions is about the distinction between: Lie groupoids: we require source and target maps to be submersions. This implies that the domain of the composition map, $G_1 \;{}_s\!\times_t G_1$, ...
Konrad Waldorf's user avatar
2 votes
0 answers
45 views

Under what conditions principal directions define an integrable distribution?

Consider a hypersurface $M^n \subset \mathbb{R}^{n+1}$ which is compact without boundary. Assume that its second fundamental form $A$ has distinct eigenvalues $\lambda_1<\ldots<\lambda_k$ (with $...
Dorian's user avatar
  • 331
2 votes
0 answers
398 views

$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]

If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of $$ \left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
Ola Sande's user avatar
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1 vote
0 answers
78 views

Metric of negative curvature on connected sum

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Riemannian manifolds of dimension $n\geq 2$. If we consider the connected sum $M=M_1\mathbin{\#}M_2$ of the two manifolds; can one get a smooth metric $g$ on $M$ ...
User5's user avatar
  • 11
2 votes
1 answer
162 views

Order of a loop around a cone point

Let $M$ be a connected orientable two-dimensional orbifold with only cone points as singular points. Assume that $M$ has genus $\geq 1$. Let $\alpha$ be a loop around an order $p$ cone point. Can we ...
RKS's user avatar
  • 531
4 votes
0 answers
297 views

Milnor’s smoothed corners technique for a product of manifolds with boundary and boundary defining functions

Let $M$ be a smooth manifold that we view as the interior of a compact manifold with boundary $\overline{M}$. Let $\rho$ be a boundary defining function for $\overline{M}$, i.e. $\rho$ is smooth, $\...
zarathustra's user avatar
2 votes
0 answers
53 views

How is the $k$-times iterative frame bundle $FF\cdots FM$ associated to the higher order frame bundle $F^k M$?

$\DeclareMathOperator\Gl{Gl}$As I understand it a natural bundle is one for which a diffeomorphism on the base space lifts to an automorphism on the total space of the bundle. It is my understanding ...
R. Rankin's user avatar
  • 230
0 votes
0 answers
35 views

Estimate the gradient (with respect to local coordinates) of a partition of unity on a manifold

Suppose $\{U_\alpha\}$ is an atlas of coordinate patches of a (noncompact) smooth manifold $M$ of dimension $n$, with coordinates $(x_\alpha^1,\dots,x_\alpha^n)$ on $U_\alpha$. Furthermore we assume ...
Anar C's user avatar
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0 answers
129 views

Noether's formula for real algebraic surfaces

Is there a version of Noether's formula for the Euler characteristic of a surface for Real algebraic surfaces? Specifically, given $X$ a real algebraic compact smooth surface, what is the relationship ...
Serge the Toaster's user avatar
-4 votes
1 answer
319 views

Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]

Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold? NOTE: PLEASE avoid the ...
Bastam Tajik's user avatar
3 votes
1 answer
311 views

What is known about the "stickiness" of a smooth manifold to its tangent space?

Consider an $m$-dimensional smooth manifold $M$ embedded in $n$-dimensional real Euclidean space $\mathbb{R}^n$ ($n>m$). Consider a point $x \in M$ of the manifold, and for simplicity, choose the ...
Smooth M's user avatar
-4 votes
1 answer
130 views

7-sphere x 4-sphere manifold and its physical significance [closed]

I am looking for sources about this manifold 7-sphere*4-sphere and its relations to theoretical physics. Where to go to read about 7-sphereX4-sphere manifold and its physical significance?
Altami's user avatar
  • 3
2 votes
0 answers
72 views

Assumptions for uniform measure of SDE on manifolds

Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
optimal_transport_fan's user avatar
2 votes
0 answers
213 views

Smooth compactification of complex varieties and uniqueness

Since I'm working in differential geometry, for the following I'm strictly interested in the smooth setting over $\mathbb{C}$ and its relation to the setting over $\mathbb{R}$. Here are a few useful ...
Paul Cusson's user avatar
  • 1,745
10 votes
1 answer
380 views

Disintegration measures and differential forms

Let $X$ and $Y$ be smooth oriented manifolds of dimension $m$ and $n$, and let $f\colon X\to Y$ a proper smooth map. There is a theorem called the "Disintegration Theorem" which says ...
Ben Webster's user avatar
  • 44.1k
4 votes
0 answers
172 views

Possible Euler characteristics of manifolds with tangential structures

Let $p:B\to BO$ be a fibration. We say that a manifold has a $B$-structure if its stable tangent bundle lifts to $B$. I am interested in the question of whether there exists, for a given even ...
Simona Vesela's user avatar
5 votes
1 answer
242 views

Clarifying a result of Klingenberg

I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each ...
E G's user avatar
  • 153
6 votes
0 answers
146 views

Extending topological vector bundles and obstruction theory

This is a question that has appeared in various forms on MathOverflow, see here and here, for example. But as opposed to these more algebraic questions, I am interested in the purely topological ...
Paul Cusson's user avatar
  • 1,745
6 votes
1 answer
250 views

Why does Bott's obstruction theorem imply the vanishing of some cohomology classes of $B\Gamma_q$?

Recall that Bott's obstruction for integrability [Bott70] asserts that: Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every ...
Ken's user avatar
  • 1,857
3 votes
1 answer
173 views

Isomorphism between tangent bundle of $S^2$ and the kernel of a bundle homomorphism

Let $S^{4n+3} \to \mathbb{H}P^n$ be the standard projection which is a fiber bundle with fiber $S^3$. By the action of $S^1$ on $S^3$ we get a fiber bundle $$ \mathbb{C}P^1 \xrightarrow{\iota} \mathbb{...
Patrick Perras's user avatar
0 votes
1 answer
147 views

How to estimate the distance between geodesics and points for Riemannian manifold with positive sectional curvature

Assume that $ M $ is a complete Riemannian manifold and there exists $ k>0 $ such that $ K(q)\geq k $ for any $ q\in M $, where $ K $ is the sectional curvature of $ M $. Let $ \gamma $ be a closed ...
Luis Yanka Annalisc's user avatar
0 votes
1 answer
103 views

Geodesic whose one end is at a ideal point

We know that every hyperbolic manifold $M$ is locally isometric to hyperbolic plane $\mathbb{H}_2$. Can we consider the inverse image, under the isometry, of the geodesic $\gamma$ that connects a ...
user avatar
5 votes
1 answer
245 views

Commutative/ symmetric second covariant derivative

Consider a smooth manifold $M$ together with an affine connection (or covariant derivative) $\nabla$ on the tangent bundle $TM$. Is it possible to have an affine connection, possibly with non-zero ...
Khaled T.'s user avatar
0 votes
0 answers
54 views

Symmetric indefinite matrix of fixed rank — manifold structure?

I have been studying symmetric indefinite matrices of fixed rank, which have been rather useful for a particular application. I wonder if there is a way to parameterise these by a smooth manifold, e.g....
turtlesandwich's user avatar
3 votes
0 answers
187 views

An almost complex structure on $\Bbb S^n$ induces a cross product on $\Bbb R^{n+1}$

It is known that the only spheres that admit an almost complex structures are $\Bbb S^2$ and $\Bbb S^6$ (Borel and Serre, 1953). In particular, $\Bbb S^4$ cannot be given an almost complex structure (...
Random's user avatar
  • 1,087
4 votes
0 answers
136 views

Triangulating piecewise-linear manifolds

Question 1: Is this the mainstream definition of a PL-manifold? Definition. A PL-manifold is a manifold with an atlas $(\varphi_i)_{i\in I}$ in which all transition maps $\varphi_j\circ\varphi_i^{-1}$ ...
Vadim's user avatar
  • 346
4 votes
0 answers
114 views

Finding inverses of certain elements in the set of normal invariants of a smooth manifold

Let, $V$ denote the Stiefel manifold of 2-frames $V_{10,2}$ . Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold. . ...
Sagnik Biswas's user avatar
4 votes
2 answers
539 views

Compactification of a product of manifolds

Let $M$ be a smooth manifold. We make the assumption that $M$ can be viewed as the interior of a compact manifold with boundary $\overline{M}$. In practice, for an explicit manifold, any ...
zarathustra's user avatar
3 votes
0 answers
55 views

Jet at a singular point or a submanifold

Let $M$ be a smooth manifold, $p\in M$ and $f\in C^\infty(M\setminus\{p\})$. We will say that $f$ has a power-law singularity at $p$ of order $\eta$ if for every smooth immersion $\gamma:(-1,1)\to M$ ...
Peter Kravchuk's user avatar
4 votes
1 answer
433 views

Detecting a "bad map" in Fintushel-Stern knot surgery

Background Let $X$ be a simply-connected smooth 4-manifold which contains a smoothly embedded torus $T$ with trivial normal bundle (in other words, $T^2\times D^2\subset X$). Let $K$ be a knot in $S^3$...
rab's user avatar
  • 159
0 votes
1 answer
112 views

A sufficient condition for a collection of open sets of a manifold to contain all open sets

Question Let $k\geq 0$ be an integer and let $M$ be a topological $n$-manifold. Let $\mathcal{U}$ be a set of open sets of $M$ which satisfies the following closure properties: (1). Let $U\subset M$ ...
Ken's user avatar
  • 1,857
5 votes
1 answer
332 views

To what extent differentiable mappings of an affine line into a manifold determine its differentiable structure? What about mappings of a plane?

If $M$ is a (real) differentiable manifold, its differentiable structure is completely determined if it is known which mappings $M\to\mathbf{R}$ are differentiable. How much can be said about the ...
Alexey Muranov's user avatar
14 votes
2 answers
2k views

Is a manifold Euclidean if its tangent bundle is Euclidean?

I'm wondering whether an $n$-dimensional manifold diffeomorphic to $\mathbb{R}^n$ if its tangent bundle is diffeomorphic to $\mathbb{R}^{2n}$. Many thanks!
Felix's user avatar
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