# 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].

1,265
questions

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### $h$-convexity requirement: does taking the directional derivative over a curve preserve inequality?

Context: The result below implies that a certain class of random variables form a non-negative supermartingale bounded by 1, with minimal assumptions on the distribution of the random variables, which ...

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1
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### Derivative norm estimates

Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$.
QUESTION. Does this norm estimate hold? ...

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### Dual space of local Sobolev space on a manifold

$\newcommand{\comp}{\mathrm{comp}}$As part of my master's thesis, I am currently learning about Sobolev spaces on manifolds. From my research online, I found out, that there are a lot of ways to ...

4
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### Property of a smooth function is true for almost every function

Consider a smooth function $f:\mathbb{R}^n\to \mathbb{R}$. We further assume $0$ is a regular value of $f$, thus $X=f^{-1}(0)$ is a smooth submanifold of dimension $n-1$. Consider the following ...

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### Identify an SDE on the sphere from its generator

I have a diffusion on the 2-sphere with expression:
$$
(L\phi)(u):=\frac{1}{2{N(u)}}\Big(f(u)\Delta_{\mathbb S^2}\phi+
2g\left( \nabla_{\mathbb S^2}\phi, \nabla_{\mathbb S^2}f\right)\Big)
$$
...

5
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1
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381
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### No analytic surjection $f:M \to N$ when $\dim(M) >\dim(N)$

Inspired by comment discussions in this MO post smooth version of splitting principle we ask:
Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is not any ...

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### Smooth version of the splitting principle

Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...

4
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168
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### When can a generalized connected sum be aspherical

Let $M$ and $N$ be compact $n$-dimensional manifolds with a common (nicely embedded) compact submanifold $S$ (we may assume that the inclusion of $S$ in $M$ and $N$ is $\pi_1$-injective). Let $M\#_S N$...

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1
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### An equivalent characterization of $\mathcal{S}'$ as distributions on the sphere

I originally asked this question on Mathstack Exchange, but I think this question is more suitable for here.
Please let me know if that is not the case so that I can delete or edit this post.
Let $\...

2
votes

1
answer

114
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### Slowly increasing smooth mappings with values in a Lie group?

Let $G$ be $l$-dimensional compact Lie group and consider any smooth $F : \mathbb{R}^n \to G$.
Then, the first-order derivative of $F$ at each $x \in \mathbb{R}^n$ can be regarded as a linear mapping $...

2
votes

1
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### Unimodular intersection form of a smooth compact oriented 4-manifold with boundary

Let $X$ be a smooth compact oriented 4-manifold with nonempty boundary. Its intersection form $$ Q_X : H^2(X,\partial X;\Bbb Z)/\text{torsion}\times H^2(X,\partial X;\Bbb Z)/\text{torsion}\to \Bbb Z$$
...

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votes

1
answer

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### Topological Properties of Subsets of $R^{m}$ induced by Smooth Manifolds in Matrix Spaces

We know that $M_{m \times n } $ is isomorphic to $R^{mn}$. Let's take a smooth manifold $\mathbb{M}$ in $R^{mn}$ and fix a point in $R^{n}$, say, $p$. Now realize the manifold $\mathbb{M}$ as a subset ...

2
votes

1
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133
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### Extending diffeomorphisms between surfaces

Suppose we have two smooth compact oriented surfaces $M_1$ and $M_2$ with boundary,both of them have genus $g$, and there are orientation preserving diffeomorphisms $\psi_1, \psi_2, \cdots, \psi_n$, ...

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### Can you recover the manifold diffeology from the diffeology of distributions?

Is it true that the manifold diffeology is the subspace diffeology heredited from the diffeology of distributions?
I wanna know the same thing for the tangent bundle.

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### Details of the proof of the inequality $ \int_{X}\left(2 r \mathrm{c}_{2}(E)-(r-1) \mathrm{c}_{1}^{2}(E)\right) \wedge \omega^{n-2} \geq 0.$

I'm trying to make sense of the following proof.
Let $E$ be a holomorphic vector bundle of rank $r$ on a compact hermitian manifold $(X, g)$. If $E$ admits an Hermite-Einstein structure then $$
\int_{...

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114
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### Defining convex sums locally on the sphere?

$S^1$ and the torus $T^2$ are spaces in which convex combinations don't make sense globally but do locally. Despite their standard representations in $\mathbf{R}^2$ and $\mathbf{R}^3$ respectively not ...

2
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### Finite dimensional manifolds as subspace of $\mathbb{R}^\mathbb{N}$

For embedded submanifold, specifically with ambient space being $\mathbb{R}^{n}$, there are many nice properties and results. Specifically there are many examples of matrix manifolds such as the ...

2
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91
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### A roof genus of high dimensional lens space

Let $p$ be a natural number, and for $i\in \{0,
..., p-1\}$,
denote the irreducible rank one complex representation of $\mathbb{Z}/p$. by $\rho_{i}$.
Let $a=(a_{1},\ldots a_{d}) $ ...

3
votes

1
answer

274
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### Unitary transformations of Vandermonde matrices forms a smooth manifold?

The space of all Vandermonde matrices $V$ with $r$ variables and degree $n$ (as below) forms an embedded submanifold of $\mathbb{R}^{(n+1) \times r}$ when $x_{i} \in \mathbb{R}$. It is naturally a ...

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answers

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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 ...

15
votes

1
answer

492
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### 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, ...

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1
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### $\infty$-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 $\...

5
votes

2
answers

375
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### 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 ...

8
votes

1
answer

222
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### 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 ...

6
votes

1
answer

240
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### 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\...

1
vote

1
answer

171
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### 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 $...

3
votes

2
answers

207
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### 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 ...

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0
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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}...

6
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0
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126
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### 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 ...

17
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2
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992
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### 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 $M$ ...

15
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191
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### 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 ...

12
votes

1
answer

373
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### 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 ...

0
votes

0
answers

59
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### 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. ...

13
votes

1
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563
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### 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 ...

5
votes

3
answers

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### 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 $\...

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0
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### 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, $G_1 \;{}_s\!\times_t G_1$, ...

2
votes

0
answers

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### 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 $...

2
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### $$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [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(...

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0
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### 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 $M$ ...

2
votes

1
answer

170
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### 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 ...

4
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0
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314
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### 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, $\...

2
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0
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### 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 ...

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votes

0
answers

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### 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 ...

0
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0
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138
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### 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 ...

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votes

1
answer

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### 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 ...

3
votes

1
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318
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### 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 ...

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votes

1
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138
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### 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?

2
votes

0
answers

75
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### 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 ...

2
votes

0
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250
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### 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 ...

10
votes

1
answer

507
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### 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 ...